Three-Regime Bayesian Updating from Daily Returns

A portfolio manager models the market as being in one of three regimes, each equally likely a priori:

• Bull: the stock goes up on any given day with probability $\frac{4}{5}$.
• Neutral: the stock goes up with probability $\frac{1}{2}$.
• Bear: the stock goes up with probability $\frac{1}{5}$.

Days are conditionally independent given the regime. Over three days the stock goes up, up, then down.

(a) What is the posterior probability of each regime?

(b) What is the conditional probability the stock goes up on Day 4?