  We study the Bayesian regret of the renowned Thompson Sampling algorithm in
contextual bandits with binary losses and adversarially-selected contexts. We
adapt the information-theoretic perspective of \cite{RvR16} to the contextual
setting by considering a lifted version of the information ratio defined in
terms of the unknown model parameter instead of the optimal action or optimal
policy as done in previous works on the same setting. This allows us to bound
the regret in terms of the entropy of the prior distribution through a
remarkably simple proof, and with no structural assumptions on the likelihood
or the prior. The extension to priors with infinite entropy only requires a
Lipschitz assumption on the log-likelihood. An interesting special case is that
of logistic bandits with $d$-dimensional parameters, $K$ actions, and Lipschitz
logits, for which we provide a $\widetilde{O}(\sqrt{dKT})$ regret upper-bound
that does not depend on the smallest slope of the sigmoid link function.
