Information-Tempered Priors for Gamma Thompson Sampling

Nearly equal pessimistic observations can make the optimal-arm posterior falsely precise in Gamma Thompson sampling (TS) with unknown means and shapes. Skipping the arm then freezes its posterior. Information-tempered matching (ITM) priors with fixed $\eta >0$ retain full support and first-order probability matching with tail $k^{-1-\eta }$. Global posterior-integral comparisons, high-shape saddlepoint control, and a small-shape carrier bound establish summable pessimism odds for these priors. For every fixed finite-arm Gamma instance with a unique optimal mean, uncapped ITM-TS is uniformly efficient and attains the Burnetas–Katehakis regret constant. Regular priors globally comparable to ITM share this guarantee. For weights $w(k)\sim C{k}^{-\tau }$, $C>0$, as $k\to \infty$, reciprocal optimism near these histories is locally integrable exactly when $\tau >1$ for every fixed sample size $n\ge 2$. With two initial pulls per arm, ordinary TS incurs $\Omega (\sqrt{T})$ lockout regret under Jeffreys' prior and a distinct $\Omega (\log T)$ coefficient constraint under the matching prior. Full-bandit experiments with 1,024 trials per instance support lower mean regret and CVaR for ITM ($\eta =.5$) than matching TS on the five-arm instance; both two-arm instances have paired intervals containing zero.