Revisiting maximum-a-posteriori estimation in log-concave models

Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation methodology in imaging sciences, where high dimensionality is often addressed by using Bayesian models that are log-concave and whose posterior mode can be computed efficiently by convex optimisation. Despite its success and wide adoption, MAP estimation is not theoretically well understood yet. The prevalent view in the community is that MAP estimation is not proper Bayesian estimation in a decision-theoretic sense because it does not minimise a meaningful expected loss function (unlike the minimum mean squared error (MMSE) estimator that minimises the mean squared loss). This paper addresses this theoretical gap by presenting a decision-theoretic derivation of MAP estimation in Bayesian models that are log-concave. A main novelty is that our analysis is based on differential geometry, and proceeds as follows. First, we use the underlying convex geometry of the Bayesian model to induce a Riemannian geometry on the parameter space. We then use differential geometry to identify the so-called natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold. For log-concave models, this canonical loss is the Bregman divergence associated with the negative log posterior density. We then show that the MAP estimator is the only Bayesian estimator that minimises the expected canonical loss, and that the posterior mean or MMSE estimator minimises the dual canonical loss. We also study the question of MAP and MSSE estimation performance in large scales and establish a universal bound on the expected canonical error as a function of dimension, offering new insights into the good performance observed in convex problems. These results provide a new understanding of MAP and MMSE estimation in log-concave settings, and of the multiple roles that convex geometry plays in imaging problems.Comment: Accepted for publication in SIAM Imaging Science

Boosting the Accuracy of Differentially-Private Histograms Through Consistency

We show that it is possible to significantly improve the accuracy of a general class of histogram queries while satisfying differential privacy. Our approach carefully chooses a set of queries to evaluate, and then exploits consistency constraints that should hold over the noisy output. In a post-processing phase, we compute the consistent input most likely to have produced the noisy output. The final output is differentially-private and consistent, but in addition, it is often much more accurate. We show, both theoretically and experimentally, that these techniques can be used for estimating the degree sequence of a graph very precisely, and for computing a histogram that can support arbitrary range queries accurately.Comment: 15 pages, 7 figures, minor revisions to previous versio

Shear and Bulk Viscosities of a Gluon Plasma in Perturbative QCD: Comparison of Different Treatments for the gg<->ggg Process

The leading order contribution to the shear and bulk viscosities, \eta and \zeta, of a gluon plasma in perturbative QCD includes the gg -> gg (22) process, gg ggg (23) process and multiple scattering processes known as the Landau-Pomeranchuk-Migdal (LPM) effect. Complete leading order computations for \eta and \zeta were obtained by Arnold, Moore and Yaffe (AMY) and Arnold, Dogan and Moore (ADM), respectively, with the inelastic processes computed by an effective g gg gluon splitting. We study how complementary calculations with 22 and 23 processes and a simple treatment to model the LPM effect compare with the results of AMY and ADM. We find that our results agree with theirs within errors. By studying the contribution of the 23 process to \eta, we find that the minimum angle \theta among the final state gluons in the fluid local rest frame has a distribution that is peaked at \theta \sim \sqrt{\alpha_{s}}, analogous to the near collinear splitting asserted by AMY and ADM. However, the average of \theta is much bigger than its peak value, as its distribution is skewed with a long tail. The same \theta behavior is also seen if the 23 matrix element is taken to the soft gluon bremsstrahlung limit in the center-of-mass (CM) frame. This suggests that the soft gluon bremsstrahlung in the CM frame still has some near collinear behavior in the fluid local rest frame. We also generalize our result to a general SU(N_c) pure gauge theory and summarize the current viscosity computations in QCD.Comment: ReVTex 4, 18 pages, 7 figures, accepted version in Phys. Rev.

Approximate Bayesian Model Selection with the Deviance Statistic

Bayesian model selection poses two main challenges: the specification of parameter priors for all models, and the computation of the resulting Bayes factors between models. There is now a large literature on automatic and objective parameter priors in the linear model. One important class are $g$-priors, which were recently extended from linear to generalized linear models (GLMs). We show that the resulting Bayes factors can be approximated by test-based Bayes factors (Johnson [Scand. J. Stat. 35 (2008) 354-368]) using the deviance statistics of the models. To estimate the hyperparameter $g$, we propose empirical and fully Bayes approaches and link the former to minimum Bayes factors and shrinkage estimates from the literature. Furthermore, we describe how to approximate the corresponding posterior distribution of the regression coefficients based on the standard GLM output. We illustrate the approach with the development of a clinical prediction model for 30-day survival in the GUSTO-I trial using logistic regression.Comment: Published at http://dx.doi.org/10.1214/14-STS510 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org