A Score-Based Approach for Training Schrödinger Bridges for Data Modelling.

A Schrödinger bridge is a stochastic process connecting two given probability distributions over time. It has been recently applied as an approach for generative data modelling. The computational training of such bridges requires the repeated estimation of the drift function for a time-reversed stochastic process using samples generated by the corresponding forward process. We introduce a modified score- function-based method for computing such reverse drifts, which can be efficiently implemented by a feed-forward neural network. We applied our approach to artificial datasets with increasing complexity. Finally, we evaluated its performance on genetic data, where Schrödinger bridges can be used to model the time evolution of single-cell RNA measurements.

Variational Bayesian Inference for Nonlinear Hawkes Process with Gaussian Process Self-Effects.

Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results.

Stochastic Control for Bayesian Neural Network Training.

In this paper, we propose to leverage the Bayesian uncertainty information encoded in parameter distributions to inform the learning procedure for Bayesian models. We derive a first principle stochastic differential equation for the training dynamics of the mean and uncertainty parameter in the variational distributions. On the basis of the derived Bayesian stochastic differential equation, we apply the methodology of stochastic optimal control on the variational parameters to obtain individually controlled learning rates. We show that the resulting optimizer, StochControlSGD, is significantly more robust to large learning rates and can adaptively and individually control the learning rates of the variational parameters. The evolution of the control suggests separate and distinct dynamical behaviours in the training regimes for the mean and uncertainty parameters in Bayesian neural networks.

A mathematical model of local and global attention in natural scene viewing.

Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two-fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.

Flexible and Efficient Inference with Particles for the Variational Gaussian Approximation.

Variational inference is a powerful framework, used to approximate intractable posteriors through variational distributions. The de facto standard is to rely on Gaussian variational families, which come with numerous advantages: they are easy to sample from, simple to parametrize, and many expectations are known in closed-form or readily computed by quadrature. In this paper, we view the Gaussian variational approximation problem through the lens of gradient flows. We introduce a flexible and efficient algorithm based on a linear flow leading to a particle-based approximation. We prove that, with a sufficient number of particles, our algorithm converges linearly to the exact solution for Gaussian targets, and a low-rank approximation otherwise. In addition to the theoretical analysis, we show, on a set of synthetic and real-world high-dimensional problems, that our algorithm outperforms existing methods with Gaussian targets while performing on a par with non-Gaussian targets.

Interacting Particle Solutions of Fokker-Planck Equations Through Gradient-Log-Density Estimation.

Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.

Optimal Decoding of Dynamic Stimuli by Heterogeneous Populations of Spiking Neurons: A Closed-Form Approximation.

Neural decoding may be formulated as dynamic state estimation (filtering) based on point-process observations, a generally intractable problem. Numerical sampling techniques are often practically useful for the decoding of real neural data. However, they are less useful as theoretical tools for modeling and understanding sensory neural systems, since they lead to limited conceptual insight into optimal encoding and decoding strategies. We consider sensory neural populations characterized by a distribution over neuron parameters. We develop an analytically tractable Bayesian approximation to optimal filtering based on the observation of spiking activity that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. Continuous distributions are used to approximate large populations with few parameters, resulting in a filter whose complexity does not grow with population size and allowing optimization of population parameters rather than individual tuning functions. Numerical comparison with particle filtering demonstrates the quality of the approximation. The analytic framework leads to insights that are difficult to obtain from numerical algorithms and is consistent with biological observations about the distribution of sensory cells' preferred stimuli.

Learning combinatorial transcriptional dynamics from gene expression data.

MOTIVATION: mRNA transcriptional dynamics is governed by a complex network of transcription factor (TF) proteins. Experimental and theoretical analysis of this process is hindered by the fact that measurements of TF activity in vivo is very challenging. Current models that jointly infer TF activities and model parameters rely on either of the two main simplifying assumptions: either the dynamics is simplified (e.g. assuming quasi-steady state) or the interactions between TFs are ignored, resulting in models accounting for a single TF. RESULTS: We present a novel approach to reverse engineer the dynamics of multiple TFs jointly regulating the expression of a set of genes. The model relies on a continuous time, differential equation description of transcriptional dynamics where TFs are treated as latent on/off variables and are modelled using a switching stochastic process (telegraph process). The model can not only incorporate both activation and repression, but allows any non-trivial interaction between TFs, including AND and OR gates. By using a factorization assumption within a variational Bayesian treatment we formulate a framework that can reconstruct both the activity profiles of the TFs and the type of regulation from time series gene expression data. We demonstrate the identifiability of the model on a simple but non-trivial synthetic example, and then use it to formulate non-trivial predictions about transcriptional control during yeast metabolism. AVAILABILITY: http://homepages.inf.ed.ac.uk/gsanguin/ CONTACT: g.sanguinetti@ed.ac.uk SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Common input explains higher-order correlations and entropy in a simple model of neural population activity.

Simultaneously recorded neurons exhibit correlations whose underlying causes are not known. Here, we use a population of threshold neurons receiving correlated inputs to model neural population recordings. We show analytically that small changes in second-order correlations can lead to large changes in higher-order redundancies, and that the resulting interactions have a strong impact on the entropy, sparsity, and statistical heat capacity of the population. Our findings for this simple model may explain some surprising effects recently observed in neural population recordings.

Expectation propagation with factorizing distributions: a Gaussian approximation and performance results for simple models.

We discuss the expectation propagation (EP) algorithm for approximate Bayesian inference using a factorizing posterior approximation. For neural network models, we use a central limit theorem argument to make EP tractable when the number of parameters is large. For two types of models, we show that EP can achieve optimal generalization performance when data are drawn from a simple distribution.

Exact solution to the random sequential dynamics of a message passing algorithm.

We analyze the random sequential dynamics of a message passing algorithm for Ising models with random interactions in the large system limit. We derive exact results for the two-time correlation functions and the speed of convergence. The de Almedia-Thouless stability criterion of the static problem is found to be necessary and sufficient for the global convergence of the random sequential dynamics.

Switching regulatory models of cellular stress response.

MOTIVATION: Stress response in cells is often mediated by quick activation of transcription factors (TFs). Given the difficulty in experimentally assaying TF activities, several statistical approaches have been proposed to infer them from microarray time courses. However, these approaches often rely on prior assumptions which rule out the rapid responses observed during stress response. RESULTS: We present a novel statistical model to infer how TFs mediate stress response in cells. The model is based on the assumption that sensory TFs quickly transit between active and inactive states. We therefore model mRNA production using a bistable dynamical systems whose behaviour is described by a system of differential equations driven by a latent stochastic process. We assume the stochastic process to be a two-state continuous time jump process, and devise both an exact solution for the inference problem as well as an efficient approximate algorithm. We evaluate the method on both simulated data and real data describing Escherichia coli's response to sudden oxygen starvation. This highlights both the accuracy of the proposed method and its potential for generating novel hypotheses and testable predictions. AVAILABILITY: MATLAB and C++ code used in the article can be downloaded from http://www.dcs.shef.ac.uk/~guido/.

Efficient statistical inference for stochastic reaction processes.

We address the problem of estimating unknown model parameters and state variables in stochastic reaction processes when only sparse and noisy measurements are available. Using an asymptotic system size expansion for the backward equation, we derive an efficient approximation for this problem. We demonstrate the validity of our approach on model systems and generalize our method to the case when some state variables are not observed.

Memory-free dynamics for the Thouless-Anderson-Palmer equations of Ising models with arbitrary rotation-invariant ensembles of random coupling matrices.

We propose an iterative algorithm for solving the Thouless-Anderson-Palmer equations of Ising models with arbitrary rotation-invariant (random) coupling matrices. In the thermodynamic limit, we prove by means of the dynamical functional method that the proposed algorithm converges when the so-called de Almeida Thouless criterion is fulfilled. Moreover, we give exact analytical expressions for the rate of the convergence.

Publisher's Note: Inverse Ising problem in continuous time: A latent variable approach [Phys. Rev. E 96, 062104 (2017)].

This corrects the article DOI: 10.1103/PhysRevE.96.062104.

Approximate Bayes learning of stochastic differential equations.

We introduce a nonparametric approach for estimating drift and diffusion functions in systems of stochastic differential equations from observations of the state vector. Gaussian processes are used as flexible models for these functions, and estimates are calculated directly from dense data sets using Gaussian process regression. We develop an approximate expectation maximization algorithm to deal with the unobserved, latent dynamics between sparse observations. The posterior over states is approximated by a piecewise linearized process of the Ornstein-Uhlenbeck type and the maximum a posteriori estimation of the drift is facilitated by a sparse Gaussian process approximation.

Inverse Ising problem in continuous time: A latent variable approach.

We consider the inverse Ising problem: the inference of network couplings from observed spin trajectories for a model with continuous time Glauber dynamics. By introducing two sets of auxiliary latent random variables we render the likelihood into a form which allows for simple iterative inference algorithms with analytical updates. The variables are (1) Poisson variables to linearize an exponential term which is typical for point process likelihoods and (2) Pólya-Gamma variables, which make the likelihood quadratic in the coupling parameters. Using the augmented likelihood, we derive an expectation-maximization (EM) algorithm to obtain the maximum likelihood estimate of network parameters. Using a third set of latent variables we extend the EM algorithm to sparse couplings via L1 regularization. Finally, we develop an efficient approximate Bayesian inference algorithm using a variational approach. We demonstrate the performance of our algorithms on data simulated from an Ising model. For data which are simulated from a more biologically plausible network with spiking neurons, we show that the Ising model captures well the low order statistics of the data and how the Ising couplings are related to the underlying synaptic structure of the simulated network.

Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations.

This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies.

Parameter estimation and inference for stochastic reaction-diffusion systems: application to morphogenesis in D. melanogaster.

BACKGROUND: Reaction-diffusion systems are frequently used in systems biology to model developmental and signalling processes. In many applications, count numbers of the diffusing molecular species are very low, leading to the need to explicitly model the inherent variability using stochastic methods. Despite their importance and frequent use, parameter estimation for both deterministic and stochastic reaction-diffusion systems is still a challenging problem. RESULTS: We present a Bayesian inference approach to solve both the parameter and state estimation problem for stochastic reaction-diffusion systems. This allows a determination of the full posterior distribution of the parameters (expected values and uncertainty). We benchmark the method by illustrating it on a simple synthetic experiment. We then test the method on real data about the diffusion of the morphogen Bicoid in Drosophila melanogaster. The results show how the precision with which parameters can be inferred varies dramatically, indicating that the ability to infer full posterior distributions on the parameters can have important experimental design consequences. CONCLUSIONS: The results obtained demonstrate the feasibility and potential advantages of applying a Bayesian approach to parameter estimation in stochastic reaction-diffusion systems. In particular, the ability to estimate credibility intervals associated with parameter estimates can be precious for experimental design. Further work, however, will be needed to ensure the method can scale up to larger problems.

The variational gaussian approximation revisited.

The variational approximation of posterior distributions by multivariate gaussians has been much less popular in the machine learning community compared to the corresponding approximation by factorizing distributions. This is for a good reason: the gaussian approximation is in general plagued by an Omicron(N)(2) number of variational parameters to be optimized, N being the number of random variables. In this letter, we discuss the relationship between the Laplace and the variational approximation, and we show that for models with gaussian priors and factorizing likelihoods, the number of variational parameters is actually Omicron(N). The approach is applied to gaussian process regression with nongaussian likelihoods.