Abstract
We consider the problem of estimating a parameter theta in Theta subseteq {mathbb {R}}^{d_{theta }} 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.
Highlights
The problem of inferring unknown parameters associated with the solution of differential equations (PDEs) is referred to as an inverse problem
Consider a Bayesian inverse problem (BIP) with unknown u ∈ X and data y ∈ Y, related through a PDE, and assume that there is an unknown parameter θ ∈ ⊆ Rdθ in the sense that one could consider the posterior density p(u, θ |y) ∝ p(y|u, θ ) p(u|θ ) p(θ )
The unknown u is treated as a nuisance parameter and the goal is to maximize the marginal likelihood of the parameters pθ (y) = pθ (y|u) pθ (u)du where we have dropped the conditioning on θ and used subscripts instead, as is classical in the literature (e.g. Cappé et al (2005)) and du is a sigma-finite measure on some measurable space (X, X )
Summary
The problem of inferring unknown parameters associated with the solution of (partial) differential equations (PDEs) is referred to as an inverse problem. In such a context, when the forward problem is well posed, the inverse problem is often ill-posed and challenging to solve, even numerically. Consider a BIP with unknown u ∈ X and data y ∈ Y, related through a PDE, and assume that there is an unknown parameter θ ∈ ⊆ Rdθ in the sense that one could consider the posterior density p(u, θ |y) ∝ p(y|u, θ ) p(u|θ ) p(θ ). Due to the strong correlation of the unknown u with respect to the parameter θ , such posterior distributions can be highly complex and very challenging to sample from, even using quite advanced Markov
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