A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation

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 (ODE, PDE), 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.Comment: Fixed some spelling errors and the author affiliations. This is the version accepted for publication by J Stat Phy

Bayesian field theoretic reconstruction of bond potential and bond mobility in single molecule force spectroscopy

Quantifying the forces between and within macromolecules is a necessary first step in understanding the mechanics of molecular structure, protein folding, and enzyme function and performance. In such macromolecular settings, dynamic single-molecule force spectroscopy (DFS) has been used to distort bonds. The resulting responses, in the form of rupture forces, work applied, and trajectories of displacements, have been used to reconstruct bond potentials. Such approaches often rely on simple parameterizations of one-dimensional bond potentials, assumptions on equilibrium starting states, and/or large amounts of trajectory data. Parametric approaches typically fail at inferring complex-shaped bond potentials with multiple minima, while piecewise estimation may not guarantee smooth results with the appropriate behavior at large distances. Existing techniques, particularly those based on work theorems, also do not address spatial variations in the diffusivity that may arise from spatially inhomogeneous coupling to other degrees of freedom in the macromolecule, thereby presenting an incomplete picture of the overall bond dynamics. To solve these challenges, we have developed a comprehensive empirical Bayesian approach that incorporates data and regularization terms directly into a path integral. All experiemental and statistical parameters in our method are estimated empirically directly from the data. Upon testing our method on simulated data, our regularized approach requires fewer data and allows simultaneous inference of both complex bond potentials and diffusivity profiles.Comment: In review - Python source code available on github. Abridged abstract on arXi

A Statistical Analysis of Santa Barbara Ambulance Response in 2006: Performance Under Load

Ambulance response times in Santa Barbara County for 2006 are analyzed using point process techniques, including kernel intensity estimates and K-functions. Clusters of calls result in significantly higher response times, and this effect is quantified. In particular, calls preceded by other calls within 20 km and within the previous hour are significantly more likely to result in violations. This effect appears to be especially pronounced within semi-rural neighborhoods

Estimating the distribution of morbidity and mortality of childhood diarrhea, measles, and pneumonia by wealth group in low- and middle-income countries

__Background:__ Equitable access to vaccines has been suggested as a priority for low- and middle-income countries (LMICs). However, it is unclear whether providing equitable access is enough to ensure health equity. Furthermore, disaggregated data on health outcomes and benefits gained across population subgroups are often unavailable. This paper develops a model to estimate the distribution of childhood disease cases and deaths across socioeconomic groups, and the potential benefits of three vaccine programs in LMICs. __Methods:__ For each country and for three diseases (diarrhea, measles, pneumonia), we estimated the distributions of cases and deaths that would occur across wealth quintiles in the absence of any immunization or treatment programs, using both the prevalence and relative risk of a set of risk and prognostic factors. Building on these baseline estimates, we examined what might be the impact of three vaccines (first dose of measles, pneumococcal conjugate, and rotavirus vaccines), under five scenarios based on different sets of quintile-specific immunization coverage and disease treatment utilization rates. __Results:__ Due to higher prevalence of risk factors among the poor, disproportionately more disease cases and deaths would occur among the two lowest wealth quintiles for all three diseases when vaccines or treatment are unavailable. Country-specific context, including how the baseline risks, immunization coverage, and treatment utilization are currently distributed across quintiles, affects how different policies translate into changes in cases and deaths distribution. __Conclusions:__ Our study highlights several factors that would substantially contribute to the unequal distribution of childhood diseases, and finds that merely ensuring equal access to vaccines will not reduce the health outcomes gap across wealth quintiles. Such information can inform policies and planning of programs that aim to improve equitable delivery of healthcare services

Iterative graph cuts for image segmentation with a nonlinear statistical shape prior

Shape-based regularization has proven to be a useful method for delineating objects within noisy images where one has prior knowledge of the shape of the targeted object. When a collection of possible shapes is available, the specification of a shape prior using kernel density estimation is a natural technique. Unfortunately, energy functionals arising from kernel density estimation are of a form that makes them impossible to directly minimize using efficient optimization algorithms such as graph cuts. Our main contribution is to show how one may recast the energy functional into a form that is minimizable iteratively and efficiently using graph cuts.Comment: Revision submitted to JMIV (02/24/13