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A Path-Integral Approach to Bayesian Inference for Inverse Problems Using the Semiclassical Approximation

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Abstract

We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations, a problem which is typically ill-posed. Regularization of these problems using \(L^2\) function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem—namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.

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Notes

  1. We will use Greek letters to denote fields

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Acknowledgments

This material is based upon work supported by the National Science Foundation under Agreement No. 0635561. JC and TC also acknowledge support from the National Science Foundation through grant DMS-1021818, and from the Army Research Office through grant 58386MA. VS acknowledges support from UCLA startup funds.

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Correspondence to Joshua C. Chang.

Appendix: Functional Taylor Approximations for the Dielectric Field Problem

Appendix: Functional Taylor Approximations for the Dielectric Field Problem

We wish to expand the Hamiltonian

$$\begin{aligned} H[\epsilon ; \rho , \varphi _{\mathrm{obs }}]&= \frac{1}{2}\sum _{m=1}^M\int \frac{\delta (\mathbf {x}-\mathbf {x}_m)}{s_m^2}\left[ \sum _{n=0}^\infty \varphi _n(\epsilon (\mathbf {x}))-\varphi _{\mathrm{obs }}(\mathbf {x})\right] ^2\,\mathrm{d}\mathbf {x}\nonumber \\&\quad +\frac{1}{2} \int \epsilon (\mathbf {x})(-\Delta )P(-\Delta )\epsilon (\mathbf {x}) \,\mathrm{d}\mathbf {x} \end{aligned}$$
(51)

about its extrema \(\epsilon ^*\). We take variations with respect to \(\epsilon (\mathbf {x})\) to calculate its first functional derivative,

$$\begin{aligned} \int \frac{\partial H}{\partial \epsilon (\mathbf {x})}\phi (\mathbf {x})\,\mathrm{d}\mathbf {x}&=\int (-\Delta )P(-\Delta )\epsilon (x) \phi (\mathbf {x})\,\mathrm{d}\mathbf {x}+\lim _{h\rightarrow 0}\frac{\,\mathrm{d}}{\,\mathrm{d} h}\frac{1}{2}\sum _{m=1}^M\int \frac{\delta (\mathbf {x}-\mathbf {x}_m)}{s_m^2}\nonumber \\&\quad \times \left[ \sum _{n=0}^\infty \varphi _n(\epsilon (\mathbf {x})+h\phi (\mathbf {x}))-\varphi _{\mathrm{obs }}(\mathbf {x})\right] ^2\,\mathrm{d}\mathbf {x}\nonumber \\&=\int (-\Delta )P(-\Delta )\epsilon \phi \,\mathrm{d}\mathbf {x}+\lim _{h\rightarrow 0}\sum _{m=1}^M\int \frac{\delta (\mathbf {x}-\mathbf {x}_m)}{s_m^2}\nonumber \\&\quad \times \left( \varphi (\mathbf {x})-\varphi _{\mathrm{obs }}(\mathbf {x}) \right) \frac{\,\mathrm{d}}{\,\mathrm{d} h}\varphi _0(\epsilon (\mathbf {x})+h\phi (\mathbf {x}))\,\mathrm{d}\mathbf {x}\nonumber \\&\quad +\lim _{h\rightarrow 0}\sum _{m=1}^M\int \frac{\delta (\mathbf {x}-\mathbf {x}_m)}{s_m^2}\left( \varphi (\mathbf {x})-\varphi _{\mathrm{obs }}(\mathbf {x}) \right) \frac{\,\mathrm{d}}{\,\mathrm{d} h}\varphi _1(\epsilon (\mathbf {x})+h\phi (\mathbf {x}))\,\mathrm{d}\mathbf {x}\nonumber \\&\quad +\underbrace{\lim _{h\rightarrow 0}\sum _{m=1}^M \int \frac{\delta (\mathbf {x}-\mathbf {x}_m)}{s_m^2}\left( \varphi (\mathbf {x})-\varphi _{\mathrm{obs }}(\mathbf {x}) \right) \sum _{n=2}^\infty \frac{\,\mathrm{d}}{\,\mathrm{d} h} \varphi _n(\epsilon (\mathbf {x})+h\phi (\mathbf {x}))\,\mathrm{d}\mathbf {x}}_{I_1} \end{aligned}$$
(52)

Let us define the quantities

$$\begin{aligned} \tilde{K}(\mathbf {y},\mathbf {z})&=\nabla _\mathbf {z}\cdot \left[ L(\mathbf {y},\mathbf {z})\nabla _\mathbf {z}\left( \frac{\phi (\mathbf {z})}{\epsilon (\mathbf {z})}\right) \right] \\ \tilde{\varphi }_0(\mathbf {x})&= -\int L(\mathbf {x},\mathbf {y})\frac{\rho (\mathbf {y})\phi (\mathbf {y})}{\epsilon ^2(\mathbf {y})}\,\mathrm{d}\mathbf {y}\\ \Psi (\mathbf {x})&=\sum _{m=1}^M \frac{\delta (\mathbf {x}-\mathbf {x}_m)}{s_m^2}\left( \varphi (\mathbf {x})-\varphi _{\mathrm{obs }}(\mathbf {x}) \right) . \end{aligned}$$

Through direct differentiation we find that

$$\begin{aligned} I_1&= \sum _{n=2}^\infty \int \Psi (\mathbf {x}) K(\mathbf {x},\mathbf {y}_n) \left( \prod _{j=1}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \tilde{\varphi }_0(\mathbf {y}_1)\,\mathrm{d}\mathbf {x}\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k\\&\quad + \sum _{n=2}^\infty \int \Psi (\mathbf {x})\tilde{K}(\mathbf {x},\mathbf {y}_n) \left( \prod _{j=1}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) {\varphi }_0(\mathbf {y}_1)\,\mathrm{d}\mathbf {x}\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k \\&\quad +\sum _{n=2}^\infty \int \Psi (\mathbf {x}){K}(\mathbf {x},\mathbf {y}_n) \sum _{k=0}^{n-1}\left( \tilde{K}(\mathbf {y}_{k+1},\mathbf {y}_k)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \varphi _0(\mathbf {y}_1)\,\mathrm{d}\mathbf {x}\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k. \end{aligned}$$

Integrating in \(\mathbf {x}\):

$$\begin{aligned} I_1&=\sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2}\int K(\mathbf {x}_m,\mathbf {y}_n) \left( \prod _{j=1}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \tilde{\varphi }_0(\mathbf {y}_1)\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k\\&\quad + \sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2} \int \tilde{K}(\mathbf {x}_m,\mathbf {y}_n) \left( \prod _{j=1}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) {\varphi }_0(\mathbf {y}_1)\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k \\&\quad +\sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2}\int K(\mathbf {x}_m,\mathbf {y}_n) \sum _{k=1}^{n-1}\left( \tilde{K}(\mathbf {y}_{k+1},\mathbf {y}_k)\prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \\&\quad \times \varphi _0(\mathbf {y}_1)\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k. \end{aligned}$$

We shift \(\phi (\cdot )\rightarrow \phi (\mathbf {x})\), and integrate-by-parts to find

$$\begin{aligned} I_1&= -\sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2}\int K(\mathbf {x}_m,\mathbf {y}_n) \left( \prod _{j=1}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \\&\times \, L(\mathbf {y}_1,\mathbf {x})\frac{\rho (\mathbf {x})}{\epsilon ^2(\mathbf {x})}\phi (\mathbf {x})\,\mathrm{d}\mathbf {x}\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k\\&+ \sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2} \int \frac{\phi (\mathbf {x})}{\epsilon (\mathbf {x})}\nabla \cdot \\&\left[ L(\mathbf {x}_m,\mathbf {x})\nabla \left( K(\mathbf {x},\mathbf {y}_{n-1})\right) \right] \left( \prod _{j=1}^{n-2} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) {\varphi }_0(\mathbf {y}_1)\,\mathrm{d}\mathbf {x}\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k \\&+\sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2}\int K(\mathbf {x}_m,\mathbf {y}_n) \sum _{k=1}^{n-1}\\&\left( \frac{\phi (\mathbf {x})}{\epsilon (\mathbf {x})}\nabla \cdot \left[ L(\mathbf {y}_{k+1},\mathbf {x})\nabla K(\mathbf {x},\mathbf {y}_{k-1}) \right] \prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^{n-2} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \varphi _0(\mathbf {y}_1)\,\mathrm{d}\mathbf {x}\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k. \end{aligned}$$

Note that all boundary terms disappear since we can take \(\phi \) to disappear on the boundary. With \(I_1\) computed, we find

$$\begin{aligned} \frac{\delta H}{\delta \epsilon (\mathbf {x})}&=(-\Delta )P(-\Delta )\epsilon (\mathbf {x})-\sum _{m=1}^M \frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2}\left[ L(\mathbf {x}_m,\mathbf {x})\frac{\rho (\mathbf {x})}{\epsilon ^2(\mathbf {x})} \right] \nonumber \\&\quad + \sum _{m=1}^M \frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2\epsilon (\mathbf {x})}\nabla \cdot \left[ L(\mathbf {x}_m,\mathbf {x})\nabla \varphi _0(\mathbf {x}) \right] \nonumber \\&\quad -\sum _{m=1}^M \frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2} \left( \frac{\rho (\mathbf {x})}{\epsilon ^2(\mathbf {x})} \right) \int K(\mathbf {x}_m,\mathbf {y}_1)L(\mathbf {x},\mathbf {y}_1) \,\mathrm{d}\mathbf {y}_1 \nonumber \\&\quad -\sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2}\int K(\mathbf {x}_m,\mathbf {y}_n) \left( \prod _{j=1}^{n-1} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \nonumber \\&\quad \times L(\mathbf {y}_1,\mathbf {x})\frac{\rho (\mathbf {x})}{\epsilon ^2(\mathbf {x})}\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k \nonumber \\&\quad + \sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2\epsilon (\mathbf {x})} \int \nabla \cdot \left[ L(\mathbf {x}_m,\mathbf {x})\nabla \left( K(\mathbf {x},\mathbf {y}_{n-1})\right) \right] \nonumber \\&\quad \times \left( \prod _{j=1}^{n-2} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) {\varphi }_0(\mathbf {y}_1)\prod _{k=1}^n\,\mathrm{d}\mathbf {y}_k\nonumber \\&\quad +\sum _{n=2}^\infty \sum _{m=1}^M\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2\epsilon (\mathbf {x})}\int K(\mathbf {x}_m,\mathbf {y}_n)\nonumber \\&\quad \sum _{k=1}^{n-1}\left( \nabla \cdot \left[ L(\mathbf {y}_{k+1},\mathbf {x})\nabla K(\mathbf {x},\mathbf {y}_{k-1}) \right] \prod _{\begin{array}{c} j=1\\ j\ne k \end{array}}^{n-2} K(\mathbf {y}_{j+1}, \mathbf {y}_j) \right) \varphi _0(\mathbf {y}_1)\prod _{k=1}^n \,\mathrm{d}\mathbf {y}_k. \end{aligned}$$
(53)

Taken to two terms in the series expansion for \(\varphi \), the first variation is

$$\begin{aligned} \frac{\delta H}{\delta \epsilon (\mathbf {x})}&\sim (-\Delta )P(-\Delta )\epsilon (\mathbf {x})+\sum _{m=1}^M \frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2\epsilon (\mathbf {x})}\nonumber \\&\times \left[ \nabla L(\mathbf {x},\mathbf {x}_m)\cdot \nabla \varphi _0(\mathbf {x}) -\frac{\rho (\mathbf {x})}{\epsilon (\mathbf {x})}\int K(\mathbf {x}_m,\mathbf {y}_1)L(\mathbf {x},\mathbf {y}_1)\,\mathrm{d}\mathbf {y}_1 \right] . \end{aligned}$$
(54)

To calculate the second-order term in the Taylor-expansion, we take another variation. Truncated at two terms in the expansion for \(\varphi \):

$$\begin{aligned} \frac{\delta ^2 H}{\delta \epsilon (\mathbf {x})\delta \epsilon (\mathbf {x}^{\prime })}= (-\Delta )P(-\Delta )\delta (\mathbf {x}-\mathbf {x}^{\prime })+\sum _{m=1}^M a_m(\mathbf {x},\mathbf {x}^{\prime }), \end{aligned}$$
(55)

where after canceling like terms,

$$\begin{aligned} a_m(\mathbf {x},\mathbf {x}^{\prime })&=\delta (\mathbf {x}-\mathbf {x}^{\prime }) \frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2\epsilon ^2(\mathbf {x}^{\prime })}\left[ \frac{2\rho (\mathbf {x}^{\prime })}{\epsilon (\mathbf {x}^{\prime })}\int K(\mathbf {x}_m,\mathbf {y}_1)L(\mathbf {x}^{\prime },\mathbf {y}_1)\,\mathrm{d}\mathbf {y}_1\right. \nonumber \\&\quad -\left. \nabla _{\mathbf {x}^{\prime }}L(\mathbf {x}^{\prime },\mathbf {x}_m)\cdot \nabla _{\mathbf {x}^{\prime }}\varphi _0(\mathbf {x}^{\prime })-L(\mathbf {x}_m,\mathbf {x}^{\prime })\frac{\rho (\mathbf {x}^{\prime })}{\epsilon (\mathbf {x}^{\prime })} \right] \nonumber \\&\quad -\frac{\varphi (\mathbf {x}_m)-\varphi _{\mathrm{obs }}(\mathbf {x}_m)}{s_m^2}\left\{ \nabla L(\mathbf {x},\mathbf {x}_m)\cdot \nabla _{\mathbf {x}^{\prime }}L(\mathbf {x},\mathbf {x}^{\prime }) \frac{\rho (\mathbf {x}^{\prime })}{\epsilon (\mathbf {x})\epsilon ^2(\mathbf {x}^{\prime })}\right. \nonumber \\&\quad +\left. \frac{\rho (\mathbf {x})}{\epsilon ^2(\mathbf {x})\epsilon (\mathbf {x}^{\prime })} \nabla L(\mathbf {x},\mathbf {x}^{\prime })\cdot \nabla _{\mathbf {x}^{\prime }}L(\mathbf {x}_m,\mathbf {x}^{\prime }) \right\} \\&\quad + \left[ \nabla L(\mathbf {x},\mathbf {x}_m)\cdot \nabla \varphi _0(\mathbf {x}) -\frac{\rho (\mathbf {x})}{\epsilon (\mathbf {x})}\int K(\mathbf {x}_m,\mathbf {y}_1)L(\mathbf {x},\mathbf {y}_1)\,\mathrm{d}\mathbf {y}_1 \right] \\&\quad \times \frac{1}{s_m^2 \epsilon (\mathbf {x})\epsilon (\mathbf {x}^{\prime })}\left[ \nabla _{\mathbf {x}^{\prime }}L(\mathbf {x}^{\prime },\mathbf {x}_m)\cdot \nabla _{\mathbf {x}^{\prime }}\varphi _0(\mathbf {x}^{\prime }) -\frac{\rho (\mathbf {x}^{\prime })}{\epsilon (\mathbf {x}^{\prime })}\int \! K(\mathbf {x}_m,\mathbf {y}_1)L(\mathbf {x}^{\prime },\mathbf {y}_1)\,\mathrm{d}\mathbf {y}_1 \!\right] \!. \end{aligned}$$

It is using this expression that we can construct an approximate probability density for our field \(\epsilon \).

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Chang, J.C., Savage, V.M. & Chou, T. A Path-Integral Approach to Bayesian Inference for Inverse Problems Using the Semiclassical Approximation. J Stat Phys 157, 582–602 (2014). https://doi.org/10.1007/s10955-014-1059-y

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