I have made some progress with my work on combining independent evidence using a Bayesian approach but eschewing standard Bayesian updating. I found a neat analytical way of doing this, to a very good approximation, in cases where each estimate of a parameter corresponds to the ratio of two variables each determined with normal error, the fractional uncertainty in the numerator and denominator variables differing between the types of evidence. This seems a not uncommon situation in science, and it is a good approximation to that which exists when estimating climate sensitivity. I have had a manuscript in which I develop and test this method accepted by the Journal of Statistical Planning and Inference (for a special issue on Confidence Distributions edited by Tore Schweder and Nils Hjort). Frequentist coverage is almost exact using my analytical solution, based on combining Jeffreys' priors in quadrature, whereas Bayesian updating produces far poorer probability matching.

Graphical causal models are an important tool for knowledge discovery because they can represent both the causal relations between variables and the multivariate probability distributions over the data. Once learned, causal graphs can be used for classification, feature selection and hypothesis generation, while revealing the underlying causal network structure and thus allowing for arbitrary likelihood queries over the data. However, current algorithms for learning sparse directed graphs are generally designed to handle only one type of data (continuous-only or discrete-only), which limits their applicability to a large class of multi-modal biological datasets that include mixed type variables. To address this issue, we developed new methods that modify and combine existing methods for finding undirected graphs with methods for finding directed graphs. These hybrid methods are not only faster, but also perform better than the directed graph estimation methods alone for a variety of parameter settings and data set sizes. Here, we describe a new conditional independence test for learning directed graphs over mixed data types and we compare performances of different graph learning strategies on synthetic data.

The classification of multivariate functional data is an important task in scientific research. Unlike point-wise data, functional data are usually classified by their shapes rather than by their scales. We define an outlyingness matrix by extending directional outlyingness, an effective measure of the shape variation of curves that combines the direction of outlyingness with conventional depth. We propose two classifiers based on directional outlyingness and the outlyingness matrix, respectively. Our classifiers provide better performance compared with existing depth-based classifiers when applied on both univariate and multivariate functional data from simulation studies. We also test our methods on two data problems: speech recognition and gesture classification, and obtain results that are consistent with the findings from the simulated data.

In the previous post, we discussed the motivation behind Maximum Likelihood Estimate and how to calculate it. We also learned a few tricks about calculating the log likelihood of a function by citing the application of monotonic functions, and how they make the entire process of estimating the critical points of a function much easier as they preserve those critical points. Put the #tails (0) and #heads (2) in the equation of theta_MLE, This result tells us that the probability of next flip being Tails is 0 (i.e., it predicts that no flip is ever gonna turn up Tails the coin is always going to show Heads), and it is glaringly obvious that this is not the case (barring the extreme case where the coin is heavily loaded). Now, this poses a big problem in the Parameter Estimation process because it does not give us the accurate probability of the next flip. We know that even a fair coin has a 25% chance of showing two Heads in a row (0.5 x 0.5 0.25).

In this paper we study the problem of noisy tensor completion for tensors that admit a canonical polyadic or CANDECOMP/PARAFAC (CP) decomposition with one of the factors being sparse. We present general theoretical error bounds for an estimate obtained by using a complexity-regularized maximum likelihood principle and then instantiate these bounds for the case of additive white Gaussian noise. We also provide an ADMM-type algorithm for solving the complexity-regularized maximum likelihood problem and validate the theoretical finding via experiments on synthetic data set.

Reinforcement learning is considered to be a strong AI paradigm which can be used to teach machines through interaction with the environment and learning from their mistakes. Despite its perceived utility, it has not yet been successfully applied in automotive applications. Motivated by the successful demonstrations of learning of Atari games and Go by Google DeepMind, we propose a framework for autonomous driving using deep reinforcement learning. This is of particular relevance as it is difficult to pose autonomous driving as a supervised learning problem due to strong interactions with the environment including other vehicles, pedestrians and roadworks. As it is a relatively new area of research for autonomous driving, we provide a short overview of deep reinforcement learning and then describe our proposed framework. It incorporates Recurrent Neural Networks for information integration, enabling the car to handle partially observable scenarios. It also integrates the recent work on attention models to focus on relevant information, thereby reducing the computational complexity for deployment on embedded hardware. The framework was tested in an open source 3D car racing simulator called TORCS. Our simulation results demonstrate learning of autonomous maneuvering in a scenario of complex road curvatures and simple interaction of other vehicles.

When faced with high frequency streams of data, clustering raises theoretical and algorithmic pitfalls. We introduce a new and adaptive online clustering algorithm relying on a quasi-Bayesian approach, with a dynamic (\emph{i.e.}, time-dependent) estimation of the (unknown and changing) number of clusters. We prove that our approach is supported by minimax regret bounds. We also provide an RJMCMC-flavored implementation (called PACBO) for which we give a convergence guarantee. Finally, numerical experiments illustrate the potential of our procedure.

We study the problem of detecting a change in the mean of one-dimensional Gaussian process data. This problem is investigated in the setting of increasing domain (customarily employed in time series analysis) and in the setting of fixed domain (typically arising in spatial data analysis). We propose a detection method based on the generalized likelihood ratio test (GLRT), and show that our method achieves nearly asymptotically optimal rate in the minimax sense, in both settings. The salient feature of the proposed method is that it exploits in an efficient way the data dependence captured by the Gaussian process covariance structure. When the covariance is not known, we propose the plug-in GLRT method and derive conditions under which the method remains asymptotically near optimal. By contrast, the standard CUSUM method, which does not account for the covariance structure, is shown to be asymptotically optimal only in the increasing domain. Our algorithms and accompanying theory are applicable to a wide variety of covariance structures, including the Matern class, the powered exponential class, and others. The plug-in GLRT method is shown to perform well for maximum likelihood estimators with a dense covariance matrix.

Causal effect estimation from observational data is an important and much studied research topic. The instrumental variable (IV) and local causal discovery (LCD) patterns are canonical examples of settings where a closed-form expression exists for the causal effect of one variable on another, given the presence of a third variable. Both rely on faithfulness to infer that the latter only influences the target effect via the cause variable. In reality, it is likely that this assumption only holds approximately and that there will be at least some form of weak interaction. This brings about the paradoxical situation that, in the large-sample limit, no predictions are made, as detecting the weak edge invalidates the setting. We introduce an alternative approach by replacing strict faithfulness with a prior that reflects the existence of many 'weak' (irrelevant) and 'strong' interactions. We obtain a posterior distribution over the target causal effect estimator which shows that, in many cases, we can still make good estimates. We demonstrate the approach in an application on a simple linear-Gaussian setting, using the MultiNest sampling algorithm, and compare it with established techniques to show our method is robust even when strict faithfulness is violated.

The pseudo-likelihood method is one of the most popular algorithms for learning sparse binary pairwise Markov networks. In this paper, we formulate the $L_1$ regularized pseudo-likelihood problem as a sparse multiple logistic regression problem. In this way, many insights and optimization procedures for sparse logistic regression can be applied to the learning of discrete Markov networks. Specifically, we use the coordinate descent algorithm for generalized linear models with convex penalties, combined with strong screening rules, to solve the pseudo-likelihood problem with $L_1$ regularization. Therefore a substantial speedup without losing any accuracy can be achieved. Furthermore, this method is more stable than the node-wise logistic regression approach on unbalanced high-dimensional data when penalized by small regularization parameters. Thorough numerical experiments on simulated data and real world data demonstrate the advantages of the proposed method.