= Solution
The probability of no recovery through month $M$ is the product of the conditional survival probabilities, so
$$
\boxed{\gamma=1-\prod_{t=1}^M(1-q_t)}.
$$
Under <Bayes estimator under squared error loss>[quadratic loss], the <Bayes estimator> is the posterior mean. Posterior independence and the mean of a <Beta distribution> give
$$
\mathbb E[1-q_t\mid X]
=\frac{1+r_t-d_t}{T+1+r_t}.
$$
Consequently
$$
\boxed{
\widehat\gamma_B
=1-\prod_{t=1}^M
\frac{1+r_t-d_t}{T+1+r_t}}.
$$
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