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)