## black box MCMC

“…back-box methods, despite using no information of the proposal distribution, can actually give better estimation accuracy than the typical importance sampling [methods]…”

Earlier this week I was pointed out to Liu & Lee’s black box importance sampling, published in AISTATS 2017. (which I did not attend). Already found in Briol et al. (2015) and Oates, Girolami, and Chopin (2017), the method starts from Charles Stein‘s “unbiased estimator of the loss” (that was a fundamental tool in my own PhD thesis!), a variation on integration by part:

$\mathbb E_p[\nabla\log p(X) f(X)+\nabla f(X)]=0$

for differentiable functions f and p cancelling at the boundaries. It also holds for the kernelised extension

$\mathbb E_p[k_p(X,x')]=0$

for all x’, where the integrand is a 1-d function of an arbitrary kernel k(x,x’) and of the score function ∇log p. This null expectation happens to be a minimum since

$\mathbb E_{X,X'\sim q}[k_p(X,X')]\ge 0$

and hence importance weights can be obtained by minimising

$\sum_{ij} w_i w_j k_p(x_i,x_j)$

in w (from the unit simplex), for a sample of iid realisations from a possibly unknown distribution with density q. Liu & Lee show that this approximation converges faster than the standard Monte Carlo speed √n, when using Hilbertian properties of the kernel through control variates. Actually, the same thing happens when using a (leave-one-out) non-parametric kernel estimate of q rather than q. At least in theory.

“…simulating n parallel MCMC chains for m steps, where the length m of the chains can be smaller than what is typically used in MCMC, because it just needs to be large enough to bring the distribution `roughly’ close to the target distribution”

A practical application of the concept is suggested in the above quote. As a corrected weight for interrupted MCMC. Or when using an unadjusted Langevin algorithm. Provided the minimisation of the objective quadratic form is fast enough, the method can thus be used as a benchmark for regular MCMC implementation.

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