Why confirmation ≠ explanation

Well, more precisely, this is about why confirmation and explanatory power cannot be explicated by the same probabilistic measure within a Bayesian framework. The argument below occurred to me while working on a paper, "A second look at the logic of explanatory power" (with Katya Tentori, now forthcoming in Philosophy of Science), wherein a debate is carried out involving Jonah Schupbach and Jan Sprenger's recent work on "The logic of explanatory power" (Philosophy of science, 78, 2011: 105-127). (By the way, I take the following note to provide some motivation for both Jonah's and Jan's project and the alternative line that Katya and I pursue.)

Why it makes sense to ask

There exists a rather effective rule of thumb to generate a decent measure E(e,h) of the explanatory power of candidate explanans h with regards to explanandum e; that is, take a plausible probabilistic measure of (incremental) confirmation C(h,e), and invert the positions of e and h. (To illustrate, Schupbach and Sprenger, 2011, readily notice that their favorite measure of explanatory power entertains just this relationship with a very interesting measure of confirmation originally defined by Kemeny and Oppenheim, 1952, and then revived by Fitelson, 2005.)
In view of these structural analogies, investigating the connection between a probabilistic measure of explanatory power E(e,h) and of confirmation C(h,e) appears appropriate, if not pressing, as a source of theoretical clarification. An instructive possibility to explore is the statement of outright identity between the two notions, of course with the caveat that the hypothesis h at issue be in some explanatory relation (to be separately defined) with evidence e. Always keeping this proviso in mind, the "reductionist" claim to identity would then be as follows:

Reduction (R).
For any e,h, C(h,e) = E(e,h).

(I'll be assuming throughout that statements are contingent and probabilities are regular.) Notice that, while admittedly daring, R seems to closely fit so-called inference to the best explanation (IBE). After all, for advocates of the IBE view, “observations support the hypothesis precisely because it would explain them” (Lipton, 2000, p. 185, emphasis added). But R is of concern even beyond that, if only because it would arguably trivialize the division of labor between two branches of formal epistemology and philosophy of science that are usually seen as distinct. For short, R is not to be dismissed too quickly, i.e., unless a relevant argument is provided to undermine it. Such an argument is sketched in what follows.

Against the reductionist claim

A compelling principle of a model of explanatory power seems to be that the better hypothesis h would succeed in explaining the occurrence of a state of affairs e the worse it would fail in explaining the occurrence of its complementary ¬e. Formally, such an inverse ordinal correlation between explanatory success and explanatory failure with regards to a pair of complementary statements e and ¬e is spelled out as follows:

Symmetry (S).
For any e,h,h*, E(e,h) >/=/< E(e,h*) iff E(¬e,h*) >/=/< E(¬e,h).

On the other hand, consider the following condition concerning confirmation:

Final probability incrementality (F).
For any h,e,e*, C(h,e) >/=/< C(h,e*) iff P(h|e) >/=/< P(h|e*).

Condition F states that, for any given hypothesis h, confirmation is an increasing function of the posterior probability conditional on the evidence at issue – a virtually unchallenged assumption in contemporary probabilistic analyses of confirmation.
Notably, the following can be proven:

Theorem. {S,F} is consistent, but {R,S,F} is not.

Relying on both S and F as sound, the theorem above discredits the reductionist claim to identity R. Apparently, probabilistic confirmation and explanatory power cannot be identified, for the two notions are constrained by genuinely distinct principles on a quite basic level. For all its tempting simplicity, thus, the reductionist thesis R turns out to be a naïve view of the connection between confirmation and explanatory power. This is not to say, of course, that there cannot be other meaningfull and systematic relationships. This does mean, however, that one natural candidate formal rendition of IBE is flawed.

References
Fitelson, B. (2005), "Inductive logic", in S. Sarkar and J. Pfeifer (eds.), Philosophy of Science. An Encyclopedia, Routledge, New York, 2005, 384-393.
Kemeny, J. and Oppenheim, P. (1952), "Degrees of factual support", Philosophy of Science, 19, 307-324.
Lipton, P. (2000), "Inference to the best explanation", in W.H. Newton-Smith (ed.), A Companion to the Philosophy of Science, Blackwell, Malden (MA), 184-193.