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Introduction

There is a log term in the \(\rho_g(\theta)\):

\[\rho_g(\theta) = -l_{NM}(T(\theta,g);g,s^2(\theta))-\frac{(\theta-T(\theta;g))^2}{2s^2(\theta)}-\frac{1}{2}\log 2\pi s^2(\theta).\]

We have shown that if we include \(-\frac{1}{2}\log 2\pi s^2(\theta)\) in the \(\rho_g(\theta)\), then \(\rho_g(\theta)\) is not really a penalty term, or at least not one that penalize the posterior mean towards prior mean. See with and without the log term in the Poisson case for comparisons.

Why this is the case?

Because in our formulation, we took a second order Taylor series expansion of the log-likelihood around posterior mean. Then we put the quadratic term into the \(\rho_g(\theta)\). So at least there’s a part of \(\rho_g(\theta)\) that belongs to the log-likelihood. The log term is from evaluating \(KL(q||g)\) where \(q(b) = g*N(z;b,s^2)/c(z)\).

In normal case the log term in likelihood and the one in “penalty” cancels out so there’s no such log term in the objective function.

If we remove the log term from \(\rho_g(\theta)\) and call the new penalty as \(\tilde\rho_g(\theta)\) then the objective is

\[-l(\theta) - \frac{1}{2}\log s^2(\theta) + \tilde\rho_g(\theta).\]

We note that \(s^2(\theta)\) is the observed fisher information at \(\theta\) so the new objective can be regarded as penalized (log-likelihood and log-information)?