= Solution
Apply the <Monte Carlo estimator> to the observable $h(s)=\exp(\sum_{i\in D}\sqrt{2+s_i})$:
$$
\boxed{\widehat\theta_T=\frac1T\sum_{t=1}^T\exp\!\left(\sum_{i\in D}\sqrt{2+S_i^{(t)}}\right).}
$$
The theoretical justification is the <ergodic theorem for a positive Harris recurrent Markov chain>, not an independent-sample law of large numbers. The <Markov chain> has finite state space, is irreducible, and has the strictly positive <posterior distribution> as invariant law, so it is positive recurrent and Harris recurrent with counting measure. The observable is bounded: $e^{K^2}\leq h(s)\leq e^{\sqrt3K^2}$. Thus the estimator converges almost surely to the posterior <expected value> from any initial configuration.
Each state also has positive self-transition probability, so the chain is an <aperiodic Markov chain>. Finiteness then gives geometric convergence and the relevant <central limit theorem>; serial dependence determines the <Markov chain Monte Carlo asymptotic variance> if error bars are required. A fixed finite burn-in can be discarded without changing consistency, but <independence> of the retained states should not be assumed.
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