This paper introduces a novel mixture model-based approach for simultaneous clustering and optimal segmentation of functional data which are curves presenting regime changes. The proposed model consists in a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associated with a cluster, and within each cluster, each piecewise polynomial component is associated with a regime (i.e., a segment). We derive two approaches for learning the model parameters. The former is an estimation approach and consists in maximizing the observed-data likelihood via a dedicated expectation-maximization (EM) algorithm. A fuzzy partition of the curves in K clusters is then obtained at convergence by maximizing the posterior cluster probabilities. The latter however is a classification approach and optimizes a specific classification likelihood criterion through a dedicated classification expectation-maximization (CEM) algorithm. The optimal curve segmentation is performed by using dynamic programming. In the classification approach, both the curve clustering and the optimal segmentation are performed simultaneously as the CEM learning proceeds. We show that the classification approach is the probabilistic version that generalizes the deterministic K-means-like algorithm proposed in H\'ebrail et al. (2010). The proposed approach is evaluated using simulated curves and real-world curves. Comparisons with alternatives including regression mixture models and the K-means like algorithm for piecewise regression demonstrate the effectiveness of the proposed approach.

The systematic biases seen in people's probability judgments are typically taken as evidence that people do not reason about probability using the rules of probability theory, but instead use heuristics which sometimes yield reasonable judgments and sometimes systematic biases. This view has had a major impact in economics, law, medicine, and other fields; indeed, the idea that people cannot reason with probabilities has become a widespread truism. We present a simple alternative to this view, where people reason about probability according to probability theory but are subject to random variation or noise in the reasoning process. In this account the effect of noise is cancelled for some probabilistic expressions: analysing data from two experiments we find that, for these expressions, people's probability judgments are strikingly close to those required by probability theory. For other expressions this account produces systematic deviations in probability estimates. These deviations explain four reliable biases in human probabilistic reasoning (conservatism, subadditivity, conjunction and disjunction fallacies). These results suggest that people's probability judgments embody the rules of probability theory, and that biases in those judgments are due to the effects of random noise.

Transposable data represents interactions among two sets of entities, and are typically represented as a matrix containing the known interaction values. Additional side information may consist of feature vectors specific to entities corresponding to the rows and/or columns of such a matrix. Further information may also be available in the form of interactions or hierarchies among entities along the same mode (axis). We propose a novel approach for modeling transposable data with missing interactions given additional side information. The interactions are modeled as noisy observations from a latent noise free matrix generated from a matrix-variate Gaussian process. The construction of row and column covariances using side information provides a flexible mechanism for specifying a-priori knowledge of the row and column correlations in the data. Further, the use of such a prior combined with the side information enables predictions for new rows and columns not observed in the training data. In this work, we combine the matrix-variate Gaussian process model with low rank constraints. The constrained Gaussian process approach is applied to the prediction of hidden associations between genes and diseases using a small set of observed associations as well as prior covariances induced by gene-gene interaction networks and disease ontologies. The proposed approach is also applied to recommender systems data which involves predicting the item ratings of users using known associations as well as prior covariances induced by social networks. We present experimental results that highlight the performance of constrained matrix-variate Gaussian process as compared to state of the art approaches in each domain.

We propose a mixture of latent trait models with common slope parameters (MCLT) for model-based clustering of high-dimensional binary data, a data type for which few established methods exist. Recent work on clustering of binary data, based on a $d$-dimensional Gaussian latent variable, is extended by incorporating common factor analyzers. Accordingly, our approach facilitates a low-dimensional visual representation of the clusters. We extend the model further by the incorporation of random block effects. The dependencies in each block are taken into account through block-specific parameters that are considered to be random variables. A variational approximation to the likelihood is exploited to derive a fast algorithm for determining the model parameters. Our approach is demonstrated on real and simulated data.

Efficient collaborative decision making is an important challenge for multiagent systems. Finding optimal joint actions is especially challenging when each agent has only imperfect information about the state of its environment. Such problems can be modeled as collaborative Bayesian games in which each agent receives private information in the form of its type. However, representing and solving such games requires space and computation time exponential in the number of agents. This article introduces collaborative graphical Bayesian games (CGBGs), which facilitate more efficient collaborative decision making by decomposing the global payoff function as the sum of local payoff functions that depend on only a few agents. We propose a framework for the efficient solution of CGBGs based on the insight that they posses two different types of independence, which we call agent independence and type independence. In particular, we present a factor graph representation that captures both forms of independence and thus enables efficient solutions. In addition, we show how this representation can provide leverage in sequential tasks by using it to construct a novel method for decentralized partially observable Markov decision processes. Experimental results in both random and benchmark tasks demonstrate the improved scalability of our methods compared to several existing alternatives.

In crowd labeling, a large amount of unlabeled data instances are outsourced to a crowd of workers. Workers will be paid for each label they provide, but the labeling requester usually has only a limited amount of the budget. Since data instances have different levels of labeling difficulty and workers have different reliability, it is desirable to have an optimal policy to allocate the budget among all instance-worker pairs such that the overall labeling accuracy is maximized. We consider categorical labeling tasks and formulate the budget allocation problem as a Bayesian Markov decision process (MDP), which simultaneously conducts learning and decision making. Using the dynamic programming (DP) recurrence, one can obtain the optimal allocation policy. However, DP quickly becomes computationally intractable when the size of the problem increases. To solve this challenge, we propose a computationally efficient approximate policy, called optimistic knowledge gradient policy. Our MDP is a quite general framework, which applies to both pull crowdsourcing marketplaces with homogeneous workers and push marketplaces with heterogeneous workers. It can also incorporate the contextual information of instances when they are available. The experiments on both simulated and real data show that the proposed policy achieves a higher labeling accuracy than other existing policies at the same budget level.

Cellular regulatory dynamics is driven by large and intricate networks of interactions at the molecular scale, whose sheer size obfuscates understanding. In light of limited experimental data, many parameters of such dynamics are unknown, and thus models built on the detailed, mechanistic viewpoint overfit and are not predictive. At the other extreme, simple ad hoc models of complex processes often miss defining features of the underlying systems. Here we propose an approach that instead constructs phenomenological, coarse-grained models of network dynamics that automatically adapt their complexity to the amount of available data. Such adaptive models lead to accurate predictions even when microscopic details of the studied systems are unknown due to insufficient data. The approach is computationally tractable, even for a relatively large number of dynamical variables, allowing its software realization, named Sir Isaac, to make successful predictions even when important dynamic variables are unobserved. For example, it matches the known phase space structure for simulated planetary motion data, avoids overfitting in a complex biological signaling system, and produces accurate predictions for a yeast glycolysis model with only tens of data points and over half of the interacting species unobserved.

Central to robot exploration and mapping is the task of persistent localization in environmental fields characterized by spatially correlated measurements. This paper presents a Gaussian process localization (GP-Localize) algorithm that, in contrast to existing works, can exploit the spatially correlated field measurements taken during a robot's exploration (instead of relying on prior training data) for efficiently and scalably learning the GP observation model online through our proposed novel online sparse GP. As a result, GP-Localize is capable of achieving constant time and memory (i.e., independent of the size of the data) per filtering step, which demonstrates the practical feasibility of using GPs for persistent robot localization and autonomy. Empirical evaluation via simulated experiments with real-world datasets and a real robot experiment shows that GP-Localize outperforms existing GP localization algorithms.

A restricted Boltzmann machine (RBM) is a probabilistic graphical model with bipartite interactions between an observed set and a hidden set of units [see Smolensky, 1986, Freund and Haussler, 1991, Hinton, 2002, 2010]. A characterizing property of these models is that the observed units are independent given the states of the hidden units and vice versa. This is a consequence of the bipartiteness of the interaction graph and does not depend on the units' state spaces. Typically RBMs are defined with binary units, but other types of units have also been considered, including continuous, discrete, and mixed type units [see Welling et al., 2005, Marks and Movellan, 2001, Salakhutdinov et al., 2007, Dahl et al., 2012, Tran et al., 2011]. We study discrete RBMs, also called multinomial or softmax RBMs, which are special types of exponential family harmoniums [Welling et al., 2005].

Recently, there has been a growing interest in modeling planning with information constraints. Accordingly, an agent maximizes a regularized expected utility known as the free energy, where the regularizer is given by the information divergence from a prior to a posterior policy. While this approach can be justified in various ways, including from statistical mechanics and information theory, it is still unclear how it relates to decisionmaking against adversarial environments. This connection has previously been suggested in work relating the free energy to risk-sensitive control and to extensive form games. Here, we show that a single-agent free energy optimization is equivalent to a game between the agent and an imaginary adversary. The adversary can, by paying an exponential penalty, generate costs that diminish the decision maker's payoffs. It turns out that the optimal strategy of the adversary consists in choosing costs so as to render the decision maker indifferent among its choices, which is a definining property of a Nash equilibrium, thus tightening the connection between free energy optimization and game theory. Keywords: bounded rationality, free energy, game theory, Legendre-Fenchel transform.