I have described this basic figure in a previous post. New to the figure are B(T), B(R|T), B(R), and B(T|R). These represent Bayesian beliefs. I know that it is more common to use the term Bayesian probabilities and the symbol P instead of B, but I want to avoid any possible confusion with frequency probabilities.

Prior to performing the next experiment, B(T) is my belief in theory/model 'T'. Like all Bayesian beliefs, it is a number between 0 and 1. Notice that this is my own subjective belief. But you will see that by using a Bayesian approach, my (changing) degree of belief will remain consistent with experiment over time.

B(R|T) is my belief that the experiment will yield observations/results 'R' assuming 'T' is true. This value will be deductively derivable from theory 'T'.

B(R) = B(R|T)B(T) + B(R|~T)B(~T). Where: B(T) + B(~T) = 1. Use this formula to calculate the degree that

*some*theory (T or ~T) could believably explain results 'R'.

Notice that theories can still overlap in predicting results and that a zero value for B(R) is possible if no prior theory could explain the results.

B(T|R) = B(R|T)B(T)/B(R). This formula conditionalizes B(T) and calculates my posterior degree of belief in theory 'T'.

Notice that if B(R) is zero then the formula is of the form 0/0 and a new theory must be developed that explains the experiment before the iterative process that is the scientific method can continue.

Notice also that if the new experiment is not independent of previous experiments, then B(R|T) = 1. (Prior theory was previously conditionalized on previous experiments.) This gives a formula of the form 1/1 and my belief will not be altered. So such experiment is useless.

Some numerical examples should make the above clear. (Note: your definition of

*likely*or

*unlikely*may vary from mine.)

**A Likely Theory Becomes Very Likely**

Prior: B(T) = .95 (likely theory) B(~T) = 1 - .95 = .05 (all other competing theory unlikely) B(R|T) = .99 (results very likely) B(R|~T) = .16 (results rather unlikely according to competing theory) Posterior: B(T|R) = .99 * .95 / [ .99 * .95 + .16 * .05] = .99 (very likely)

**A Likely Theory Becomes Neutral**

Prior: B(T) = .95 (likely) B(~T) = 1 - .95 = .05 (competing theory unlikely) B(R|T) = .05 (unlikely results) B(R|~T) = .99 (but strongly predicted by competing theory) Conditioned: B(T|R) = .05 * .95 / [ .05 * .95 + .99 * .05] = .49 (neutral)

**An Unlikely Theory Becomes Neutral**

Prior: B(T) = .05 (unlikely) B(~T) = 1 - .95 = .95 (competing theory likely) B(R|T) = .99 (but unlikely theory highly confirmed) B(R|~T) = .05 (and competing theory did not predict result) Posterior: B(T|R) = .05 * .99 / [ .05 * .99 + .05 * .95] = .51 (neutral)

Thought you'd appreciate this paper: Towards a Bayesian Account of Explanatory Power

ReplyDeleteAbstract:This paper defends an explication of explanatory power in Bayesian terms. It is argued that explanation and confirmation are intimately related, both on the qualitative and the quantitative level. Thus, the success of Bayesian confirmation theory can be used to argue for the transfer of Bayesian methods to an analysis of the quantitative, epistemic dimension of scientific explanations. The starting point is Hempel’s (1965) observation that good explanations rationalize their explanandum, making the concept suitable for a Bayesian analysis. Thereby, the paper does not only reveal the deep structural similarities between confirmation and explanation, but also makes a contribution to better understanding explanatory reasoning in science, particularly statistics. We obtain a subjectivist theory of explanatory power which is better suited to describe statistical practice than several objectivist proposals that flourished in the past.From the intro:

...lots of sophisticated proposals have been made, but none of them succeeded at rebutting the flood of objections and counterexamples. There is no consensus about whether and how a scientific explanation can be characterized, using a set of precise and unambiguous conditions.