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Posts Tagged ‘bayesian inference’

Strevens on Induction

In induction on 04/08/2011 at 4:38 pm

When two theories are empirically equivalent, their likelihoods relative to any given body of evidence are equal. Thus the difference in anyone’s subjective probabilities for the theories must be due entirely to the difference in the prior probabilities that were assigned to the theories before any evidence came in. Bayesian confirmation theory preserves a prior bias towards simplicity, but it implements no additional bias of its own. …

Bayesian confirmation theory does impose an objective constraint on inductive inference, in the form of the likelihood lover’s principle, but this is not sufficient to commit the Bayesian to assuming the uniformity of nature, or the superiority of “non-grueish” vocabulary or simple theories. The first of these failures, in particular, implies that [Bayesian confirmation theory] does not solve the problem of induction in its old-fashioned sense.

If the old-fashioned problem of induction cannot be solved, what can we nevertheless say about [Bayesian confirmation theory]’s contribution to the justification of induction? There are two kinds of comments that can be made. First, we can identify unconditional, though relatively weak, constraints that [Bayesian confirmation theory] puts on induction, most notably the likelihood lover’s principle. Second, we can identify conditional constraints on induction, that is, constraints that hold given other, reasonable, or at least psychologically compelling, assumptions. We can say, for example, that if we assign low priors to grueish hypotheses, [Bayesian confirmation theory] directs us to expect a future that resembles the past. This is, remember, considerably more than we had before we began. (Michael Strevens, Notes on Bayesian Confirmation Theory [.pdf], 66)

Strevens is admirable, for he is upfront about the inadequacies of Bayesianism. That said, Strevens at times overstates his case. For instance, tet me simplify the bolded passages: “The evidence for any particular theory is underdetermined … however, if we reject theories that are incompatible with our assumption that the future will resemble the past, we will expect a future that resembles the past.” At least two problems for Strevens:

(1) The future does not resemble the past in all domains: black swans, white ravens — in fact, all falsified scientific theories — should give us pause before assuming something that is demonstrably false when applied to all domains. Therefore, if the future resembles the past in only some domains, why we should assume that the future will resemble the past in any particular domain?

(2) Even if the future should resemble the past in this particular domain, it does not follow that any theory that assumes a future that resembles the past is true, for while ‘grueish hypotheses’ are ruled out, there may still be an alternative theory that follows from that same assumption that the future will resemble the past that is in fact true. After all, it’s happened in the past (Einstein replacing Newton).

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