Papers › Gibbs Flow for Approximate Transport with Applications to Bayesian Computation
Gibbs Flow for Approximate Transport with Applications to Bayesian Computation
Jeremy Heng, Arnaud Doucet, Yvo Pokern
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Let π₀ and π₁ be two distributions on the Borel space (ℝᵈ,ℬ(ℝᵈ)). Any measurable function T:ℝᵈ→ℝᵈ such that Y=T(X)∼π₁ if X∼π₀ is called a transport map from π₀ to π₁. For any π₀ and π₁, if one could obtain an analytical expression for a transport map from π₀ to π₁, then this could be straightforwardly applied to sample from any distribution. One would map draws from an easy-to-sample distribution π₀ to the target distribution π₁ using this transport map. Although it is usually impossible to obtain an explicit transport map for complex target distributions, we show here how to build a tractable approximation of a novel transport map. This is achieved by moving samples from π₀ using an ordinary differential equation with a velocity field that depends on the full conditional distributions of the target. Even when this ordinary differential equation is time-discretized and the full conditional distributions are numerically approximated, the resulting distribution of mapped samples can be efficiently evaluated and used as a proposal within sequential Monte Carlo samplers. We demonstrate significant gains over state-of-the-art sequential Monte Carlo samplers at a fixed computational complexity on a variety of applications.
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