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Does the concept of a Perfect Bayesian Equilibrium apply only to incomplete games with a common prior / consistent belief?

In both Bonanno's "Game Theory" and Osborne's "A Course in Game Theory", the definition of a Perfect Bayesian Equilibrium relies on the game being in extensive form. My understanding is that this can only be done for games with a common prior / consistent belief.

However, I feel like it should be possible to define a "similar" equilibrium concept to the Perfect Bayesian Equilibrium / Weak Sequential Equilibrium in the case of inconsistent beliefs, where sequential rationality is combined with Bayesian updating (at least on the equilibrium path).

Does such an equilibrium concept occur? If not, are there any refinements for the Bayesian Nash Equilibrium in the context of incomplete games with an inconsistent belief that are similar in flavor?

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    $\begingroup$ Perhaps look at the robustness literature? They tend to look at models where there are small differences in beliefs and see if results differ massively from standard results. $\endgroup$
    – Walrasian Auctioneer
    Jan 24 at 15:18
  • $\begingroup$ @WalrasianAuctioneer Do you have any references for this? $\endgroup$
    – oswinso
    Jan 24 at 20:24
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    $\begingroup$ Perhaps start at Kajii Morris 1997? $\endgroup$
    – Walrasian Auctioneer
    Jan 25 at 16:25

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