Agents acting in the natural world aim at selecting appropriate actions based on noisy and partial sensory observations. Many behaviors leading to decision mak- ing and action selection in a closed loop setting are naturally phrased within a control theoretic framework. Within the framework of optimal Control Theory, one is usually given a cost function which is minimized by selecting a control law based on the observations. While in standard control settings the sensors are assumed fixed, biological systems often gain from the extra flexibility of optimiz- ing the sensors themselves. However, this sensory adaptation is geared towards control rather than perception, as is often assumed. In this work we show that sen- sory adaptation for control differs from sensory adaptation for perception, even for simple control setups. This implies, consistently with recent experimental results, that when studying sensory adaptation, it is essential to account for the task being performed.

We introduce a nonparametric approach for estimating drift functions in systems of stochastic differential equations from incomplete observations of the state vector. Using a Gaussian process prior over the drift as a function of the state vector, we develop an approximate EM algorithm to deal with the unobserved, latent dynamics between observations. The posterior over states is approximated by a piecewise linearized process and the MAP estimation of the drift is facilitated by a sparse Gaussian process regression.

We propose an approximate inference algorithm for continuous time Gaussian-Markov process models with both discrete and continuous time likelihoods. We show that the continuous time limit of the expectation propagation algorithm exists and results in a hybrid fixed point iteration consisting of (1) expectation propagation updates for the discrete time terms and (2) variational updates for the continuous time term. We introduce corrections methods that improve on the marginals of the approximation. This approach extends the classical Kalman-Bucy smoothing procedure to non-Gaussian observations, enabling continuous-time inference in a variety of models, including spiking neuronal models (state-space models with point process observations) and box likelihood models. Experimental results on real and simulated data demonstrate high distributional accuracy and significant computational savings compared to discrete-time approaches in a neural application.

Expectation Propagation (EP) provides a framework for approximate inference. When the model under consideration is over a latent Gaussian field, with the approximation being Gaussian, we show how these approximations can systematically be corrected. A perturbative expansion is made of the exact but intractable correction, and can be applied to the model's partition function and other moments of interest. The correction is expressed over the higher-order cumulants which are neglected by EP's local matching of moments. Through the expansion, we see that EP is correct to first order. By considering higher orders, corrections of increasing polynomial complexity can be applied to the approximation. The second order provides a correction in quadratic time, which we apply to an array of Gaussian process and Ising models. The corrections generalize to arbitrarily complex approximating families, which we illustrate on tree-structured Ising model approximations. Furthermore, they provide a polynomial-time assessment of the approximation error. We also provide both theoretical and practical insights on the exactness of the EP solution.

Restricted Boltzmann Machines (RBMs) are generative models which can learn useful representations from samples of a dataset in an unsupervised fashion. They have been widely employed as an unsupervised pre-training method in machine learning. RBMs have been modified to model time series in two main ways: The Temporal RBM stacks a number of RBMs laterally and introduces temporal dependencies between the hidden layer units; The Conditional RBM, on the other hand, considers past samples of the dataset as a conditional bias and learns a representation which takes these into account. Here we propose a new training method for both the TRBM and the CRBM, which enforces the dynamic structure of temporal datasets. We do so by treating the temporal models as denoising autoencoders, considering past frames of the dataset as corrupted versions of the present frame and minimizing the reconstruction error of the present data by the model. We call this approach Temporal Autoencoding. This leads to a significant improvement in the performance of both models in a filling-in-frames task across a number of datasets. The error reduction for motion capture data is 56\% for the CRBM and 80\% for the TRBM. Taking the posterior mean prediction instead of single samples further improves the model's estimates, decreasing the error by as much as 91\% for the CRBM on motion capture data. We also trained the model to perform forecasting on a large number of datasets and have found TA pretraining to consistently improve the performance of the forecasts. Furthermore, by looking at the prediction error across time, we can see that this improvement reflects a better representation of the dynamics of the data as opposed to a bias towards reconstructing the observed data on a short time scale.

In this paper we show the identification between stochastic optimal control computation and probabilistic inference on a graphical model for certain class of control problems. We refer to these problems as Kullback-Leibler (KL) control problems. We illustrate how KL control can be used to model a multi-agent cooperative game for which optimal control can be approximated using belief propagation when exact inference is unfeasible.

We consider the problem of Bayesian inference for continuous-time multi-stable stochastic systems which can change both their diffusion and drift parameters at discrete times. We propose exact inference and sampling methodologies for two specific cases where the discontinuous dynamics is given by a Poisson process and a two-state Markovian switch. We test the methodology on simulated data, and apply it to two real data sets in finance and systems biology. Our experimental results show that the approach leads to valid inferences and nontrivial insights.

Bayesian filtering of stochastic stimuli has received a great deal of attention re- cently. It has been applied to describe the way in which biological systems dy- namically represent and make decisions about the environment. There have been no exact results for the error in the biologically plausible setting of inference on point process, however. We present an exact analysis of the evolution of the mean- squared error in a state estimation task using Gaussian-tuned point processes as sensors. This allows us to study the dynamics of the error of an optimal Bayesian decoder, providing insights into the limits obtainable in this task. This is done for Markovian and a class of non-Markovian Gaussian processes. We find that there is an optimal tuning width for which the error is minimized. This leads to a char- acterization of the optimal encoding for the setting as a function of the statistics of the stimulus, providing a mathematically sound primer for an ecological theory of sensory processing.

We present a novel approach to inference in conditionally Gaussian continuous time stochastic processes, where the latent process is a Markovian jump process. We first consider the case of jump-diffusion processes, where the drift of a linear stochastic differential equation can jump at arbitrary time points. We derive partial differential equations for exact inference and present a very efficient mean field approximation. By introducing a novel lower bound on the free energy, we then generalise our approach to Gaussian processes with arbitrary covariance, such as the non-Markovian RBF covariance. We present results on both simulated and real data, showing that the approach is very accurate in capturing latent dynamics and can be useful in a number of real data modelling tasks.

A series of corrections is developed for the fixed points of Expectation Propagation (EP),which is one of the most popular methods for approximate probabilistic inference. These corrections can lead to improvements of the inference approximation orserve as a sanity check, indicating when EP yields unrealiable results.