Bayesian Approach to the Philosophy of Science

Bayesian Approach to the Philosophy of Science

Bayes’s Theorem: P(h/e) = P(h) · P(e/h)

P(e)

·  P(h/e): the (posterior) probability of hypothesis h in light of the evidence e

·  P(h): the (prior) probability of h in the absence of the evidence e

·  P(e/h): the probability of the evidence e assuming that h is correct

·  P(e): the probability of e in the absence of any assumption regarding the truth of h

Features of Bayes’s Theorem

1.  is a mathematical theorem provable from the basic laws of probability

2.  enables us to assign non-zero probabilities to well-confirmed theories

3.  enables us to update probabilities of hypotheses in light of new evidence

Objective vs. Subjective Bayesianism

·  Objective Bayesianism: The probabilities are the probabilities that rational individuals ought to assign in light of the objective facts.

o  problem—Where does the initial prior probability P(h) come from?

·  Subjective Bayesianism: The probabilities represent subjective degrees of belief.

o  Even if the initial prior probabilities P(h) are different for different individuals, the posterior probabilities P(h/e) will tend to converge. (pp. 180-181)

Implications of Subjective Bayesianism:

1.  Repeating an experiment under the same circumstances does not confirm a theory to as high a degree as the original experiment did.

2.  Experimental confirmations of a research program (Lakatos) are more important than apparent falsifications of the program.

3.  The Duhem-Quine problem is solved by feeding in the appropriate prior probabilities, calculating the posterior probabilities, and dropping assumptions when their probabilities become too low.

4.  A theory is better confirmed by various kinds of evidence than by evidence of a particular kind.

5.  Evidence that is used to construct a theory can sometimes legitimately be used as evidence supporting that theory.

6.  The main reason that ad hoc hypotheses should be rejected is that Bayesian posterior probability calculations show that they are implausible—even when they are independently testable.

Objections to Subjective Bayesianism:

1.  Since there is no appeal beyond one’s initial (subjective) prior probabilities, some scientific disputes may not be rationally resolvable. (p. 188)

2.  Since there is no appeal beyond one’s initial (subjective) prior probabilities, it is possible for a scientist to believe a theory so strongly that no evidence could ever force him/her to abandon the theory. (pp. 189-190)

3.  The idea that whether a particular scientific theory represents “progress” in science depends on our access to private degrees of belief is “implausible.” (p. 189)

4.  Since much scientific research is collaborative, it is unclear whose private degrees of belief are relevant in determining how “progressive” the research is. (p. 189)

5.  When the evidence predates the theory, it may be necessary to assign counterfactual degrees of belief in order to measure the prior probabilities and calculate the posterior probabilities. (pp. 190-191)

6.  Subjective Bayesianism makes no provision for evaluating evidence, experimental technique, and other nuts-and-bolts considerations.