s^2 posterior term still seems not right.
I'm still not convinced that the posterior s^2 calculation in the model (and paper) are right. As shown in the paper, it works out to the SSR of the entire data (both current and past) divided by the size of the entire data.
That's fine, when there is no weakening.
Under weakening, the expression is amended to undo the cumulative weakening that got applied to V^{-1} (by multiplying by W^2): this seems wrong. We have the prior's estimate of s^2, based on \nu_prior observations, and now we have n new observations: up to the previous period, s^2 posterior was our best estimate of \sigma^2; now we're getting a new posterior beta based on that and new data: but the past residuals seem like they will (in general) be too large, since they receive lower weight in determining beta.
I'm not entirely sure what the right thing is, however: just dropping the third term doesn't seem right because of the argument in the pre-weakening description: it would give make the one-by-one s^2 results different from the all-at-once regression results.
I'm thinking, rather, that not changing the weakening result at all might be the best way to go: then the difference in betas, multiplying V^{-1}, captures the decreasing importance of past observations--but they still enter the equation fairly directly through the s^2_prior (which isn't weakened). So essentially it would be only the difference resulting from the changing beta that is deweighted, which actually seems appropriate: the s^2 calculation would become a weighted sum of new SSRs, old SSRs, and the change in old SSRs with the deweighting applied via V^{-1}.
I've changed it in the paper, and should do the same in the code.