Bayes' Theorem and Theory Appraisal

1. Bayes' Conditional Probabilities

Is there a way of salvaging Bayes' Theorem in order to make it useful for philosophers of science as a way of quantifying the degree of support that a piece of observational evidence lends a scientific theory? As an advocate of Imre Lakatos' Methodology of Scientific Research Programs (see Lakatos [1978]), I would like to think so. However, I'm not supremely confident this can be done, despite the best efforts of Bayesian theorists.

Bayes' Theorem, as proved by Thomas Bayes in 1763, is stated as:

where	'P(A/B)' = 'probability of A conditional on B'
	'P(A)'    = 'probability of A independent of B'
	'P(B/A)' = 'probability of B conditional on A'
	'P(B)'    = 'probability of B independent of A'

A simple example illustrating conditional probability is the following:

An urn contains three balls; two black balls and one red ball. What is the probability of drawing a red ball following the drawing of a black ball where the black ball is not put back in the urn?

let	'a' = 'red ball drawn'
	'b' = 'black ball drawn'

The number of possible outcomes for the first drawing ('sample space') is 3 (red ball, black ball, black ball). The probability of getting a red ball ('event') on the first draw is:

P(a) = 1/3

The probability of getting one of the two black balls on the first draw is:

P(b) = 2/3

After drawing a black ball first, there remain one red ball and one black ball in the urn (sample space = 2). The probability of drawing a red ball next is:

P(a/b) = 1/2

What is the probability of drawing a black ball on the second draw when a red ball is drawn first? Following the first draw, there remain two black balls in the urn (sample space = 1). So, the probability of getting a black ball on the second draw is:

P(b/a) = 1/1 = 1

These answers accord with that given by the application of Bayes' Theorem. Inserting our calculated values into the equation:

2. Bayes' Theorem in the Philosophy of Science

Bayes' Theorem, as applied to the evaluation of the probability of scientific theories based on a piece of evidence, is the following:

where	'P(h/e)' = 'probability conferred on hypothesis h by evidence e'
	'P(h)'    = 'prior probability of h independent of evidence e'
	'P(e/h)' = 'probability conferred on e on the assumption of h'
	'P(e)'    = 'probability ascribed to e independent of h'

A subjective interpretation of the prior probability of h based on the personal degrees of beliefs held by actual scientists is wholly inadequate for a claimed objective account of theory appraisal. I'm interested in seeing how Bayes' Theorem can be married to an objectivist account such as Lakatos' Methodology of Scientific Research Programmes (see my Allan [2016]).

Using the example of balls drawn from an urn again, imagine an urn containing a fixed number of red and black balls, with r red balls and b black balls. (See Siegmund [2023] for just such an example.) Picture scientists disagreeing on the composition of the urn, believing different hypotheses about the composition (h1, h2, h3). The scientists perform an experiment (e) drawing a number of balls from the urn. Applying Bayes' Theorem, they calculate the a posteriori probability that the composition of the urn matches their hypothesis (h). They each calculate P(h/e) for their favoured hypothesis. The first problem is that the probabilities they calculate for the competing hypotheses h1, h2, h3 differ as they differ in the a priori probability they assign to each hypothesis. As Siegmund [2023] concludes:

The weakness, as indicated above, is that different people may choose different subjective probabilities for the composition of the urn a priori and hence reach different conclusions about its composition a posteriori.

3. An Objective Interpretation of Bayes' Theorem

Perhaps the solution here is to apply the well-known 'principle of indifference' in predicating prior probabilities. So, if there are two competing hypotheses, assign P(h) = 0.5 to each hypothesis. If there are four competing hypotheses, assign the value of 0.25 to each.

Applying this principle of charity even to a seeming highly implausible theory, such as creationism or flat earth theory, will still yield an informative result if evidence e is deducible from hypothesis h (plus background assumptions plus initial conditions) (that is, P(e/h) is 1 or close to 1) and e is truly novel, being unexpected in the absence of assuming h true (that is, P(e) is close to P(h)). Where there is a small and finite number of competing hypotheses being considered, the denominator of Bayes' Theorem (P(e)) becomes (in the case of only two competing hypotheses):

P(e/h1).P(h1) + P(e/h2).P(h2)

Where Darwin's evolutionary theory (P(h1)) and the special creation theory (P(h2)) are both afforded a prior probability of 0.5, the finding of transitional fossils in sedimentary layers boosts the a posteriori probability of Darwin's theory significantly while at the same time starkly reducing the a posteriori probability of creation theory.

In this way, perhaps this move is successful in applying Bayes' Theorem to theory appraisal that is both faithful to the actual history of science and preserves an objective interpretation of prior probability P(h).

Perhaps Popper's objection that all theories have a prior probability of 0 does not apply to theory appraisal as theories being evaluated by scientists are explanatory of a class of phenomena in terms of forces and micro- or macro-structures. Scientific theories are not simply a set of premises from which an observation statement is deduced. Yes, there are infinitely many sets of premises from which can be deduced grue-like observation statements. But Goodman's grue predicate is not embedded within an explanatory theory that postulates just how a grue-like emerald changes its colour at some arbitrarily selected time. (For a discussion of Goodman's grue paradox, see Losee [2001: 184–6].)

4. Outstanding Problems

There are other well-known problems with applying Bayes' Theorem to theory appraisal. I'm not sure these have a satisfactory solution.

Firstly, I'm not sure probability rules applying to games of chance, such as flipping a coin, throwing a dice, betting on a horse race, and so on, can be applied to the seemingly quite different realm of evaluating explanatory hypotheses.

Secondly, Bayes' Theorem may be able to quantify the strength of evidence given by a confirmed novel prediction. However, there is no natural way for it to give an account of the strength given by a postdiction; that is, that given by a fact that was known at the time of theory construction, but not used in the construction of the theory (e.g., how the known advance of the perihelion of the planet Mercury lent weight to Einstein's General Theory of Relativity). (See, for example, Losee [2001: 224].)

Finally, with the solution I propose above, how much weight ought to be given to previous successes of a theory? Ought a battle-hardened theory keep its a posteriori probability when confronted with a new challenger? Or ought all competing theories be reset to a level playing field with an assigned prior probability P(h) where P(h) = 1/(number of rival theories)?

Copyright © 2023 Leslie Allan