Multi-Analyst Signal Fusion and Sequential Updating

A trading desk receives directional predictions from three independent analysts. The market will either go "up" or "down", each with prior probability $\frac{1}{2}$. Each analyst independently (given the true state) predicts the correct direction with the following probabilities:

• Analyst 1: accuracy 0.8
• Analyst 2: accuracy 0.7
• Analyst 3: accuracy 0.9

(a) All three analysts predict "up". What is $P(\text{up} \mid \text{all three say up})$?

(b) Analysts 1 and 2 predict "up", but Analyst 3 predicts "down". What is $P(\text{up} \mid U_1, U_2, D_3)$?

(c) Show that the posterior in part (b) can be computed by sequential Bayesian updating — updating on one analyst's signal at a time — and that the final answer does not depend on the order of updates. Demonstrate this by computing the posterior via two different orderings.