U-calibration asks one online probability forecaster to have low regret for every bounded proper loss, including losses unknown when the forecasts are made. For nontrivial $K\geq2$, the optimal worst-case rate is $\Theta( \sqrt{\min\{K,T\}T})$, while the smooth-loss class has minimax time rate $\Theta(\log T)$. A recent simultaneous algorithm attains both time rates but leaves a polynomial gap in $K$. We show that the classical Bayesian bootstrap closes this gap. At round $t$, independently assign exponential weights to the past outcomes and predict their normalized weighted histogram. Equivalently, if $c_{t-1}$ is the vector of past outcome counts, sample $P_t\sim\operatorname{Dirichlet}(c_{t-1})$ on the observed face of the simplex. For every outcome sequence with terminal counts $n_1,\ldots,n_K$, observed support size $m$, and every proper loss in $[-1,1]$, the expected regret is at most $6\sum_{i=1}^K\sqrt{n_i}\leq6\sqrt{mT}\leq6\sqrt{\min\{K,T\}T}.$ For every $\beta$-smooth loss, the same forecasts have regret at most the regret of empirical follow-the-leader plus $(\beta/2)\log T$, and therefore $O(\beta\log T)$ for bounded smooth proper losses. The proof uses three elementary facts: a one-count Dirichlet update has total variation $O(1/\sqrt{c_i})$, size-biasing an exponential weight adds an independent exponential weight, and weighted be-the-leader holds simultaneously for all proper losses. A class-aggregation reduction gives the matching dimension-capped lower bound. The algorithm is loss-agnostic, horizon-free, proper, and minimax-optimal for the worst bounded loss while retaining the minimax-optimal smooth-class time rate.
Keywords: U-calibration, online forecasting, proper losses, Bayesian bootstrap, Dirichlet distribution, minimax regret