In a previous post, I argued for restricting the Law of Likelihood[1] to mutually exclusive hypotheses and claimed that this restriction would not exclude cases of genuine scientific interest because scientists don’t test hypotheses against one that are not mutually exclusive.

Chandler (2013) argues for this restriction as well. He considers the objection that Forster and Sober (2004) claim that in the context of model selection, scientists sometimes say that data favor a given model over a logically weaker one. For instance, they might say that a set of observations of a variable Y taken for different X values that fall roughly along a straight line favor the model (LIN) according to which Y is a linear function of X plus a noise term over the model (QUAD) according to which Y is a quadratic function of X plus a noise term. But (LIN) is a special case of (QUAD) obtained by setting the coefficient of the X^{2} term in (QUAD) to zero. Thus, the scientists who speak in this way are speaking of an evidential favoring relation between compatible hypotheses. As a result, restricting the Law of Likelihood to apply only to mutually exclusive hypotheses excludes cases of genuine scientific interest.

Chandler’s response to this objection is that, “as Forster and Sober use the term, ‘E favors model M_{1} over model M_{2}’ is actually shorthand for ‘E favours the likeliest (in the technical sense) disjunct L(M_{1}) of model M_{1} over the likeliest distinct of model M_{2},’ with L(M_{1})∩L(M_{2})=∅.“ This reply may be true, but it does address the whole problem. Forster and Sober may use the phrase “E favors model M_{1} over model M_{2}” in this way, but scientists use it in other ways as well. For instance, they might use the phrase “E favors (LIN) over model (QUAD)” to mean their data favors the claim that the element of (QUAD) that is closest to the true model in some sense has a nonzero coefficient for the X^{2} term over its negation. If they are a bit more sophisticated then they will realize that it is implausible in typical applications that the coefficient of the X^{2} term is *exactly *zero. What they might mean instead is that their data favors over its negation the hypothesis that a nonzero coefficient for the X^{2} term is necessary for producing a statistically adequate curve, meaning roughly a curve the residuals of which look like white noise to a degree that is adequate for their aims.

On both of these interpretations, “E favors (LIN) over (QUAD)” actually means “E favors (LIN) over (QUAD)(LIN).” In general, when scientists talk about evidence favoring one model over another when those models are nested, they typically mean either that it favors the best element of the first over the second or that it favors the smaller model over the set-theoretic difference of the larger model minus the smaller model or vice versa. Unless there are cases in which scientists clearly mean what they say when they speak of evidence favoring one model over another when those models are nested, such examples do not indicate that there are cases of scientific interest that involve testing compatible hypotheses against one another.

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