Low Markov Research Articles

Jensen's and Cantelli's inequalities with imprecise previsions

We investigate how basic probability inequalities can be extended to an imprecise framework, where (precise) probabilities and expectations are replaced by imprecise probabilities and lower/upper previsions. We focus on inequalities giving information on a single bounded random variable X, considering either convex/concave functions of X (Jensen's inequalities) or one-sided bounds such as (X≥c) or (X≤c) (Markov's and Cantelli's inequalities). As for the consistency of the relevant imprecise uncertainty measures, our analysis considers coherence as well as weaker requirements, notably 2-coherence, which proves to be often sufficient. Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of consistency for the related lower/upper previsions. In the case of coherent imprecise previsions, the corresponding Cantelli's inequalities make use of Walley's lower and upper variances, generally ensuring better bounds.

A sequence-based extension of mean-field annealing using the forward/backward algorithm: Application to image segmentation

Optimization based on mean-field theory, including mean-field annealing (MFA), is widely used for discrete optimization (label assignment) problems defined on the pixel sites of an image. One formulation of MFA is via maximum entropy, where one seeks the joint distribution over the (random) assignments subject to an average level of cost. MFA is obtained by assuming the assignments at each pixel are statistically independent, given the observed image. Alternatively, we make the less restrictive assumption of independent row labelings. The independence assumption means that at each step, MFA optimizes over only one pixel, whereas our method jointly optimizes over an entire row, i.e., our method is less greedy. In principle, an MFA extension could be developed that explicitly re-estimates the row labeling distributions, but such an approach is, in practice, infeasible. Even so, we can indirectly implement this re-estimation, by re-estimating quantities that determine the row labeling distributions. These quantities are the a posteriori site probabilities, re-estimable via the well-known forward/backward (F/B) algorithm. Thus, our algorithm, which descends in the ME Lagrangian/free energy, consists of iterative application of F/B to the image rows (columns). At convergence, maximum a posteriori site labeling is performed. Our method was applied to segmentation of both real and synthetic noise-corrupted images. It achieved lower Markov random field model potentials and better segmentations compared with other methods, and, in high noise, standard MFA.