Lipschitz continuity of probability kernels in the optimal transport framework

In Bayesian statistics, a continuity property of the posterior distribution with respect to the observable variable is crucial as it expresses well-posedness, i.e., stability with respect to errors in the measurement of data. Essentially, this requires analyzing the continuity of a probability kernel or, equivalently, of a conditional probability distribution with respect to the conditioning variable. Here, we tackle this problem from a theoretical point of view. Let (X,dX) be a metric space, and let B(Rd) denote the Borel σ-algebra on Rd. Let π(·|·) : B(Rd) × X → [0,1] be a dominated probability kernel, i.e. of the form π(dθ|x) = g(x,θ)π(dθ) for some suitable function g : X × Rd → [0, +∞). We provide general conditions ensuring the Lipschitz continuity of the mapping X ∋ x → π(·|x) ∈ P(Rd ) when the space of probability measures P(Rd ) on (Rd , B(Rd )) is endowed with a metric arising within the optimal transport framework, such as a Wasserstein metric. In particular, we prove explicit upper bounds for the Lipschitz constant in terms of Fisher-information functionals and weighted Poincaré constants, obtained by exploiting the dynamic formulation of the optimal transport. Finally, we give some illustrations on noteworthy classes of probability kernels, and we apply the main results to improve on some open questions in Bayesian statistics, dealing with the approximation of posterior distributions by mixtures and posterior consistency.