Parameters estimation for spatio-temporal maximum entropy distributions: application to neural spike trains

We propose a numerical method to learn Maximum Entropy (MaxEnt) distributions with spatio-temporal constraints from experimental spike trains. This is an extension of two papers [10] and [4] who proposed the estimation of parameters where only spatial constraints were taken into account. The extension we propose allows to properly handle memory effects in spike statistics, for large sized neural networks.Comment: 34 pages, 33 figure

A Bayesian Filtering Algorithm for Gaussian Mixture Models

A Bayesian filtering algorithm is developed for a class of state-space systems that can be modelled via Gaussian mixtures. In general, the exact solution to this filtering problem involves an exponential growth in the number of mixture terms and this is handled here by utilising a Gaussian mixture reduction step after both the time and measurement updates. In addition, a square-root implementation of the unified algorithm is presented and this algorithm is profiled on several simulated systems. This includes the state estimation for two non-linear systems that are strictly outside the class considered in this paper

KL-optimum designs: theoretical properties and practical computation

In this paper some new properties and computational tools for finding KL-optimum designs are provided. KL-optimality is a general criterion useful to select the best experimental conditions to discriminate between statistical models. A KL-optimum design is obtained from a minimax optimization problem, which is defined on a infinite-dimensional space. In particular, continuity of the KL-optimality criterion is proved under mild conditions; as a consequence, the first-order algorithm converges to the set of KL-optimum designs for a large class of models. It is also shown that KL-optimum designs are invariant to any scale-position transformation. Some examples are given and discussed, together with some practical implications for numerical computation purposes.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s11222-014-9515-

Optimal projection of observations in a Bayesian setting

Optimal dimensionality reduction methods are proposed for the Bayesian inference of a Gaussian linear model with additive noise in presence of overabundant data. Three different optimal projections of the observations are proposed based on information theory: the projection that minimizes the Kullback-Leibler divergence between the posterior distributions of the original and the projected models, the one that minimizes the expected Kullback-Leibler divergence between the same distributions, and the one that maximizes the mutual information between the parameter of interest and the projected observations. The first two optimization problems are formulated as the determination of an optimal subspace and therefore the solution is computed using Riemannian optimization algorithms on the Grassmann manifold. Regarding the maximization of the mutual information, it is shown that there exists an optimal subspace that minimizes the entropy of the posterior distribution of the reduced model; a basis of the subspace can be computed as the solution to a generalized eigenvalue problem; an a priori error estimate on the mutual information is available for this particular solution; and that the dimensionality of the subspace to exactly conserve the mutual information between the input and the output of the models is less than the number of parameters to be inferred. Numerical applications to linear and nonlinear models are used to assess the efficiency of the proposed approaches, and to highlight their advantages compared to standard approaches based on the principal component analysis of the observations

Sequential Monte Carlo with kernel embedded mappings: the mapping particle filter

In this work, a novel sequential Monte Carlo filter is introduced which aims at an efficient sampling of the state space. Particles are pushed forward from the prediction to the posterior density using a sequence of mappings that minimizes the Kullback-Leibler divergence between the posterior and the sequence of intermediate densities. The sequence of mappings represents a gradient flow based on the principles of local optimal transport. A key ingredient of the mappings is that they are embedded in a reproducing kernel Hilbert space, which allows for a practical and efficient Monte Carlo algorithm. The kernel embedding provides a direct means to calculate the gradient of the Kullback-Leibler divergence leading to quick convergence using well-known gradient-based stochastic optimization algorithms. Evaluation of the method is conducted in the chaotic Lorenz-63 system, the Lorenz-96 system, which is a coarse prototype of atmospheric dynamics, and an epidemic model that describes cholera dynamics. No resampling is required in the mapping particle filter even for long recursive sequences. The number of effective particles remains close to the total number of particles in all the sequence. Hence, the mapping particle filter does not suffer from sample impoverishment