Abstract
Consider a reference Markov process with initial distribution $\pi_0$ and
transition kernels $\M_t\_tın1:T$, for some $TınN$. Assume
that you are given distribution $\pi_T$, which is not equal to the marginal
distribution of the reference process at time $T$. In this scenario,
Schrödinger addressed the problem of identifying the Markov process with
initial distribution $\pi_0$ and terminal distribution equal to $\pi_T$
which is the closest to the reference process in terms of Kullback--Leibler
divergence. This special case of the so-called Schrödinger bridge problem can
be solved using iterative proportional fitting, also known as the Sinkhorn
algorithm. We leverage these ideas to develop novel Monte Carlo schemes, termed
Schrödinger bridge samplers, to approximate a target distribution $\pi$ on
$R^d$ and to estimate its normalizing constant. This is achieved by
iteratively modifying the transition kernels of the reference Markov chain to
obtain a process whose marginal distribution at time $T$ becomes closer to
$\pi_T = \pi$, via regression-based approximations of the corresponding
iterative proportional fitting recursion. We report preliminary experiments and
make connections with other problems arising in the optimal transport, optimal
control and physics literatures.
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