Relation between Bayesian Nash equilibrium and Perfect Bayesian equilibrium

Question

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 $\phi_i(\theta_{-i}|\theta_i)$?

Answer 1

The difference between BNE and PBE is that BNE assumes best response to any reachable posterior beliefs, whilst PBE assumes best response to all possible posterior beliefs.

In the definition of BNE the reachable posterior belief depends on the strategy profile of other players. Since in PBE any posterior belief is considered, there is no need to consider the strategies of other players when considering the posterior beliefs.