# Relation between Bayesian Nash equilibrium and Perfect Bayesian equilibrium

In Bayesian Nash equilibrium's (BNE) definition, players calculate expected payoffs accroding to the posterior $$\phi_i(\theta_{-i}|\theta_i)$$, namely, they don't update their beilief according to other players' strategies. This makes me hard to understand how perfect Bayesian equilibrium (PBE) is a refinement of BNE, as in PBE players' belief are consistent with strategies.

In particular, I don't understand the proof of this proposition in Steven Tadelis' textbook Game Theory: An Introduction (p.311):

This is my puzle. $$\sigma^*$$ is a BNE where the players' beliefs are the posterior $$\phi_i(\theta_{-i}|\theta_i)$$. How do we know player i's strategy $$\sigma_i^*$$ is a best response to the belief $$\mu^*$$ and others' strategies $$\sigma^*_{-i}$$, where $$\mu$$ is consistent with Baye's rule and others' strategies and hence different from $$\phi_i(\theta_{-i}|\theta_i)$$?