Traditional analyses of the curve fitting problem maintain that the data do not indicate what form the fitted curve should take. Rather, this issue is said to be settled by prior probabilities, by simplicity, or by a background theory. In this paper, we describe a result due to Akaike [1973], which shows how the data can underwrite an inference concerning the curve’s form based on an estimate of how predictively accurate it will be. We argue that this approach throws light on the theoretical virtues of parsimoniousness, unification, and non ad hocness, on the dispute about Bayesianism, and on empiricism and scientific realism.
Curve fitting is a two-step process. First one selects a family of curves (or the form that the fitted curve must take). Then one finds the curve in that family (or the curve of the required form) that most accurately fits the data. These two steps are universally supposed to answer to different standards. The second step requires some measure of goodness-of-fit. The first is the context in which simplicity is said to play a role. Intrinsic to this two-step picture is the idea that these different standards can come into conflict. Maximizing simplicity usually requires sacrifice in goodness-of-fit. And perfect goodness-of-fit can usually be achieved only by selecting a complex curve.
This view of the curve fitting problem engenders two puzzles. The first concerns the nature and justification of simplicity. What makes one curve simpler than another and why should the simplicity of a curve have any relevance to our opinions about which curves are true? The second concerns the relation of simplicity and goodness-of-fit. When these two desiderata conflict, how is a trade-off to be effected? A host of serious and inventive philosophical proposals notwithstanding, both these questions remain unanswered.
If it could be shown that a single criterion for selecting a curve gives due weight to both simplicity and goodness-of-fit, then the two problems mentioned above for traditional analyses of the curve fitting problem would fall into place. It would become clear why simplicity matters (and how it should be measured). In addition, simplicity and goodness-of-fit would be rendered commensurable by representing each in a common currency. In what follows we describe a result in statistics, stemming from the work of Akaike [1973], [1974], which provides this sort of unified treatment of the problem, in which simplicity and goodness-of-fit are both shown to contribute to a curve’s expected accuracy in making predictions.1
In this section, we present the basic concepts that are needed to formulate the curve-fitting problem and to solve
it. To begin with, we need to describe the kinds of hypotheses that curves represent and the relationship of those
curves to the data we have available. A ‘deterministic’ curve is a line in the X/Y plane; it represents a
function, which maps values of X (the independent variable) onto unique values of Y (the dependent variable).2 For
example, Figure 1 depicts two such curves; each says that Y is a linear function of X. Each of these curves may 1
There is a growing technical literature on the subject. Linhart & Zucchini [1986] surveys the earlier work of
statisticians. Researchers in computer science have used the concept of‘shortest data descriptions’ to
warrant the trade-off between simplicity and goodness of fit.
See Rissanen [1978], [1989], or more recently,
Wallace and Freeman [1992]. While there are criteria in the literature that are quantitatively different from Akaike’s,
there is a measure of agreement in the way they define simplicity and goodness-of-fit. We have focused on Akaike’s
seminal work because he motivates his criterion in a general and perspicuous manner.
2 The idea that there is just one independent variable is a simplifying assumption adopted for ease of exposition.
The results we will describe generalize to any number of independent
variables.
be obtained by fixing the values of the parameters α0 and α1 in the following
equation:
Y = α0 + α1 X .
The two curves in Figure 1 are equally simple, we might say, because each is a straight line and each is obtained from a functional form in which there are just two adjustable parameters. These two curves belong to a family of curves? namely, the set of all straight lines. We will be talking about both specific curves and families of curves in what follows, so it will be important to keep the distinction between them in mind. In fact, it will turn out that there is no need to define the simplicity of a specific curve; all that is needed is the notion of the simplicity of a family of curves, and this Akaike’s approach provides.
Observed value of X
Curve 2’s prediction of Y
X
Y
Curve 2
Curve 1
The observed value of
Y
Curve 1’s prediction of Y
FIGURE 1
Suppose the true specific curve determined the outcomes of the observations we make. Then, if Curve 1 were true, the set of data points we obtain would have to fall on a straight line (i.e., on the straight line depicted by Curve 1 itself). But we will suppose that the observation process involves error. Even if Curve 1 were true, it is nonetheless quite possible that the data we obtain will not fall exactly on that curve. It may be impossible to say when any particular data point will fall above or below the true curve - only that it should ‘tend’ to be close. To represent this possibility of error, we associate a probability distribution with each curve. This distribution tells us how probable it is that the Y-value we observe for a given X-value will be ‘close’ to the curve. The most probable outcome is to obtain a Y-value that falls exactly on the true curve. Locations that are further off the curve have lower probabilities (symmetrically above and below) of being what we observe.
To make this idea concrete, suppose that we are interested in plotting the location of a planet as it moves across
the sky. In this case, the X-axis represents time and the Y-axis represents location. The true curve is the actual,
unique trajectory of the planet. But our observation of the planet’s motion is subject to error. Even if Curve 1
in Figure 1 describes the planet’s true trajectory, it nonetheless is possible that we should obtain data that
fail to fall exactly on that
curve.
So there are two factors that influence the observations we make. There is the planet’s actual trajectory; and there is the process of observation, which is subject to error. If the planet’s trajectory is a straight line, we can combine these two influences into a single expression:
(LIN) Y = α0 + α1 X + σ U.
The last addend represents the influence of error. Here, of course, Y doesn’t represent the planet’s
actual location, but represents its apparent location.3
Now consider the data points depicted in Figure 1. If
Curve 1 were true, it is possible that we should obtain the data before us. But the same is true of Curve 2; if it
were true, it also could have generated the data at hand. Although this is a similarity between the two curves, there
nonetheless is a difference: the probability of obtaining the data, if Curve 1 is true, exceeds the probability of
obtaining the data, if Curve 2 were true: p(Data/Curve 1) > p(Data/Curve 2).4
Statisticians use the technical
term likelihood to describe this difference; they would say that Curve 1 is more likely than Curve 2, given the data
displayed. It is important to note that the likelihood of a hypothesis is not the same thing as its probability; don’t
confuse p(Data/Curve 1) with p(Curve 1/Data).
In a sense, Curve 1 fits the data better than Curve 2 does. The
standard way to measure this goodness-of-fit is by a curve’s sum of squares (SOS). As depicted in Figure 1, we
compute the difference between the Y-value of a data point and the Y-value on the curve directly above or below it. We
square this difference and then sum the same squared differences for each data point. Curve 1 has a lower SOS value
than Curve 2, relative to the data in Figure 1. Comparing SOS values is a way to compare likelihoods. Notice that if
we were to increase the number of data points, the SOS values for both curves would almost certainly go up.5
We
can use the concept of SOS to reformulate the curve-fitting problem.
Given a set of data, how are we to decide
which curve is most plausible? If minimizing the SOS value were our sole criterion, we would almost always prefer
bumpier curves over smoother ones. Even though Curve 1 is rather close to the data depicted in Figure 1, we could draw
a more complex curve that 3 Alternatively, the error term can be given a physical, instead of an epistemological,
interpretation, if one wishes to represent the idea that nature itself is stochastic. In that case, Y would represent
the planet’s ‘mean’ position. This difference in interpretation will not affect our subsequent
discussion.
4 When random variables are continuous, the likelihood is defined in terms of probability densities rather than
probabilities. A lower case p is a probability density, while the upper case
P is reserved for probabilities.
5 The SOS value for a curve can’t go down as the data set is enlarged; it would stay the same, if, improbably enough, the new data points fell exactly on the curve. Also note that a curve’s likelihood will decline as the data set is enlarged, even if the new points fall exactly on the curve.
passes exactly through those data points. The practice of science is to not do this. Even though a hypothesis with
more adjustable parameters would fit the data better, scientists seem to be willing to sacrifice goodness-of-fit if
there is a compensating gain in simplicity. The problem is to understand the rationale behind this behavior.
Aesthetics to one side, the fundamental issue is to understand what simplicity has to do with truth.
The universal
reaction to this problem among philosophers has been to think that the only thing the data tell you about the problem
at hand is given by the SOS values. The universal refrain is that ‘if we proceed just on the basis of the data,
we will choose a curve that passes exactly through the data points.’ This interpretation means that giving weight
to simplicity involves an extraempirical consideration. We thereby permit considerations to influence us other than
the data at hand. Giving weight to simplicity thus seems to embody a kind of rationalism; a consistent empiricist must
always opt for bumpy curves over smooth ones.
The elementary framework developed so far allows us to show that
this standard reaction is misguided. Let us suppose that the curve in Figure 2 is true.
Now consider the data that
this true curve will generate. Since we assume that observation is subject to error, it is overwhelmingly probable
that the data we obtain will not fall exactly on that true curve. An example of such a data set, obtained from the
true curve, also is depicted in Figure 2. Now suppose we draw a curve that passes exactly through those data points.
Since the data points do not fall exactly on the true curve, such a best-fitting curve will be false. If we think of
the true curve as the ‘signal’ and the deviation from the true curve generated by errors of observation as
‘noise,’ then fitting the data perfectly involves confusing the noise with the signal. It is overwhelmingly
probable that any curve that fits the data perfectly is false.
Of course, this negative remark does not provide a
recipe for disentangling signal from noise. We know that any curve with perfect fit is probably false, but this does
not tell us which curve we should regard as true. What we would like is a method for separating the ‘trends’
in the data from the random deviations from those trends generated by error. A solution to the curve fitting problem
will provide a method of this sort.
X
Y
H
FIGURE 2
To explain Akaike’s proposal, we need to introduce a precise definition of how successful a curve is in
identifying the trend behind the data. In addition to talking about a curve’s distance from a particular data
set, we need a way to measure a curve’s distance from the true curve. A constraint on this new concept is already
before us: a curve that is maximally close to the data (because it passes exactly through all the data points) is
probably not going to be maximally close to the truth. Closeness to the truth is different from closeness to the data.
How should the concept of closeness to the truth be defined?
Let us suppose that Curve 1 in Figure 1 is true. We
want a way to measure how close Curve 2 is to this true curve. Curve 1 has generated the data set displayed in the
figure, and we can use the SOS measure to describe how close Curve 2 is to these data points. The idea is to define
the distance of Curve 2 from Curve 1 in terms of the average distance of Curve 2 from the data generated by Curve 1.
So, imagine that Curve 1 generates new data sets, and each time we measure the distance of Curve 2 from the generated
data set. We repeat this procedure indefinitely, and we compute the average distance that Curve 2 has with respect to
data sets generated by the true Curve 1. Remember that this average is computed over the space of possible data sets,
rather than actual data sets.6 This allows us to define distance from the truth as follows:
Distance from the true
curve (T) of curve C = df
Average[SOS of C, relative to data set D generated by T] −
Average[SOS of T,
relative to data set D generated by T].
First, note that the distance from the true curve is relative to the
process of data generation; it depends on the method of generating the array of X-values whose associated Y-values the
curves are asked to predict.7 Second, note that the true curve, T, is the curve that is closest to the truth (its
distance from the truth is 0) according to this definition. However, the average SOS value of the true curve T,
relative to the data sets that T generates, is nonzero. This is because of the role of error; on average, even the
true curve won’t fit the data perfectly.
We now define the concept of distance from the truth for families of
curves.
The above definition defines what it means for Curve 2 to be a certain distance from the true curve. But
what would it mean to describe how close to the true curve the family of straight lines (LIN) is? Here’s the
idea: Let’s think of two data sets, D1 and D2, each generated by the true curve T. First, we find the specific
curve within the family that fits D1 best. Then we compute the SOS of that curve relative to the second data set D2.
Imagine carrying out this procedure
6 Statisticians mark this distinction by using the term ‘expected value’ rather than ‘average value.’ We have chosen not to do this because the psychological connotations of the word ‘expected’ may mislead some readers.
7 The X-arrays for the predicted data do not have to be the same as the X-array for the actual data, but both must be generated by the same stochastic process.
again and again for different pairs of data sets. The average SOS obtained in this way is the family’s distance
from the truth:
Distance from the true curve (T) of family F = df
Average[SOS of L1(F), relative to data set
D2 generated by T] −
Average[SOS of T, relative to data set D2 generated by T].
Here L1(F) is the best
fitting (‘likeliest’) member of the family F, relative to data set D1.8
Our definition of a family’s
distance from the truth is intended to measure how accurate the predictions will be that the best fitting curve in a
family generates.
Consider the family of straight lines (LIN) and the data displayed in Figure 1.
How close is
the family (LIN) to the truth? We can imagine finding the straight line that best fits the data at hand. The question
we’d like to answer is how accurately that particular straight line will predict new data. The average distance
from the truth of best fitting curves selected from that family is the distance of the family from the truth:
Distance from the true curve (T) of family F = Average[Distance of best fitting curves in F from the truth T].
Our
interest in the distance of families from the truth stems from this equality.
Families are of interest because
they are instruments of prediction; they make predictions by providing us with a specific curve–viz. the curve in
the family that best fits the data.9
If the true curve is in fact a straight line, (LIN) will of course be very
close to the truth (though the distance will be nonzero).10 But if the truth is highly nonlinear, (LIN) will perform
poorly as a device for predicting new data from old data. Let us move to a more complicated family of curves and ask
the same questions. Consider (PAR), the family of parabolic equations:
(PAR) Y = β0 + β1 X + β2 X 2
+ σ U.
Specific parabolas will be c-shaped or 1-shaped curves. Notice that (LIN) is a subset of (PAR). If
the true specific curve is in (LIN), it also will be in (PAR).
However, the converse relation does not hold.
So if (LIN) is true, so is (PAR) (but not conversely). This may lead one to 8 The definition of distance from the
truth of a specific curve C is a special case of the definition for a family of curves F. A family is a set of curves;
when a family contains just one curve, its best fitting member is just that curve itself.
9 In the kinds of example we consider, there will be a unique curve in a family that fits the data best when the number of data points exceeds the number of adjustable parameters.
10 A family can be literally true (by including the true curve) and still have a non-zero distance from the truth because other curves in the family (including L(F)) will be closer than the true curve to the actual data.
expect that PAR must be at least as close to the truth as (LIN) is. However, this is not so! Let’s suppose that
the true curve is, in fact, a straight line. This will generate sets of data points that mostly fail to fall on a
straight line. Fitting a straight line to one set of data points will provide more accurate predictions about new data
than will fitting a parabolic curve to that set. To be sure, for each data set, the best fitting parabola will be
closer to the data than the best fitting straight line. But this leaves open how well these two curves will predict
new data. (LIN) will be closer to the truth (in the sense defined) than (PAR) is, if the truth is a straight line.
Curves that fit a given data set perfectly will usually be false; they will perform poorly when they are asked to make
predictions about new data sets.
Perfectly fitting curves are said to ‘overfit’ the data. This fact
about specific curves is reflected in our definition of what it means for a family to be close to the truth. If (LIN)
is closer to the truth than (PAR) is, then a straight line hypothesis fitted to one data set will do a better job of
predicting new data than a parabolic curve fitted to the same data, at least on average. In this case, the more
complex family is disadvantaged by the greater tendency of its best fitting case, L(PAR), to overfit the data.
The
definitions just given of closeness to the truth do not show how that quantity is epistemologically accessible. To
apply these definitions and compute how close to the truth a curve C (or a family F) is, one must know what the truth
(T) is. Nonetheless we can use the concept of closeness to the truth to reformulate the curve-fitting problem and to
provide it with a solution.
All families with at least one free parameter are able to reduce their least SOS by
fitting to random fluctuations in the data. This is true of low dimensional families as well, though to a lesser
degree. For example, the data in Figure 1 were generated by a straight line, but random fluctuations in the data
enable a parabola to fit it better than any straight line. This shows that the phenomenon of overfitting is
ubiquitous.11 Thus, there are two reasons why the least SOS goes down as we move from lower to higher dimensional
families: (a) Larger families generally contain curves closer to the truth than smaller families. (b) Overfitting: The
higher the number of adjustable parameters, the more prone the family is to fit to noise in the data. Our promised
reformulation of the curve fitting problem is this: We want to favour larger families if the least SOS goes down
because of factor (a), but not if its decline is largely due to (b). If only we could correct the SOS value for
overfitting, then the corrected SOS value would be an unbiased indication of what we are interested in?viz. the
distance from the true curve.
11 This is the same overfitting problem that plagues general purpose learning devices like neural networks. Moody
[1992] and Murata et al. [1992] are working on generalizing the Akaike
framework to apply to artificial neural
networks. See Forster [1992b] for further details. It is interesting that there is such a fundamental connection
between neural learning and the philosophy of science (Churchland [1989]).
At this point, we will simply state Akaike’s theorem, without attempting to work through the mathematical
argument that establishes its correctness. (See the Appendix A for a non-technical explanation of the assumptions
needed, and Appendix B for the proof of the theorem in a special case. The most thorough, and accessible, technical
treatment is found in Sakamoto et al.[1986].) Akaike [1973] discovered a way of estimating the size of the overfitting
factor. The procedure is fallible, of course, but it has the mathematical property of providing an unbiased estimate12
of the comparative distances of different families from the truth under favourable conditions (see Appendix A). The
amazing thing about Akaike’s result is that it renders closeness to the truth epistemologically accessible; the
estimate turns on facts that we can readily ascertain from the family itself and from the single data set we have
before us:
Estimated[(Distance from the truth of family F) = SOS[L(F)] + 2k σ2 + Constant.
L(F) is the
member of the family that fits the data best, k is the number of adjustable parameters that the family contains, and
σ2 is the variance (degree of spread) of the distribution of errors around the true curve. The last term on the
right hand side is common to all families, and so it drops out in comparative judgments.
The first term on the
right hand side, SOS[L(F)], is what we have been calling the least SOS for the family. It represents what empiricists
have traditionally taken to exhaust the testimony of evidence. The second term corrects for the average degree of
overfitting for the family. Since overfitting has the effect of reducing the SOS, any correction should be positive.
That this correction is proportional to k, the number of adjustable parameters,13 reflects the intuition that
overfitting will increase as we include more curves that are able to mould themselves to noise in the data. That the
expected degree of overfitting also is proportional to σ2 is plausible as well - the bigger the error deviations
from the true curve, the greater the potential for misleading fluctuations in the data. Also note that if there is no
error (σ2 = 0), then the estimate for the distance from the truth reduces to the least SOS value. The
postulation of error is essential if 12 ‘Unbiased’ means that its average performance will center on the
true value of the quantity being estimated. Note that an unbiased estimator can have a wide or narrow variance, which
measures how much the estimate ‘bounces around’ on average. Unbiasedness is only one desideratum for ‘good’
estimators.
13 In our running example, (LIN) contains two adjustable parameters and (PAR) contains three.
The number of
adjustable parameters is not a merely linguistic feature of the way a family is represented. For example, Y = α
+ β X + γ X is one way of representing (LIN), but k is still 2, because there is a reparameterization (viz.
α′ = α, β ′ = (β + γ), and γ ′ = (β − γ)) such
that Y = α ′ + β ′ X. In contrast, the dimension of the family Y = α + β X + γ
Z is 3 because there is no such reparameterization.
simplicity (as measured by k) is to be relevant to our estimates concerning what is true.14
We will use the term
‘predictive accuracy’ to describe how close to the truth a curve or family is. ‘Accuracy’ is a
synonym for ‘closeness to the truth’, while the term ‘predictive’ serves to remind the reader that
the concept is relativized to the process by which the true curve generates new data. Instead of using SOS as a
measure of distance, we use the log of the likelihood to measure closeness to the data (the greater the
log-likelihood, the smaller the distance from the data).
Thus, we define the predictive accuracy of a curve C,
denoted by A(curve C), as the average log-likelihood of C per datum. The predictive accuracy of a family F is the
average predictive accuracy of its best fitting curves.15 This leads to a more general statement of Akaike’s
Theorem, since the log-likelihood applies to cases, like coin tossing examples, in which the SOS value is not defined.
Recalling the connection between the low SOS value of a specific curve and its high likelihood, the general statement
of Akaike’s theorem is as follows:
Akaike’s Theorem: Estimated[A(family F)] = (1/N)
[log-likelihood(L(F)) − k], where N is the number of data points.16 We no longer need to assume that the error
variance, σ2, is known, for the error variance may be treated as another adjustable parameter.17
14 We
regard the total absence of error as radically implausible. Even if nature were completely deterministic, there still
would be observational errors. And even then, there still would be lawless deviations from any ‘curve’ that
limits itself to an impoverished stock of independent variables. For example, it may be that the temperature at a
particular place and time is determined. A curve that truly captures the dependence of temperature on the time of day
and time of year will not predict the temperature exactly because there are other relevant factors.
The data will
behave as randomly as if the world were indeterministic. From an epistemological point of view, this is all that
matters. Forster [1988b] and Harper [1989] examine the role of this third kind of error (arising from the action of
other variables) in the ‘exact’ science of astronomy.
15 This average is computed as follows: Take a data set D1 generated by the true curve T, and note the predictive
accuracy of the best curve L1(F) in F relative to D1. Imagine that this procedure is repeated with new data sets D2,
D3, ..., each time noting the predictive values of
the curves L2(F), L3(F), ... Now take the average of all these
values.
16 The factor (1/N) arises from the fact that we prefer to define accuracy as the average per datum log-likelihood, so that the accuracy of a hypothesis does not change when we consider the prediction of data sets of different sizes.
17 When σ2 is treated as unknown, a curve (by itself) no longer confers a probability on the data.
Literally speaking, a curve is a family of probability distributions?one for each numerical value of σ2. From
now on we will understand a ‘curve’ to be associated with some specific numerical value of σ2. Also
note that Akaike’s estimate of predictive accuracy of a family of‘curves’ in which σ2 is a free
parameter is related to the least SOS value for the family by a different formula (Sakamoto et al. [1986], p.170):
Estimate[A(Family FN)] = − (1/2)log[SOS(B(F))/N] − kN/N + constant, where FN is the higher dimensional
family obtained from F by making σ2 adjustable. Here, SOS(B(F)) is the least SOS for the original family F, and
kN is the dimension of the final family. For LIN and PAR, kN = k + 1.
This theorem, we believe, provides a solution to the curve-fitting problem. It explains why fitting the data at hand is not the only consideration that should affect our judgment about what is true. The quantity k is also relevant; it represents the bearing of simplicity. A family F with a large number of adjustable parameters will have a best member L(F) whose likelihood is high; however, such a family will also have a high value for k. Symmetrically, a simpler family will have a lower likelihood associated with its best case, but will have a low value for k. Akaike’s theorem shows the relevance of goodness-of-fit and simplicity to our estimate of what is true. But of equal importance, it states a precise rate-of-exchange between these two conflicting considerations; it shows how the one quantity should be traded off against the other. We emphasize that Akaike’s theorem solves the curve-fitting problem without attributing simplicity to specific curves; the quantity k, in the first instance, is a property of families.18
A special case of Akaike’s result is worth considering. Suppose one has a set of data that falls fairly evenly around a straight line. In this case the best fitting straight line will be very close to the best fitting parabola. So L(LIN) and L(PAR) will have almost the same SOS values. In this circumstance, Akaike’s theorem says that the family with the smaller number of adjustable parameters is the one we should estimate to be closer to the truth. A simpler family is preferable if it fits the data about as well as a more complex family. Akaike’s theorem describes how much of an improvement in goodness-of-fit the move to a more complicated family must provide for it to make sense to prefer the more complex family. A slight improvement in goodness-of-fit will not be enough to justify the move to a more complex family. The improvement must be large enough to overcome the penalty for complexity (represented by k).
Another feature of Akaike’s theorem is that the relative weight we give to simplicity declines as the number of data points increases. Suppose that there is a slight parabolic bend in the data, reflected in the fact that the SOS value of L(PAR) is slightly lower than the SOS value of L(LIN). Recall that the absolute value of these quantities depends on the number of data points. With a large amount of data our estimate of how close a family is to the truth will be determined largely by goodness-of-fit and only slightly by simplicity. But with smaller amounts of data, simplicity plays a more determining role. Only when a nonlinear trend in the data is ‘statistically significant’ should that regularity be taken seriously. This is an intuitively plausible idea that Akaike’s result explains.
It is not at all standard to think that the curve fitting problem is related intimately
18 Thus, the problems of defining the simplicity of curves described by Priest [1976] do not undermine Akaike’s proposal.
to the problem of explaining why unified theories are preferable to disunified ones. The former problem usually is associated with ‘inductive’ inference, the latter with ‘inference to the best explanation.’ We are inclined to doubt that there really are such fundamentally different kinds of nondeductive inference (Forster [1986], [1988a], [1988b]; Sober [1988b], [1990a], [1990b]).19 In any case, Akaike’s approach to curve fitting provides a ready characterization of the circumstances in which a unified model is preferable to two disunified models that cover the same domain.20
It is always a substantive scientific question whether two data sets should be encompassed by a single theory or
different theories should be constructed for each. Should celestial and terrestrial motion be given a unified
treatment or do the two sets of phenomena obey different laws? In retrospect, it may seem obvious that these two kinds
of motion should receive the same theoretical treatment. But this is the wisdom of hindsight; individual phenomena do
not
have written on their sleeves the other phenomena with which they should be coalesced.
Traditional
approaches to this problem make the allure of unification something of a mystery.21 Given two data sets D1 and D1, a
unified model Mu
19 William Whewell [1840] described the process of curve fitting as a special case of a process of conceptualization called the ‘colligation of facts’ (Forster [1988b]). He then referred to the process that leads to the unification of disparate curve fitting solutions as the ‘consilience of inductions.’ On our view, both of these processes are seen as aspects of a single kind of inferential procedure. Bogen and Woodward [1988] argue that the inferential relationship of observation to theory has two parts: of observation to phenomena and of phenomena to theory.
Again, it is not clear to us that these relationships are fundamentally different in kind.
20 We will follow statistical practice and reserve the term ‘model’ for a family of hypotheses, in which each hypothesis includes a specific statement about the distribution of errors (so that likelihoods are well defined). A model leaves the values of some parameters unspecified. In applying the term to astronomy, we need only assume that some assumption about the form of the error distribution is included (e. g. that the distribution is Gaussian, as was assumed in Gauss’s own application of the method of least squares to astronomy?see Porter [1986]). The variance of the distribution may be left as an adjustable parameter. The important point to notice is that distinguishing models from curves, or from abstract ‘theories’, is now critical to the philosophy of science, since Akaike’s framework only provides a way of defining the simplicity of models.
21 Friedman [1983], like some of the authors he cites (p. 242), describes unification as the process of reducing the
number of independent theoretical assumptions. Of course, a model that assumes principles A, B, and C is made more
probable if these assumptions are whittled down to just A and B. However, as Friedman realizes, head counting will not
deliver this verdict when the postulates of one model fail to be a subset of the postulates of the other.
Friedman
suggests (e.g., pp. 259-60) that a unified model receives more ‘boosts’ in confirmation than a model of
narrower scope. If model Mu covers domains D1 and D2, whereas model M1 covers only domain D1, then Mu can receive a
confirmational boost from both data sets, whereas M1 can receive a boost only from D1. Two points need to be made
about this proposal. First, although Mu receives two boosts whereas M1 receives only one, the conjunction M1 and M2
receives two boosts as well. Here M2 is a model that aims to explain only the data in D2. The conjunction M1&M2 is
a disunified model. If one wishes to explain the virtues of unification, one should compare Mu with this conjunction,
not Mu with M1. The second point is that ‘boosts’ in probability are increases in probability, not the
absolute values
might be constructed that seeks to explain them both. Alternatively, a disunified pair of models M1 and M2 also might
be constructed, each theory addressing a different part of the total data. If M1 fits D1 at least as well as Mu does,
and if M2 fits D2 at least as well as Mu does, what reason could there be to prefer Mu over the conjunction of M1 and
M2? The temptation is to answer this question by invoking some consideration that lies outside of what the evidence
says. One might appeal to the allegedly irreducible scientific goal of unification or to the connection of unification
with simplicity.
The problem posed by the question of goodness-of-fit is a real one, since the combined data set
D1 and D2 often will be more heterogeneous than either subpart is on its own. This engenders a conflict between
unification and goodness-of-fit; a unified theory that encompasses both data sets will fit the data less well than a
conjunction of two separate theories, each tailor-made to fit only a single data set. However, just as in the curve
fitting problem, this conflict can be resolved. Once again, the key is to correct for the fact that disunified
theories are more inclined to overfit the data than their unified counterparts are.
For example, consider the two
data sets represented in Figure 3 and the following three models:
(Mu) The X and Y values in D1 and D2 are related
by the function
Y = α0 + α1X + α2X2 + σ U.
(M1) The X and Y values in D1 are related
by the function
Y = β0 + β1X + σ U.
(M2) The X and Y values in D2 are related by the
function
Y = γ0 + γ1X + σ U.
Since each data set is close to collinear, M1 will be more
likely than Mu with respect to D1 and M2 will be more likely than Mu with respect to D2. However, what happens when we
use Akaike’s Theorem to compare Mu with the conjunction M1 and M2, relative to the combined data? Notice that Mu
has four free parameters, whereas the conjunction M1 and M2 has five. If its assumptions apply (see Appendix A),
Akaike’s Theorem entails that Mu may be more thus attained. The fact that Mu receives two boosts while M1
receives only one is quite consistent with Mu’s remaining less probable than M1. Friedman (pp. 143-4) recognizes
this problem. His solution is to argue that deriving M1 from a unified theory Mu renders M1 more plausible than it
would be if M1 were not so derivable. We note that this claim, even if it could be sustained, does not show why Mu is
more plausible than M1 and M2, where the unified model and its disunified competitor are incompatible. In addition,
the fact that M1 is more plausible in one scenario than it is in another does not bear on the question of how
plausible Mu is.
In addition to these specific problems with Friedman’s proposal, we also wish to note that
its basic motivation is contrary to what we learn from Akaike’s framework. Friedman seeks to connect unification
with paucity of assumptions; as we will see in what follows, unified models
impose more constraints than their
disunified counterparts.
X
Y
D1
D2
FIGURE 3
predictively accurate even though its best case is less likely than the best
case of
M1 and M2. The best fitting case of the disunified theory would have to have a
log-likelihood at least
1 unit greater than the best fitting case of the unified
model if the disunified model were to be judged
predictively superior. This is
not true for the data in Figure 3. We conclude that estimated accuracy explains
why a unified model is (sometimes) preferable to its disunified competitor. At
least for cases that can be
analyzed in the way just described, it is gratuitous to
invoke ‘unification’ as a sui generis constraint
on theorizing.
The history of astronomy provides one of the earliest examples of the problem
at hand. In
Ptolemy’s geocentric astronomy, the relative motion of the earth and
the sun is independently replicated
within the model for each planet, thereby
unnecessarily adding to the number of adjustable parameters in his
system.
Copernicus’s major innovation was to decompose the apparent motion of the
planets into their
individual motions around the sun together with a common
sun-earth component, thereby reducing the number of
adjustable parameters. At
the end of the non-technical exposition of his programme in De Revolutionibus,
Copernicus repeatedly traces the weakness of Ptolemy’s astronomy back to its
failure to impose any principled
constraints on the separate planetary models.
In a now famous passage, Kuhn ([1957], p.181) claims that the
unification or
‘harmony’ of Copernicus’ system appeals to an ‘aesthetic sense, and that alone’.
Many
philosophers of science have resisted Kuhn’s analysis, but none has made
a convincing reply. We present the
maximization of estimated predictive
accuracy as the rationale for accepting the Copernican model over its
Ptolemaic
rival. For example, if each additional epicycle is characterized by 4 adjustable
parameters, then
the likelihood of the best basic Ptolemaic model, with just
twelve circles, would have to be e20 (or more than 485
million) times the
likelihood of its Copernican counterpart with just seven circles for the evidence
to favour
the Ptolemaic proposal.22 Yet it is generally agreed that these basic
22 If the log-likelihood is penalized by
subtracting k, then the likelihood is penalized by
multiplying it by a ‘decay factor’ e−k.
How
to Tell when Simpler Theories will Provide More Accurate Predictions 15
models had about the same degree of fit
with the data known at the time. The
advantage of the Copernican model can hardly be characterized as merely
aesthetic; it is observation, not a prioristic preference, that drives our choice of
theory in this
instance.23
4. CAUSAL MODELING
Newton’s first Rule of Reasoning in Philosophy in Principia was that
‘we are to
admit no more causes of natural things than such as are both true and sufficient
to explain
their appearances.’ Here Newton gives voice to a version of
Ockham’s razor -- explanations that
postulate fewer causes should be preferred
over explanations that postulate more. Although this injunction is
often thought
to be quite separate from the criterion of evidential support, some everyday
applications of the
rule can be given a simple representation in Akaike’s
framework.
The entries in the following table
represent the probabilities that an event C
has, given the four combinations of the putative causes A and B:
P(C / − )
A −A
B w + a + b + i w + b
−B w + a w
Next we define a characteristic function χA:
χA = 1 if A occurs
χA = 0 if A does not occur.
Ditto for the definition of χB.
We now can formulate three hypotheses
about the probability that C has in
these four possible circumstances:
(INT) P(C / χA = xA , χB = xB)
= w + axA + bxB + ixAxB
(ADD) P(C / χA = xA , χB = xB) = w + axA + bxB
(SING) P(C / χA = xA ,
χB = xB) = w + axA.
(SING) says that only a single cause (namely A) makes a difference in whether C
occurs. (ADD) says that two causes play a role and that their relationship is
additive. (INT) says that there are
two causes whose contributions are
interactive (i.e., nonlinear or nonadditive). The hypotheses are listed in
order of
increasing parsimoniousness?one cause is simpler than two, and an additive
23 Forster (1988b) and
Harper (1989) argue that the subsequent impact of Kepler and Newton
may be understood in the same terms.
16
Malcolm Forster and Elliott Sober
model with two causes is simpler than an interactive model for those two causes.
As in the curve fitting problem, it is standard to understand causal modeling as
a problem with two parts. First
one selects a hypothesis about the form the
causal relationship is to take; then one finds the best hypothesis of
that form by
estimating parameter values. Rather than solving the first problem by appeal to
simplicity, our
approach shows how estimated predictive accuracy can be
brought to bear from the beginning. Suppose one has a
large and equal number
of observations for each of the four treatment cells. Let the empirical
frequencies of
C in those four cells be:
P(C / − )
A −A
B 0.5 0.2
−B 0.5 0.2
The three hypotheses now have the same best case, namely one in which w =
0.2, a = 0.3, b =
0, and i = 0. Recall that the estimated predictive accuracy of
each model is 1/N times its maximum log-likelihood
minus k/N. This means
that when one model is a special case of another and they have the same best
case, the
model of lower dimensionality has greater estimated predictive
accuracy. It follows that (SING) has greater
estimated predictive accuracy than
(ADD) and (ADD) has greater estimated predictive accuracy than (INT). For
the
data just given, predictive accuracy explains why it is vain to postulate more
causes when fewer
suffice.24 And as in our discussion of unification, it is
possible to adjust the data set so as to provide a
rationale for favouring a
hypothesis of greater complexity.
5 THE PROBLEM OF AD HOCNESS
The bugbear of ad
hoc hypotheses has traditionally been raised within the
framework of a hypothetico-deductive philosophy of
science. Predictions can be
deduced from theories only with the help of auxiliary hypotheses. On this view,
we
test a theory by observing whether its predictions are true. However, the
Quine-Duhem thesis states that the core
theory may always be shielded from
refutation by making after-the-fact adjustments in the auxiliary hypotheses, so
that correct predictions are deduced. The classic example of this is Ptolemaic
astronomy, where the model may
always be amended in the face of potential
refutation by adding another circle?so much so that the expression
‘adding
24 In this example, it isn’t just that fewer causes are preferable to more; in addition, we have
shown why an additive model for two causes is preferable to an interactive model of those two
causes. Counting
causes is a special case of the more general consideration of dimensionality.
Forster [1988b] argues that Newton
was sensitive to this wider conception.
How to Tell when Simpler Theories will Provide More Accurate Predictions
17
epicycles to epicycles’ has become synonymous with ‘ad hocness’. Although we
reject the
hypothetico-deductive picture of science, we do accept the usual
conclusion that there is an important distinction
to be drawn between reasonable
revision and ad hoc evasion.
Philosophers of science have recognized that
protection of the core theories by
post hoc revision is not always bad. The example usually cited is Leverrier’s
postulation of Neptune’s existence to protect Newtonian mechanics from the
anomalous wiggles in Uranus’
orbit. The problem is to understand the
epistemological grounds for distinguishing good from bad revisions of
auxiliary
hypotheses (which Lakatos [1970] refers to as the protective belt). As is
customary, we reserve the
term ‘ad hoc’ for revisions of the bad kind, but reject
the ad hominem or historicist construal of the
term. Ad hocness, if it is relevant
to questions of evidence, has nothing to do with the motives of the person
advocating the hypothesis, or with historical sequences of theories and their
evidence.25
Lakatos [1970]
notes, with approval, that Leverrier’s amendment of the prior
Newtonian planetary model produced novel
predictions; he introduces the
derogatory term ‘degenerating’ for research programmes that fail to do
this. But
there are at least two problems with this approach. Musgrave [1974] warns that
a careless reading of
the term ‘novel’ may tempt us into a view of confirmation
in which historical contingencies are given
undue emphasis. The second defect
in Lakatos’s idea is that it fails to distinguish estimated predictive
success from
predictive power. It is obvious that predictive power is important, for without it
there can be
no predictive success. But predictive power is not enough to
indicate that model revisions are of the good kind.
For example, the continued
addition of epicycles in Ptolemy’s astronomy is not degenerate in
Lakatos’s
sense. Each addition leads to novel predictions about the future positions of the
planets. What
we need is a measure of the predictive success that these
additions can be expected to bring, and this is what
Akaike’s idea of estimated
predictive accuracy provides.
Our proposal is that a research programme is
degenerative just in case loss in
simplicity is not compensated by a sufficient gain in fit with data. Of course,
the
fit will always improve, but the improvement may not be enough to increase the
estimated predictive value.
Established research programmes often achieve considerable predictive
success, so why do some researchers put
their money on an undeveloped
programme? First note that on our proposal there is no impediment for new
programmes to take over the predictive successes of old ones. There is no
‘problem of old evidence’ (Glymour [1980], Eells [1985]), since estimated
25 We do not rule out the
possibility that historical or psychological circumstances may
sometimes be a reliable indication of ad hocness.
Our only point is that these circumstances
do not make a theory ad hoc, anymore than a barometer makes it
rain.
18 Malcolm Forster and Elliott Sober
predictive accuracy does not depend on the historical sequence of
discovery.
But further, it is perfectly understandable that researchers may decide where to
invest their
energy by formulating a judgment about projected predictive
success, and the degree to which current programmes
are degenerating is thus a
relevant consideration.26
6 THE SUB-FAMILY PROBLEM
While this explication of
Lakatos’ notion is a point in favour of our approach,
there is another type of ad hocness that is a threat to
Akaike’s programme. A
literal reading of Akaike’s Theorem is that we should use the best fitting
curve
from the family with the highest estimated predictive value. However, for any
such family, it is
possible to construct an ad hoc family of curves with the same
best fitting curve, with yet higher estimated
predictive accuracy: Fix one or
more of the adjustable parameters at their maximum likelihood values. Each
subfamily, so constructed, will have the same best case. At the end of the
procedure, we obtain a zero dimensional
family whose only member is the best
fitting curve of the original family. The Akaike estimate of the
predictive
accuracy of this singleton family is just the log-likelihood of the curve. If this is
allowed, then
we are pushed back towards selecting complicated curves that fit
the data exactly. We call this the sub-family
problem.27
Our resolution of this problem returns us to an idea described in Section 2: If a
curve fits the
data so well that it looks ‘too good to be true’, then it probably is.
In order to spell this out, we
now describe a theorem (stronger than Akaike’s)
that characterizes the behaviour of the error in estimating
the predictive
accuracy of families. The error of the estimated predictive accuracy of family F,
or
Error[Estimated(A(F))], is defined as the difference between Akaike’s
estimate of the predictive accuracy of
family F and the true predictive accuracy
of that family. Notice that the true predictive accuracy is constant?it
does not
depend on which hypothetical data set generated by the truth happens to be the
actual data set. On
the other hand, the estimated predictive accuracy of F does
depend on the actual data?it is what statisticians
call a random variable. So
Error[Estimated(A(F))] also depends on the data, and the following theorem
describes this dependence by decomposing it into the sum of three errors:28
26 The Akaike approach also finesses
the problem of ‘Kuhn loss’: Superceding theories do not
always carry over all the successes of their
predecessor. For example, Cartesian vortex theory
‘explains’ why all planets revolve around the sun in the same direction, whereas Newton’s
theory
dismisses this as a mere coincidence. Within Akaike’s framework, the losses are
weighed against the gains in
the common currency of likelihoods.
27 The reader should not be misled into thinking that the subfamily problem is
a problem for
Akaike’s criterion alone; it arises for any proposal that measures simplicity by the paucity of
parameters.
28 The result we are about to describe is close to, but not identical with, equation (4.55) in
Sakamoto et al. ([1986], p.77). Similar formulae were originally proven in Akaike [1973].
See Forster [1992a] for
further explanation.
How to Tell when Simpler Theories will Provide More Accurate Predictions 19
The Error
Theorem:
Error[Estimated(A(F))] =
Residual Fitting Error + Common Error + Sub-family Error.
It is
important to remember that these errors are not errors of prediction - they
are errors in the estimation of
predictive accuracy. This is why the Error
Theorem might be called a ‘meta-theorem’ - it is a theorem
about the ‘meaning’
of Akaike’s Theorem. However, it rests on the same assumptions as Akaike’s
Theorem (see Appendix A).
Akaike’s Theorem states that the average of Error[Estimated(A(F))] over all
possible data sets generated by the truth is zero, which is to say Akaike’s
estimate of predictive accuracy
is statistically unbiased.29 ‘Statistically unbiased’
means that its average performance will center on
the true value of the quantity
being estimated; it is a minimal requirement for ‘good’ estimators.
Akaike’s
estimate conforms to this standard, but sometimes fails to meet another
desideratum, which we
will refer to as epistemic unbiasedness. We shall now
explain the distinction in terms of an example.
First,
consider a standard example of a statistically unbiased estimate: the
measurement of the mass of an object. For
this measurement, the deviation from
the true mass value is determined by a symmetrical error distribution centred
on
the true mass value, so that it is just as probable that the measured value is below
the true value as it
is above the true value. The measured value of mass is a
statistically unbiased estimate of the true mass. But now
suppose that we modify
this estimate by adding +10 or −10 depending on whether a fair coin lands heads
or tails, respectively. Supposed that the measured value of mass was 7 kg, and
the fair coin lands heads. Then the
new estimate is 17 kg. Surprisingly, this new
estimate is also a statistically unbiased estimate of the true mass!
The reason is
that in an imagined series of repeated instances, the +10 will be subtracted as
often as it is
added, so that the value of the average value of the modified
estimate will still be equal to the true mass value.
However, we know that the
modified estimate is an overestimate in this instance, because we know that the
coin
landed heads. If the coin had landed tails, then the estimate would have
been −3 kg, and would have been
known to be an underestimate. In either case,
we say that the modified estimate is epistemically biased. In sum,
the
unmodified measurement value is a statistically and epistemically unbiased
estimate of the mass, while the
modified estimate is statistically unbiased, but
epistemically biased. Other things being equal, we prefer an
estimate that is
epistemically unbiased.
With this distinction in hand, the Error Theorem is able to explain
the
limitations of Akaike’s method. Here is a brief overview of our analysis:
29 Statistical unbiasedness
is really a property of the formula for obtaining the estimate, rather
than the particular value of the estimator.
20 Malcolm Forster and Elliott Sober
First, the common error is the same for all families (hence its name); it
cancels
out when we make comparisons, and has no effect on model selection. It will
not be mentioned again.
Second, the Residual Fitting Error is statistically and
epistemically unbiased. But the Subfamily Error has a
peculiar property. It is
statistically unbiased (as is required by Akaike’s Theorem); however, it is not
always free of epistemic bias. Sometimes Akaike’s estimate displays an
epistemic bias, and this bias is
highlighted by the subfamily problem. A careful
analysis of the Subfamily Error will reveal the source and nature
of the problem.
We begin by filling in some background. One of the assumptions of these
theorems is that there
is some complex K-dimensional family of hypotheses
(curves) that includes the true hypothesis, and that every
family F that we may
wish to consider is a subfamily of this superfamily (which we will call K).
Every
hypothesis under consideration may be represented as a point in the
parameter space of K. This space may be
treated as a K-dimensional vector
space. So, if we imagine that our coordinate frame is centered on the Truth
(where else?), then various hypotheses may be located in different directions, as
shown in Figure 4. The two
vectors shown are particularly important because
the subfamily error is equal to the dot product, or scalar
product, of these two
vectors. The first vector is the one to L(K), the best fitting curve in K. Clearly
this
vector will move around when we consider different data sets generated by
the truth. In fact, its tip falls just
as probably on one point as on any other on the
circle shown, although its length will vary as well. The other
vector is fixed. It
is the vector from the truth, T, to the hypothesis in the family F that is closest to
T
(viz. the most predictively accurate hypothesis in F). Now, the dot product is
the product of the lengths of these
two vectors times the cosine of the angle
between them. The cosine factor is +1 if the vectors are parallel, 0 if
they are
orthogonal, −1 if they are anti-parallel, and in between for in between angles.
The Akaike
estimate for a low dimensional family whose best fitting case is
close to the data (and such families are the
dangerous ‘pretenders’, for they
‘unfairly’ combine high log-likelihoods with small penalties for complexity)
The hypothesis in F
T
closest to T
L(K), representing the data
in parameter space
FIGURE 4
How to Tell when Simpler
Theories will Provide More Accurate Predictions 21
exhibits an epistemic bias, as we now explain. The most
predictively accurate
hypothesis in such small families will also be close to the data, and therefore
close to
L(K). The danger is that the tips of the two vectors (whose dot product
is equal to the subfamily error) will be
close together. Then the cosine factor is
close to +1 and the subfamily error is large and positive. To illustrate
this aspect
of the relationship of Akaike’s Theorem and the Error Theorem, consider the
following
example. Suppose we have a very large data set that exhibits strong
linearity. We wish to estimate the predictive
accuracies of L(LIN) and
L(POLY-n), where POLY-n is the family of n-degree polynomials with n
parameters free,
and L(F) is obtained by using the data to single out the best
fitting curve in family F.30 We may apply Akaike’s
Theorem to (LIN) and
(POLY-n) directly, or we can apply it to the singleton families containing just
L(LIN)
and L(POLY-n), respectively. The surprising fact - that the ad hoc
Akaike estimate for L(POLY-n) is surely an
overestimate of the predictive
accuracy of L(POLY-n) - may have been anticipated from the fact that unreliable
ad hoc comparisons of L(POLY-n) and L(LIN) will always favour L(POLY-n),
because it is always closer to the data.
In sum, both the direct and the ad hoc
method of accuracy estimation are statistically unbiased (as required by
Akaike’s
Theorem), but the ad hoc application of Akaike’s method yields an estimate that
we know is
too high. The ad hoc application yields an estimate that is
epistemically biased.31
We have now unpacked our
slogan about a curve’s looking ‘too good to be
true’ to provide deeper insights into the source and
solution of the subfamily
problem: The Akaike estimates of the predictive accuracy of L(F) obtained by
viewing
L(F) as the best fitting case in the ad hoc hierarchy of subfamilies of F
tend to be too high. Indeed, that is
exactly what we observe?the Akaike
estimate of L(F) increases steadily as we move down the hierarchy towards
the
singleton subfamily. In sum: We have good reason not to trust the Akaike
accuracy estimates for ad hoc
subfamilies constructed by fixing adjustable
parameters at their maximum likelihood values. We emphasize that this
has
nothing to do with when subfamilies are constructed, or who constructs them.
Our analysis of the Error
Theorem has been brief and necessarily incomplete.
Much more research is needed on the management of errors in
Akaike’s method
of model selection. Our aim has been to give the reader a taste for the heuristic
power
of Akaike’s framework in addressing such foundational questions. We
close by pointing out that the resolution
we have sketched depends (like
Akaike’s Theorem) on the existence of prediction errors, for otherwise the
vector
30 Remember (from Section 2) that we are interested in estimating the predictive accuracy of a
family
only because it also provides an estimate of the predictive accuracy of its best fitting
curve.
31 Although
the estimate is known to be too high, given the data at hand, the Akaike estimate of
the predictive accuracy of
that same singleton family relative to other data sets generated by
the true ‘curve’ will be too low. On
average, of course, the estimate will be centred on the true
value.
22 Malcolm Forster and Elliott Sober
to L(F) would be 0 and there would no subfamily errors for any family.
7. THE BEARING ON BAYESIANISM
The
fundamental principle behind Akaike’s method is that we should aim to
select hypotheses that have the
greatest predictive accuracy. Since the truth has
the maximum possible predictive accuracy and accuracy is a
measure of
‘closeness’, Akaike’s recipe aims to move us towards the truth. In contrast, the
central thesis of
the kind of Bayesianism we will criticize here is that hypotheses
should be compared as to their probability of
truth.32
In this section, we examine the possibility that Akaike’s method might be
recast in a Bayesian
framework. Since our argument is many-faceted, we
provide a brief summary here. We criticize two different
Bayesian proposals
that promise to yield a solution to the curve fitting problem. The first Bayesian
strategy
is to focus on families?show that the best families by Akaike’s
standards are the most probable families, and
then give a Bayesian justification
for selecting the best fitting case. The second approach is to bypass families,
and show how the most accurate individual hypotheses end up with higher
posterior probabilities. After criticizing
these suggestions, we end the section by
suggesting that Bayesian methods may be useful for assessing the risks in
applying Akaike’s criterion.
The key element of any Bayesian approach is the use of Bayes’ Theorem,
which says that the probability of any hypothesis H given any data is
proportional to its prior probability times
its likelihood: p(H/Data) ∝ p(H) ×
p(Data/H). However, it is an unalterable fact about probabilities
that (PAR) is
more probable than (LIN), relative to any data you care to describe. No matter
what the
likelihoods are, there is no assignment of priors consistent with
probability theory that can alter the fact that
p(PAR/Data) ≥ p(LIN/Data). The
reason is that (LIN) is a special case of (PAR). How, then, can Bayesians
explain the fact that scientists sometimes prefer (LIN) over (PAR)?33
Bayesians might propose to address this
problem as follows. Instead of (LIN)
32 The problems we will enumerate for Bayesianism in what follows apply with
equal force to
what might be called incremental Bayesianism. This doctrine has no interest in assigning
absolute values to prior and posterior probabilities, but seeks only to make sense of differences
or ratios that
obtain between these quantities. If H1 and H2 are both confirmed by the data,
both P(H1/Data)/P(H1) and
P(H2/Data)/P(H2) are greater than unity. To compare these ratios
to find out which hypothesis received the larger
boost, we need to evaluate the likelihood ratio
P(Data/H1)/P(Data/H2). When the hypotheses are single curves, the
better fitting hypothesis
automatically receives the higher boost. When the hypotheses are families, evaluating
this
ratio leads to the problems we will describe in connection with Bayesian approaches to
defining the
likelihood of families.
33 One might seek to evade this conclusion by saying that (LIN) and (PAR) are embedded
in
different theoretical contexts, that this difference gives rise to differences in meaning between
their
respective theoretical parameters, and that it follows from this that (PAR) is not entailed
by (LIN). Although we
are prepared to grant that this might be plausible in certain special
cases, we doubt that this is an adequate
response in general.
How to Tell when Simpler Theories will Provide More Accurate Predictions 23
and (PAR),
let us consider (LIN) and (PAR*), where (PAR*) is some subset of
(PAR) from which (LIN) has been removed. Since
(LIN) and (PAR*) are
disjoint, nothing prevents us from ordering their prior probabilities as we see fit.
In
response, we note that this ad hoc maneuver does not address the problem of
comparing (LIN) versus (PAR), but
merely changes the subject. In addition, it
remains to be seen how Bayesians can justify an ordering of priors for
the
hypotheses thus constructed and how they are able to make sense of the idea that
families of curves (as
opposed to single curves) possess well defined likelihoods.
Rosenkrantz [1977] and Schwarz [1978] independently
argued for a proposal
of the first kind?ignoring the problems of logical entailment, they seek to
compare the
likelihoods of families of curves.34 So consider some family of
curves F with dimension k. The idea is to define
the average likelihood of the
family in terms of some prior weighting of the members of the family,
p(Curve/F).35
If p(Curve/F) is strictly informationless, then it is easy to see that p(Data/F) =
0. Almost
every curve in the family will be very far from the data. This means
that if we accord equal weight to every curve
in F, the average likelihood of F
will be zero. What if we let p(Curve/F) be ‘almost’ informationless?
This
means that we divide the curves in the family into two subsets -- within one
subset (which includes
curves close to the data points), we let the weights be
equal and nonzero; outside this volume, we let the weights
be zero. We
illustrate this proposal by returning to the examples of (LIN) and (PAR), where
the error variance
σ2 is known. For (LIN), we specify a volume V1 of parameter
values for α0 and α1 within which
the likelihoods are non-negligible. For PAR,
we specify a volume V2 of parameter values for β0, β1, and
β3 with the same
characteristic. If we let boldface α and β range over curves in (LIN) and
(PAR)
respectively, the average likelihoods of those families then may be expressed
approximately as
follows:
p(Data/LIN) = (1/V1) I⋅⋅⋅I p(Data/α,LIN) dα
p(Data/PAR) = (1/V2) I⋅⋅⋅I
p(Data/β,PAR) dβ,
where the integration is restricted to the subsets of curves with non-zero
weights.
Note that as larger and larger volumes are taken into account, the average
likelihoods approach zero
(as the weighting become more strictly
informationless).
How are these two likelihoods to be compared? The
volume V1 has two
dimensions in parameter space; the volume V2 has three. Although Rosenkrantz
34 They ignore
the entailment problem by comparing only the likelihoods of families; they
bracket the Bayesian comparison of
posterior probabilities.
35 Here, the ‘average likelihood’ is an average over the members of a family of
curves, and the
Data are fixed. In contrast, the ‘average log-likelihoods’ we discussed in previous
sections
were averages of the log-likelihood of a single curve with respect to many (hypothetical) data
sets.
24 Malcolm Forster and Elliott Sober
[1977] and Schwarz [1978] do not formulate their analysis in terms
of the
volumes V1 and V2, their proposal is equivalent to setting V1 = V2. This is one
way to render
commensurable the volumes of different dimensionality that
appear in the likelihood expressions.36
The trouble
is that the proposal is not invariant under reparameterization.
Consider the following pair of equations:
(LIN) Y = α0 + α1 X + σ U
(LIN′) Y = (α0′)/3 + (α1′/2) X + σ
U.
These equations define exactly the same family of straight lines. Yet, the
proposal entails that the latter
has 6 times the average likelihood of the former.37
Let us now turn to another strategy that Bayesians might
pursue in finding a
solution to the weighting problem. This is to let p(α/LIN) be equal to some
informative probability p(α/LIN,E0). Here the weighting scheme is a posterior
probability, constructed on
the basis of some evidence E0 that was acquired
before the Data. The difficulty with this proposal is that it only
pushes the
problem back a step. One still has to make sense of the average likelihood
p(E0/LIN). This requires
us to evaluate quantities of the form p(α/LIN).
Eventually, this must lead the Bayesian back to the quest
for informationless (or
almost informationless) priors, which we have discussed already.38 In light of
these
considerations, we think it is highly questionable that this first Bayesian
36 The ad hocness of any such
assumption is noted by Aitkin [1991], who refers his readers to
Lindley [1957].
37 The reader can most easily
grasp this result by considering the problem of integrating a
function f(x), where f(x) = 1 between the limits 0
and 1, and f(x) = 0 elsewhere. Clearly,
f (x)dx 1
∞
−∞
∫ = .
Yet if we transform coordinates such that xN = 6x, while equating g(xN) and f(x) for
corresponding
values of x and xN, we obtain
g(x)dx 6
∞
−∞
∫ ′ ′= .
38 Nevertheless, Schwarz [1978] has pressed ahead and derived an interesting
asymptotic
expression for the average likelihood (with the V term omitted). Under conditions similar to
those
for Akaike’s Theorem,
Log(Average Likelihood of F) = log p(Data/L(F)) − (logN) k/2 + other terms ,
where L(F) is the maximum likelihood hypothesis in F, N is the number of data, and k is the
dimension of F. The
‘other terms’ are negligible for large N. The resulting recipe for model
selection is often referred to
as the Bayesian Information Criterion, or BIC for short. We will
not evaluate the criterion here. But we deny that
it is securely grounded in the Bayesian
framework, for the reasons we have given. In that regard, it is
interesting to note that the same
criterion has been independently derived from quite different principles by
Akaike [1977] and
Rissanen [1978], [1989].
How to Tell when Simpler Theories will Provide More Accurate
Predictions 25
approach?in which families of curves are the objects of investigation?can
provide a
satisfactory treatment of the curve fitting problem.39
So let us consider a Bayesian who compares the
probabilities of particular
curves. The problem here is that there seems to be no principled way for
estimated
predictive accuracies to affect the estimated probability of their truth.
For such a Bayesian is bound by Bayes’
Theorem, which says that the posterior
probability of such a particular hypothesis is proportional to the prior
probability
times the likelihood relative to the total evidence:
p(Curve/Data) = p(Curve) p(Data/Curve)
/p(Data) .
The likelihood term, p(Data/Curve), simply measures the goodness-of-fit, so the
only vehicle for
including any estimate of the predictive value of the curve is in
the prior probability, p(Curve). In order to
replicate the Akaike result, we would
need
p(Curve) = p(Data) e−k ,
where p(Data) is merely a
normalization factor. But we do not see how a
Bayesian can justify assigning priors in accordance with this
scheme.
The problem is not avoided by adopting a subjectivist approach that eschews
the need for objective
justification. The problem is deeper than that. The
trouble is that a particular curve, as opposed to a family of
curves, cannot be
assigned a value of k on a priori grounds. After all, any curve is a member of
many families
of different dimensions. While this problem for Akaike arises in
the guise of the subfamily problem, the proposed
solution was to distrust
subfamilies that have a special relationship with the data. However, no
comparable
solution is available to the Bayesians because the determination of k
must be made independently of the data.
Thus, Bayesians must find an entirely
different kind of solution to the subfamily problem,
39 However, Aitkin
[1991] has a different ‘average likelihood’ proposal, which allegedly solves
the curve fitting problem.
He computes the average by weighing each curve in the family by
its posterior probability p(Curve/Data), given all
the available data. A theorem based on the
same assumptions as Akaike’s Theorem shows that:
Log(Aitkin
Average Likelihood of F) = Log-likelihood(L(F)) − (k/2)log2 .
Since log2 is less than 1 (the logarithms are
to base e), Aitkin imposes less than 1/2 of
Akaike’s penalty for complexity. This is already an uncomfortable
consequence because the
Error Theorem shows that (PAR) will be chosen over (LIN) by Aitkin’s criterion more
often
than not even when (LIN) is true. But the real problem is that the criterion is just ‘pulled out of
a hat.’ What will families of greater average posterior likelihood provide for us? Will they
tend to bring us
closer to the truth, or give us more accurate predictions, or what? Aitkin
provides no answers to these questions.
Given that Aitkin’s proposal does not have more
fundamental principles to fall back on, how does he cope with
the subfamily problem? There
is no analogue to the Error Theorem for Aitkin because there is no sense in which
average
likelihood is in error if it is not estimating anything. Also see the commentaries immediately
following Aitkin’s paper, including one by Akaike.
26 Malcolm Forster and Elliott Sober
and we fail to
see how this can be done.40
Our diagnosis of the problem is that Bayesianism is unable to capture the
proper
significance of considering families of curves. We work with families
because they deliver the most reliable
estimates of the predictive accuracy of a
few curves; namely their best fitting cases. There is no reason to
suspect that
such an enterprise can be construed as maximizing the probability that these best
fitting cases
are true. Why should we be interested in the probability of these
curves’ being true, when it is intuitively
clear that no curve fitting procedure will
ever deliver curves that are exactly true? If we have to live with
false
hypotheses, then it may be wise to lower our sights, and aim at hypotheses that
have the highest
possible predictive accuracy. Thus, the brand of Bayesianism
most popular amongst philosophers is founded on too
narrow a conception of the
scientific enterprise.41
Having said all that, we do not draw the rash conclusion
that Bayesian
methodology is irrelevant to Akaike’s new predictive paradigm. There are many
Bayesian
solutions to practical statistical problems. However, Akaike’s
reconceptualization of statistics does
recommend that the foundations of
Bayesian statistics require rethinking.42 A positive suggestion may be that
Bayesian methods can help determine the probability that one hypothesis is more
predictively accurate than
another. In that way, Bayesian methods might be
usefully brought to bear on the problem of assessing the
reliability of estimated
accuracies, for that appears to be an important and open area of research.
8.
EMPIRICISM AND REALISM
One virtue of our approach is that it makes clear what the simplicity of a curve
has to
do with the reasons one might have for believing it. Popper [1959] argued
that simpler curves are more
falsifiable; Sober [1975] suggested that simpler
curves are more informative. These proposals, and others like
them,43 make it
40 In this respect, we think it is instructive to consider the recent attempt by Jefferys and
Berger
[1992] to provide a Bayesian rationale for Ockham’s razor. We criticize their proposal in
Sober
and Forster [1992].
41 It is easy to construct examples which show that maximizing probability of truth is
different
from maximizing closeness to the truth. A common example is the use of averages to estimate
a
discrete number, say the number of children in an American family. An estimate of 1.9
children has less
probability of being true in any case than an estimate of 2, but may be
predictively more accurate
nevertheless.
42 Akaike [1985] shows how the rule of Bayesian conditionalization, as a method of updating
probabilities, may be understood in terms of maximizing expected predictive accuracy.
43 Turney [1990]
demonstrates that simpler families of curves are more stable. Roughly, the
instability of a family of curves,
relative to the data, is the expected ‘distance’ (measured by
the SOS) of a new best fitting curve from
the old best fitting curve when the data are perturbed
in accordance with the known error distribution. Turney’s
measure of instability takes one step
How to Tell when Simpler Theories will Provide More Accurate Predictions
27
difficult to say why one ought to believe simpler curves rather than their more
complex competitors. In
contrast, the analysis we have proposed greatly
simplifies the task of justification. When a simpler curve is more
plausible than
its more complex alternatives, this is because it has a higher estimated predictive
accuracy.
We believe that our account of curve fitting is good news for empiricism,
although it does not
accord with what has been said by many empiricists. The
idea that some sui generis criterion of simplicity is
relevant to judging the
plausibility of hypotheses is deeply inimical to empiricism. For empiricism,
hypothesis evaluation should be driven by data, not by a priori assumptions
about what a ‘good’
hypothesis should be like. Empiricists often take this point
to heart and conclude that simplicity is a merely
pragmatic virtue, one having to
do with the usefulness of hypotheses, but not with their plausibility (cf. e.g.,
Van
Fraassen [1980], pp. 87-89). The embarrassing thing about this dismissal of
simplicity is that it applies
not just to highly theoretical hypotheses, but to quite
mundane empirical generalizations of the sort that figure
in some curve fitting
problems. In these contexts, skepticism about simplicity threatens to lead the
empiricist down the garden path to skepticism about induction (Sober [1990a]).
Empiricists therefore should
welcome the idea that curve fitting does not require
a sui generis criterion of simplicity. This does not show
that some form of
radical empiricism is true. Rather, we draw the more modest conclusion that the
data tell
you more than you may have thought.44
Although our goal has been to show how the simplicity of a curve can reflect
important facts about its predictive accuracy, we do not claim that all uses of
simplicity and parsimony in
science reduce to purely evidential considerations.
We do not deny that scientists often have pragmatic reasons
for using simpler
curves instead of more complex ones. However, we would insist that these
pragmatic
considerations not be confused with evidential ones. Monolithic
theories about simplicity and parsimony?which
claim that these considerations
are never evidential or that they are never merely pragmatic?should be
replaced
by a more pluralistic approach. At least in the context of the curve fitting
problem, Akaike’s
technical result provides a benchmark that identifies the
degree to which simplicity has evidential significance.
Any further weight
accorded to simplicity, we suspect, derives from pragmatic considerations.
Our analysis
supports the idea that the simplicity of a family of curves is an
towards estimating the degree of overfitting, as
we have characterized it. However, in our
opinion, his paper does not show why stability should be relevant to the
question of what to
believe. We also note that Turney leaves open the justification for trade offs between
simplicity and goodness-of-fit. Akaike’s Theorem is more general than Turney’s theorem in
any case?it is
not restricted to the standard curve fitting situation, and does not assume a
known error variance.
44 For the
bearing of this thesis on traditional arguments against the existence of component
forces in Newtonian physics,
see Forster [1988b].
28 Malcolm Forster and Elliott Sober
epistemic epiphenomenon.45 Sometimes simpler curves
are to be preferred over
more complicated ones, but the reason for this is not that simplicity is an
epistemic
end-in-itself. At other times, more complex curves are to be preferred
over simpler alternatives, but this is not
because the irreducible demands of
simplicity are overwhelmed by more weighty considerations of some other
sort.
Whether a simpler curve is preferable to some more complex alternative, or the
reverse is true, has
nothing to do with simplicity and everything to do with
predictive accuracy.
Our brand of empiricism is not
antithetical to the realist view that science aims
at the truth,46 in the same sense that archers aim at the bull’s-eye
even when
they have no hope of hitting it. In the past, the curve fitting problem has posed a
dilemma: Either
accept a realist interpretation of science at the price of viewing
simplicity as an irreducible and a prioristic
sign of truth and thereby eschew
empiricism, or embrace some form of anti-realism. Akaike’s solution to
the
curve fitting problem dismantles the dilemma. It now is possible to be a realist
and an empiricist at the
same time.
Popper [1968] initiated a realist program that takes the ‘disastrous metainduction
’ (Laudan [1984]) seriously - all of our scientific theories in the past
have been false, so it is likely
that all of our theories in the future will also be
false. Even granting this prediction of failure, it may make
sense to claim that
our theories aim at the truth if we could (1) define a measure of
closeness-to-the-truth,
and (2) show how theory choice could be viewed as
implementing some method that would, more often than not, take
us closer to
the truth. Proposed solutions to the problem of defining verisimilitude have
never gained wide
acceptance,47 and the second part of the programme is seldom
discussed.
We have already described predictive
accuracy as a measure of closeness to
45 This thesis complements the view of parsimony developed in Sober [1988b],
[1990b]. It also
might be formulated in terms of the idea of screening off: Simplicity is correlated with
plausibility, but only because simplicity also is correlated with predictive accuracy. Once the
estimated
predictive accuracy of a hypothesis is held fixed, its simplicity has nothing further to
contribute to an
assessment of its plausibility.
46 We do not claim that this is the only aim of science. We agree with
sociologists of science
that a complete account of the practice of science must include an account of pragmatic
and
social values. Modern theories of decision making are well equipped to model scientific
practice in terms
of pragmatic, social, and evidential considerations, in a way that is
compatible with realism (Hooker [1987]).
However, we do oppose those extremists who
believe that internal evidential considerations play no role in the
social dynamics of science.
47 Popper’s original definition of verisimilitude was formulated in terms of the
deductive
consequences of theories; fatal flaws were detected independently by Tichý [1974] and by
Miller [1974]. Tichý [1974] presents an alternative definition of his own, which Miller [1974]
shows to be
language dependent. Miller [1975] also argues that the intuitive notion of
accuracy of prediction is also subject
to the same kind of language variance. Good’s [1975]
reply to Miller’s paper contains a brief
explanation of why a probabilistic definition of
accuracy, like Akaike’s, is not susceptible to Miller’s
objection. See Forster [1992a] for
further discussion.
How to Tell when Simpler Theories will Provide More
Accurate Predictions 29
the truth. To that extent, Akaike’s approach revitalizes Popper’s
programme.48
However, we suspect that those neo-Popperians who seek some grand
metaphysical definition of
closeness to the truth will be disappointed with a
notion of predictive accuracy defined by reference to a
specified domain of
inquiry.49 Nonetheless, we are convinced that any definition of verisimilitude
must be
limited in this way if we are primarily interested in epistemological
questions. In any event, the important point
is that Akaike’s Theorem lays the
epistemological foundation for our progress towards the truth in this
domainrelative
sense.
In spite of our sympathy for Popper’s quest for a concept of verisimilitude, we
nonetheless reject hypothetico-deductivism, on which the Popperian programme
is founded.50 The
hypothetico-deductivist strategy has been to adopt an
idealized model of science in which there are no
probabilistic errors in the data,
to use this error-free idealization to solve various philosophical problems, and
then to add an account of error as an afterthought.51 Our analysis suggests that
many central problems in the
philosophy of science are not decomposable in this
way. Simplicity and unification are relevant to our judgments
about what is
truth-like only to the extent that observing and inferring are subject to error.
9. APPENDIX A:
THE ASSUMPTIONS BEHIND AKAIKE’S THEOREM
There are three kinds of assumption behind the proof of Akaike’s
Theorem.
First, there is a ‘uniformity of nature’ assumption that says that the true curve,
whatever
it is, remains the same for both the old and the new data sets
considered in the definition of predictive
accuracy. The second kind of
assumption consists of mathematically formulated conditions that ensure the
‘asymptotic normality’ of the likelihood function (viz. the likelihood viewed as a
function of parameter
values). These assumptions contribute to proving various
central limit theorems in mathematical statistics. The
final assumption is that
the sample size (the amount of data) is large enough to ensure that the likelihood
function will approximate its asymptotic properties. It is the second assumption
that requires the most
explaining. We first say what the ‘normality’ assumption
48 This perspective also is relevant to
Cartwright’s [1983] argument that the proliferation of
mutually incompatible models in physics is a reason to
reject realism. This is an
embarrassment to a realist who interprets all (viable) models as true. On the other
hand, our
brand of realist is only interested in interpreting hypotheses as being more or less
close-to-the-truth. A plurality of models is conducive to a more modest realist programme.
49 We note in this
connection that there are philosophical issues raised by the concept of
prediction that are not addressed by
Akaike’s notion of predictive accuracy.
50 Note that hypothetico-deductivism, as we understand it, is not
rescued by the fact that
probabilistic assertions about future data are deduced from scientific hypotheses.
For
hypothetico-deductivism demands that at least some of the deductive consequences of our
theories are
observations, but we do not observe probabilities.
51 See Forster [1994] for a discussion of how this bears on
Hempel’s raven paradox.
30 Malcolm Forster and Elliott Sober
is, and describe the pivotal role it has
played in statistics.
The normal, or Gaussian, probability distribution is easily recognized in its
one
dimensional form by its characteristic bell shape. In its more general
multivariate form, the normal distribution
has come to play a pivotal role in
experimental and theoretical statistics. In experimental statistics, error
distributions (in the estimation of parameter values) are found to be
approximately normal, especially for large
data sets. According to Cramér
([1946], p.231), ‘Such is the case, e.g., with the distributions of errors
of
physical and astronomical measurements, a great number of demographical and
biological distributions, etc.’
In fact, the assumption that measurement errors are
normally distributed around a mean value is so widespread in
science that it is
often referred to as the law of errors. On the theoretical side, ‘the central limit
theorem affords a theoretical explanation of these empirical facts.’ In a
somewhat humorous tone, Cramér
([1946], p.232) sums up by quoting Lippman
as saying: ‘everyone believes in the law of errors, the
experimenters because
they think it is a mathematical theorem, the mathematicians because they think it
is an
experimental fact,’ and adds that ‘both parties are perfectly right, provided
that their belief is not
too absolute.’
Mathematically, these assumptions are difficult to state explicitly, not just
because they
are mathematically esoteric, but also because there are various
ways in which the assumptions may be weakened (see
Cramér [1946]). For this
reason, mathematical statisticians almost always vaguely refer to the
assumptions as
‘certain regularity conditions.’ They would certainly not make
the brazen claim that these conditions
hold for all real scientific models, and we
follow their lead here. However, we do wish to say that the conditions
are not
unduly restrictive. There is no need to assume that the error distributions
associated with the
observational data are themselves approximately bell-shaped.
The standard coin tossing example illustrates the
point. The assumed ‘error’
distribution is the binomial distribution (the probability getting the high
value is
p, while the probability of the low value is (1−p)), yet the distribution for the
p-estimates
is asymptotically normal. The second point is that asymptotic
normality is not restricted to models that are
linear in their parameters. For
example, suppose that the product α β occurs in one of the equations of
the
model. If ˆ α and ˆβ are their maximum likelihood estimates and the values of α
and β are sufficiently close to these estimates, then we may write: α β = (αˆ +
Δα)( ˆβ+ Δβ) ≈ ˆ α ˆβ + ˆ α Δβ +
ˆβ Δα. Here, ˆ α and ˆβ are constants, and the
nonlinear product is now
linear in the new, transformed, parameters Δα and Δβ.
This approximation will be valid
because the region of non-negligible
likelihoods becomes more narrowly concentrated around the best estimates
as
the sample size increases. The same argument applies to other sufficiently
smooth nonlinear equations, such
as Y = sin(αX + β), and so on.
Perhaps the most restrictive assumption is that the sample size be
large. This
How to Tell when Simpler Theories will Provide More Accurate Predictions 31
does not mean merely
that the total data set is large, but that there is enough data
within the domain of each parameter. For example,
the approximate normality
of the model M1 and M2 in Section 3 requires that both of the data sets D1 and D2
are sufficiently large.
10 APPENDIX B: A PROOF OF A SPECIAL CASE OF AKAIKE’S
THEOREM
Suppose that we
are sampling from a target population of values of a random
variable X (e.g. the population of possible
measurements of the mass of an
object) with mean μ* (the true mass) and variance σ2 (the error of
measurement),
where the true probability distribution p for the values x of the random variable
X is normal,
or Gaussian. That is,
( ) ( )2
2 2
1 exp 1 *
2 2
p x x μ
πσ σ
= − −
.
Now consider a
hypothesis (‘curve’) that (falsely) asserts that the mean is μ. The
hypothesis in question asserts
that the probability distribution for measured
values of X is
( ) ( )2
2 2
1 exp 1
2 2
q x x
μ
πσ σ
= − −
.
Hypotheses like
q(x) form a family of hypotheses, each of which corresponds to
a particular value of the parameter μ. Thus, it
is notationally convenient to
denote the hypothesis itself by μ. (It will be clear from the context when μ
is the
parameter, the parameter value, or the hypothesis in the family corresponding to
a parameter value.)
The simplicity of a family of hypotheses (referred to by
statisticians as a model) is measured by the number of
adjustable parameters; in
this case there is only one (μ).
If we accept this family of hypotheses, the next
step is to find the best fitting
hypothesis, and this is the hypothesis that confers the highest probability
(density) on the data (i.e. has the maximum likelihood out of all the members of
the family). We denote the
maximum likelihood hypothesis (which is also
μ* μ x
p (x) q (x)
FIGURE 5
32 Malcolm Forster and Elliott Sober
the maximum log-likelihood
hypothesis) byμˆ . How willμˆ , obtained from past
data, fare in the prediction of new data drawn
from the same population? For
any particular datum x, we might measure the accuracy with which it is
predicted
by its goodness-of-fit; viz. the log-likelihood, log p(x). But we are
really interested in the ‘average datum’
drawn from the population, so we define
the predictive accuracy (A for ‘accuracy’) of an arbitrary
hypothesis μ to be:
A(μ) = df E*(log q(x)),
where q(x) is the probability distribution in the family
corresponding to the
parameter value μ, and E* is the expected value calculated with respect to the
true
hypothesis (μ*). That is,
A(μ) p(x)logq(x)dx
∞
−∞
= ∫ .
Note that A(μ) is the expected log-likelihood per datum for a data set of
arbitrary
size N. From the diagram, it is intuitively clear that a distribution q(x) with
central point μ
that is far from the true value μ* is not going to do so well in
predicting data randomly sampled from the true
population. By the same token,
p(x) is going to do the best job of fitting the data it generates. The
following
result gives this intuitive fact a quantitative representation:
A(μ) = A(μ*) − ½
(μ −μ*)2/σ 2. (1)
Proof: The log of
( )2
2
exp 1
2
x μ
σ
− −
is clearly equal to
− ½ (μ −μ*)2/σ 2.
But,
(x − μ)2 = (x − μ* − (μ
−μ*))2 = (x −μ*)2 - 2(x − μ)(μ −μ*) + (μ −μ*)2.
When we take
expectations and simplify the result follows. This completes the
proof.
Since (1) holds for any hypothesis in
the family, it surely holds for the
hypothesis that best fits the past data. Thus,
( ) ( ) ( )1 2 2
2 Aμˆ=Aμ*−
μ−μ* σ .
While interesting, this result is still epistemologically unhelpful because we
don’t
know A(μ*) and we don’t know the value of μ*. The second problem is
surmounted in the following way. We
may estimate A(μˆ) by the expected value
of the right hand side, where the expected value is taken over
the maximum
likelihood estimateμˆ . That is,
Estimate of ( ) ( ) ( )1 2 2
2 Aμˆ=E*Aμ−
μˆ−μ* σ.
How to Tell when Simpler Theories will Provide More Accurate
Predictions 33
But the central limit theorem tells us that the expected sum of squared deviations
of an
estimate of μ from its true value is just σ2/N, where N is the number of data
in the sample from which
the estimate is taken (the number of ‘past data’).
Thus, we have
Estimate of A(μˆ ) = A(μ*)
− ½ /N. (2)
The only remaining problem is to estimate A(μ*). Again the qualitative facts are
clear. Ifμˆ is the best fitting hypothesis relative to past data, then it fits the past
data better than
any other hypothesis (by definition), and therefore it fits better
than μ*. Thus, if l(μˆ)is the
log-likelihood of the best fitting hypothesis, then
l(μˆ)> l(μ*) andE*(l(μˆ ) N) >E*(l(μ*)
N) = df A(μ*). The question as to how
much greater is answered by the following result (without proof):
A(μ*)
= 1
2 E*(l(μˆ ) N) − N. (3)
If we now combine (2) and (3) we get:
Estimate of A(μˆ)=E*(l(μˆ)−1)N.
Since l(μˆ)−1 is clearly an unbiased estimate of E*( l(μˆ)−1), we finally arrive at
the main result, as it applies to this example:
Akaike [1973]: Estimate of A(μˆ ) = (1/N )[ l(μˆ)
−1].
That is, if we are interested in the predictive accuracy of the best fitting
hypothesis from the
family, we should not judge its accuracy by its
goodness-of-fit, for that estimate is usually biased towards being
too high. An
unbiased estimate is obtained by using a corrected measure of goodness-of-fit.
The important fact
is that this result generalizes (surprisingly well) to a variety
of conditions, and to examples of models with
many adjustable parameters. If k
is the number of adjustable parameters in a model, then we may state Akaike’s
theorem in its general form:
Akaike [1973]: Estimate of A(μˆ ) = (1/N )[ l(μˆ) −k].
This
is the formula that quantifies the trade-off between simplicity (the number
of adjustable parameters) and
goodness-of-fit (the maximum log-likelihood).
Department of Philosophy
University of Wisconsin, Madison
53706
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