Markov chain Monte Carlo algorithms are a popular and well-studied methodology that can be used to draw samples from posterior distributions. Over the past few years these algorithms have been extended to tackle problems where the model likelihood is intractable (Beaumont, 2003). Andrieu and Roberts (2009) showed that within the Metropolis-Hastings algorithm, if the likelihood is replaced with an unbiased estimate, then the sampler still targets the correct stationary distribution. Andrieu et al. (2010) extended this work further to create a class of 1 Markov chain algorithms that use sequential Monte Carlo methods, also known as particle filters. Current implementations of pseudo-marginal and particle Markov chain Monte Carlo use random-walk proposals to update the parameters (e.g., Golightly and Wilkinson, 2011; Knape and de Valpine, 2012) and shall be referred to herein as particle random-walk Metropolis algorithms. Random walk-based algorithms propose a new value from some symmetric density centred on the current value.

We learn the structure of a Markov Network between two groups of random variables from joint observations. Since modelling and learning the full MN structure may be hard, learning the links between two groups directly may be a preferable option. We introduce a novel concept called the \emph{partitioned ratio} whose factorization directly associates with the Markovian properties of random variables across two groups. A simple one-shot convex optimization procedure is proposed for learning the \emph{sparse} factorizations of the partitioned ratio and it is theoretically guaranteed to recover the correct inter-group structure under mild conditions. The performance of the proposed method is experimentally compared with the state of the art MN structure learning methods using ROC curves. Real applications on analyzing bipartisanship in US congress and pairwise DNA/time-series alignments are also reported.

Partition functions of probability distributions are important quantities for model evaluation and comparisons. We present a new method to compute partition functions of complex and multimodal distributions. Such distributions are often sampled using simulated tempering, which augments the target space with an auxiliary inverse temperature variable. Our method exploits the multinomial probability law of the inverse temperatures, and provides estimates of the partition function in terms of a simple quotient of Rao-Blackwellized marginal inverse temperature probability estimates, which are updated while sampling. We show that the method has interesting connections with several alternative popular methods, and offers some significant advantages. In particular, we empirically find that the new method provides more accurate estimates than Annealed Importance Sampling when calculating partition functions of large Restricted Boltzmann Machines (RBM); moreover, the method is sufficiently accurate to track training and validation log-likelihoods during learning of RBMs, at minimal computational cost.

A Markov chain is a random process with the property that the next state depends only on the current state. For example: If you have the choice of red or blue twice the process would be Markovian if each time you chose the decision had nothing to do with your choice previously (see diagram below). How can Markov Chains help us? To start with we need to define some basic terminology. The changes of state within the system are called transitions, and the probabilities associated with various state-changes are called transition probabilities.

Probabilistic models can be defined by an energy function, where the probability of each state is proportional to the exponential of the state's negative energy. This paper considers a generalization of energy-based models in which the probability of a state is proportional to an arbitrary positive, strictly decreasing, and twice differentiable function of the state's energy. The precise shape of the nonlinear map from energies to unnormalized probabilities has to be learned from data together with the parameters of the energy function. As a case study we show that the above generalization of a fully visible Boltzmann machine yields an accurate model of neural activity of retinal ganglion cells. We attribute this success to the model's ability to easily capture distributions whose probabilities span a large dynamic range, a possible consequence of latent variables that globally couple the system. Similar features have recently been observed in many datasets, suggesting that our new method has wide applicability.

In this post, we will see the main technical ideas in the analysis of the mixing time of evolutionary Markov chains introduced in a previous post. We start by introducing the notion of the expected motion of a stochastic process or a Markov chain. In the case of a finite population evolutionary Markov chain, the expected motion turns out to be a dynamical system which corresponds to the infinite population evolutionary dynamics with the same parameters. Surprisingly, we show that the limit sets of this dynamical system govern the mixing time of the Markov chain. In particular, if the underlying dynamical system has a unique stable fixed point (as in asexual evolution), then the mixing is fast and in the case of multiple stable fixed points (as in sexual evolution), the mixing is slow.

Exponential-family harmoniums (EFHs), which extend restricted Boltzmann machines (RBMs) from Bernoulli random variables to other exponential families (Welling et al., 2005), are generative models that can be trained with unsupervised-learning techniques, like contrastive divergence (Hinton et al., 2006; Hinton, 2002), as density estimators for static data. Methods for extending RBMs--and likewise EFHs--to data with temporal dependencies have been proposed previously (Sutskever and Hinton, 2007; Sutskever et al., 2009), the learning procedure being validated by qualitative assessment of the generative model. Here we propose and justify, from a very different perspective, an alternative training procedure, proving sufficient conditions for optimal inference under that procedure. The resulting algorithm can be learned with only forward passes through the data--backprop-through-time is not required, as in previous approaches. The proof exploits a recent result about information retention in density estimators (Makin and Sabes, 2015), and applies it to a "recurrent EFH" (rEFH) by induction. Finally, we demonstrate optimality by simulation, testing the rEFH: (1) as a filter on training data generated with a linear dynamical system, the position of which is noisily reported by a population of "neurons" with Poisson-distributed spike counts; and (2) with the qualitative experiments proposed by Sutskever et al. (2009).

The following interview is one of many included in the report. As part of our ongoing series of interviews surveying the frontiers of machine intelligence, I recently interviewed Brendan Frey. Frey is a co-founder of Deep Genomics, a professor at the University of Toronto and a co-founder of its Machine Learning Group, a senior fellow of the Neural Computation program at the Canadian Institute for Advanced Research, and a fellow of the Royal Society of Canada. His work focuses on using machine learning to understand the genome and to realize new possibilities in genomic medicine. Brendan Frey: I completed my Ph.D. with Geoff Hinton in 1997.

We consider partially observable Markov decision processes (POMDPs) with a set of target states and positive integer costs associated with every transition. The traditional optimization objective (stochastic shortest path) asks to minimize the expected total cost until the target set is reached. We extend the traditional framework of POMDPs to model energy consumption, which represents a hard constraint. The energy levels may increase and decrease with transitions, and the hard constraint requires that the energy level must remain positive in all steps till the target is reached. First, we present a novel algorithm for solving POMDPs with energy levels, developing on existing POMDP solvers and using RTDP as its main method. Our second contribution is related to policy representation. For larger POMDP instances the policies computed by existing solvers are too large to be understandable. We present an automated procedure based on machine learning techniques that automatically extracts important decisions of the policy allowing us to compute succinct human readable policies. Finally, we show experimentally that our algorithm performs well and computes succinct policies on a number of POMDP instances from the literature that were naturally enhanced with energy levels.

Human communication takes many forms, including speech, text and instructional videos. It typically has an underlying structure, with a starting point, ending, and certain objective steps between them. In this paper, we consider instructional videos where there are tens of millions of them on the Internet. We propose a method for parsing a video into such semantic steps in an unsupervised way. Our method is capable of providing a semantic "storyline" of the video composed of its objective steps. We accomplish this using both visual and language cues in a joint generative model. Our method can also provide a textual description for each of the identified semantic steps and video segments. We evaluate our method on a large number of complex YouTube videos and show that our method discovers semantically correct instructions for a variety of tasks.