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We consider the problem of estimating a parameter associated to a Bayesianinverse problem. Treating the unknown initial condition as a nuisanceparameter, typically one must resort to a numerical approximation of gradientof the log-likelihood and also adopt a discretization of the problem in spaceand/or time. We develop a new methodology to unbiasedly estimate the gradientof the log-likelihood with respect to the unknown parameter, i.e. theexpectation of the estimate has no discretization bias. Such a property is notonly useful for estimation in terms of the original stochastic model ofinterest, but can be used in stochastic gradient algorithms which benefit fromunbiased estimates. Under appropriate assumptions, we prove that our estimatoris not only unbiased but of finite variance. In addition, when implemented on asingle processor, we show that the cost to achieve a given level of error iscomparable to multilevel Monte Carlo methods, both practically andtheoretically. However, the new algorithm provides the possibility for parallelcomputation on arbitrarily many processors without any loss of efficiency,asymptotically. In practice, this means any precision can be achieved in afixed, finite constant time, provided that enough processors are available.

Unbiased Estimation of the Gradient of the Log-Likelihood in Inverse Problems