(Originally written November 14, 2006 in Book 8)
Chapter 12: The Bayesian Approach
[Introduction]
The reliability and practical use of science's predictions in recent history may highlight that philosophers of science have exaggerated the fallibility of theories.
The Bayesians have gravitated to this view and stated that something has gone radically wrong in philosophy of science.
Bayesian are followers of Thomas Bayes, an 18th century mathematician who proved a probability theory.
[Bayes' Theorem]
The theorem is about conditional probabilities. Conditional probabilities are those propositions who depend on the evidence bearing on those propositions.
Bayes' theorem is a method of prescribing how probabilities are to be changed in the light of new evidence.
Bayes' Theorem:
P(h/2) = P(h) P(e/h)/P(e)
P(h) is prior probability of hypothesis 'h'
P(h/e) is posterior probability: the probability of 'h' in light of 'e'
Bayes' theorem indicates that prior probability is to be changed by a scaling factor in light of evidence 'e'.
The weight of evidence 'e' will strengthen or weaken the original probability. Thus P(h) will become P(h/e) and P(h/e) will become more or less probable than P(h) based on the evidence 'e'.
'e' will be 1 if it proves 'h'.
'e will be 0 if it disproves 'h'.
If it neither proves more disproves 'h' its value will be greater than 0 and less than 1
"The extent to which some evidence supports a hypothesis is proportional to the degree to which the hypothesis predicts the evidence" (Chalmers, 176).
If the evidence is confirmed the hypothesis is confirmed and the opposite (and the in between).
The Bayesians allow and need auxiliary assumptions in their theory.
Bayes' theorem is a theorem. It takes for granted some minimal assumptions about probability.
The minimum assumptions are probability calculus (which is widely accepted).
[Subjective Bayesianism]
The Bayesians disagree on a fundamental question concerning the nature of the probabilities involved.
According to objective Bayesians the probabilities represent probabilities that rational agents ought to subscribe to in the light of the objective situation.
Objective Bayesians hold that probabilities are distributed equally and then the Bayes' theorem is used to modify the probabilities in the light of the evidence.
A major problem with this approach is how to ascribe objective prior probability to hypotheses.
Where is a list of hypotheses to be found in any given field? A possible list could be infinite. In such a case all probabilities will be 0 and Popperian falsficationism is proved right.
Subjective Bayesian is different to the objective versions.
Scientists take certain things for granted and thus probability is not initially evenly distributed. Probabilities are assigned based on subjective beliefs.
The subjective Bayesians take the degrees of belief in hypotheses that scientists take as a matter of fact as the basis for the prior probabilities in their Bayesian calculations.
"Bayesianism makes a great deal of sense in the context of gambling" (Chalmers, 179).
Linehan - I just thought it was funny.
The degree of belief held by a scientist is analogous to the odds on a horse in a fair race.
Not all Bayesians will make the same choice between alternatives when applying the Bayesian calculus to science.
Any attempt to understand science and scientific reasoning in terms of subjective beliefs of scientists would seem to be disappointing for those who seek an objective account of science.
Bayesians insist that the Bayesian theory constitutes an objective theory of scientific inference.
The Bayesians see their approach as similar to logic. Logic doesn't care where the premises come from, only if the conclusion flows from the premises.
Bayesians can take the argument further and state that while scientists can be subjective in assigning prior probability, the Bayes' theorem, if applied correctly on the evidence will bring scientist's to the same conclusion regardless of their starting points.
Applications of the Bayesian Formula
There is a low of diminishing return in science that states once a theory has been confirmed by an experiment once, repeating the same experiment under the same circumstances will not be taken as confirming a theory to as high a degree as the first experiment did.
The Bayesian formula captures the essence of the law of diminishing return.
IF the theory "T" predicts the experimental result "E" then the probability of P(E/T) is 1.
The probability of T is to be increased in the light of a positive result E each time the experiment is performed, consequentially the probability of a theory being correct will increase by a smaller amount each time it is performed.
The Bayesians claim to be able to capture the rationale of Lakatos' ideal that confirmations, not falsifications are the key to scientific development.
Lakatos' 'methodological decisions' seemed plausible in his account, but he gave no proof for it. The Bayesians provided a basis for it.
Bayesianism also helps to eliminate ad hoc modifications.
[Critique of subjective Bayesianism]
One criticism is that by embracing subjective probabilities is too much of a concession to be able to attribute probabilities to theories.
There are two major problems with Bayesianism's approach. A Bayesian must know what any scientist felt about a degree of belief.
1) Gaining access to a scientist's degrees of belief is problematic
2) The implausibility of private beliefs having anything to do with a superiority of a theory or another one is also problematic
These problems are intensified when a collaborative work produces results. Whose beliefs are to be the factor in this account?
The extent to which degrees of belief are dependent upon prior probabilities is also another problem.
The subjectiveness of the starting point of the Bayesians makes it impossible for it to be an objective scientific method. If the starting point is gone all that is left is the Bayes' theorem which without science proves nothing.
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