Find a Perfect Bayesian Equilibrium with a game that includes Nature I want to determine all pure-strategy Perfect Bayesian Equilibria for this task, but I cannot get very far. Side note: First number is payoff for A, second number payoff for player B.
For the last stage, I know that upper A will always choose to exit, and the bottom A will always choose to pass.
But I don't know how to proceed from there? I have ascribed beliefs for player B being in the upper and bottom node (beta and 1-beta, respectively), but am not sure what to do with it.
I tried assuming that Nature goes 'up' and in that case beta=1. Then B would choose pass because it has a greater payoff than exiting. That's where I get confused.
You've rightly identified A's optimal actions at his last decision nodes (shown as red arrows in the figure below). Next, you need to

conjecture a pair of actions at A's initial nodes,
use those actions to determine B's belief at her information set,
find B's best response(s) to that belief, and
check whether A's initial actions are optimal with respect to B's best response(s).

If the answer at step 4 is &quot;yes&quot;, then you have a PBE; if not, repeat the steps with a different conjecture.
In your particular example, there is no pure strategy PBE. I will nevertheless go through an example just to demonstrate the above approach:

Suppose A chooses &quot;pass&quot; at both initial information sets (blue arrows).
Using Bayes' rule, B's belief must be $0.9$ at the top node and $0.1$ at the bottom node.
Based on this belief, B's expected payoff is $1$ from choosing &quot;exit&quot;, and $0.9(3)+0.1(-5)=2.2$ from choosing &quot;pass&quot;. So her best response is to choose &quot;pass&quot; (green arrows).
Given that B would choose &quot;pass&quot;, A's best response after Nature chooses the top path would be &quot;exit&quot; (purple arrow), since $4&gt;3$.

Thus we do not have a PBE involving A choosing &quot;pass&quot; at both initial nodes. You can go through the rest of A's possible action combinations at his initial nodes to verify that there is no PBE with pure strategies.