Fix a strategy 1r for an extensive form game and beliefs µ that are consistent with 1r in the sense of sequential equilibrium.

(a) In any proper subgame of the game, we can speak of applying Bayes' rule within the subgame at information sets h in the subgame that are reached with positive probability, conditional on the subgame being reached Prove that µ is consistent with Bayes' rule in every proper subgame, in just this fashion.

(b) Define an almost-proper subgame to be an information set h and all the successor nodes to nodes in h, which will be denoted S(h), with the property that if x E S( h) and x' E h( x), then x' E S( h). For an almost-proper subgame whose "root" is h, given a specification of beliefs over nodes in h and a strategy profile 1r, one can use Bayes' rule to compute beliefs on information sets in S(h) that are reached if 1r is played, conditional on h being reached according to beliefs µ on h. Show that µ is consistent with Bayes' rule in every almost-proper subgame in this sense.