# Bernardo (1979) paper, section 3.2

In section 3.2 of his paper, "Reference Posterior distribution for Bayesian Inference" (on 10 Dec, 2018) he writes

$$H(p^*(\theta/z))=-\int p^*(\theta/\hat{\theta})log( p^*(\theta/\hat{\theta}))d\theta,$$

$$=-log(p^*(\hat{\theta}|\hat{\theta}))+o(1),$$

$$=K(\hat{\theta})+o(1),$$ where $$H(.)$$ denotes the Shannon's entropy, $$p^*$$ denotes the asymptotic distribution, $$z$$ denotes the set of data. Here, he assumed that the asymptotic posterior distribution depends upon data only through maximum likelihhod estimate $$\hat{\theta}$$.

Further, he writes that since for large number of samples the likelihood $$p(z|\theta)$$ will also concentrate around its maximum $$\hat{\theta}$$, we have $$\int p(z|\theta)K(\hat{\theta})dz=K(\theta)+o(1).$$ This last equation is what I do not understand. Since $$K(\hat{\theta})$$ is just a constant in case, I think that the right side should be equal to $$K(\hat{\theta})$$ instead of $$K(\theta)$$.

Please tell me what is wrong with my thinking in this case?