Unbiased estimation of the gradient of the log-likelihood in inverse problems

Abstract: We consider the problem of estimating a parameter θ∈Θ⊆Rdθ associated with a Bayesian inverse problem. Typically one must resort to a numerical approximation of gradient of the log-likelihood and also adopt a discretization of the problem in space and/or time. We develop a new methodology to unbiasedly estimate the gradient of the log-likelihood with respect to the unknown parameter, i.e. the expectation of the estimate has no discretization bias. Such a property is not only useful for estimation in terms of the original stochastic model of interest, but can be used in stochastic gradient algorithms which benefit from unbiased estimates. Under appropriate assumptions, we prove that our estimator is not only unbiased but of finite variance. In addition, when implemented on a single processor, we show that the cost to achieve a given level of error is comparable to multilevel Monte Carlo methods, both practically and theoretically. However, the new algorithm is highly amenable to parallel computation.

Citation

Statistics and Computing, volume 31, issue 3, page 21

Publisher

Springer US

URI

http://hdl.handle.net/10034/624546

Type

article

Description

From Springer Nature via Jisc Publications Router
History: received 2020-03-10, registration 2021-01-07, accepted 2021-01-07, pub-electronic 2021-03-03, online 2021-03-03, pub-print 2021-05
Publication status: Published
Funder: King Abdullah University of Science and Technology; doi: http://dx.doi.org/10.13039/501100004052; Grant(s): Baseline
Funder: Alan Turing Institute; doi: http://dx.doi.org/10.13039/100012338