We consider the problem of learning the preferences of a heterogeneous population by observing choices from an assortment of products, ads, or other offerings. Our observation model takes a form common in assortment planning applications: each arriving customer is offered an assortment consisting of a subset of all possible offerings; we observe only the assortment and the customer's single choice. In this paper we propose a mixture choice model with a natural underlying low-dimensional structure, and show how to estimate its parameters. In our model, the preferences of each customer or segment follow a separate parametric choice model, but the underlying structure of these parameters over all the models has low dimension. We show that a nuclear-norm regularized maximum likelihood estimator can learn the preferences of all customers using a number of observations much smaller than the number of item-customer combinations. This result shows the potential for structural assumptions to speed up learning and improve revenues in assortment planning and customization. We provide a specialized factored gradient descent algorithm and study the success of the approach empirically.

We introduce Bayesian Poisson Tucker decomposition (BPTD) for modeling country--country interaction event data. These data consist of interaction events of the form "country $i$ took action $a$ toward country $j$ at time $t$." BPTD discovers overlapping country--community memberships, including the number of latent communities. In addition, it discovers directed community--community interaction networks that are specific to "topics" of action types and temporal "regimes." We show that BPTD yields an efficient MCMC inference algorithm and achieves better predictive performance than related models. We also demonstrate that it discovers interpretable latent structure that agrees with our knowledge of international relations.

The EM algorithm is a novel numerical method to obtain maximum likelihood estimates and is often used for practical calculations. However, many of maximum likelihood estimation problems are nonconvex, and it is known that the EM algorithm fails to give the optimal estimate by being trapped by local optima. In order to deal with this difficulty, we propose a deterministic quantum annealing EM algorithm by introducing the mathematical mechanism of quantum fluctuations into the conventional EM algorithm because quantum fluctuations induce the tunnel effect and are expected to relax the difficulty of nonconvex optimization problems in the maximum likelihood estimation problems. We show a theorem that guarantees its convergence and give numerical experiments to verify its efficiency.

In this series we've been using the empirical Bayes method to estimate batting averages of baseball players. Empirical Bayes is useful here because when we don't have a lot of information about a batter, they're "shrunken" towards the average across all players, as a natural consequence of the beta prior. When players are better, they are given more chances to bat! (Hat tip to Hadley Wickham to pointing this complication out to me). That means there's a relationship between the number of at-bats (AB) and the true batting average. For reasons I explain below, this makes our estimates systematically inaccurate.

This paper presents a novel method for controlling teams of unmanned aerial vehicles using Stochastic Optimal Control (SOC) theory. The approach consists of a centralized high-level planner that computes optimal state trajectories as velocity sequences, and a platform-specific low-level controller which ensures that these velocity sequences are met. The planning task is expressed as a centralized path-integral control problem, for which optimal control computation corresponds to a probabilistic inference problem that can be solved by efficient sampling methods. Through simulation we show that our SOC approach (a) has significant benefits compared to deterministic control and other SOC methods in multimodal problems with noise-dependent optimal solutions, (b) is capable of controlling a large number of platforms in real-time, and (c) yields collective emergent behaviour in the form of flight formations. Finally, we show that our approach works for real platforms, by controlling a team of three quadrotors in outdoor conditions.

Search-And-Tracking (SaT) is the problem of searching for a mobile target and tracking it once it is found. Since SaT platforms face many sources of uncertainty and operational constraints, progress in the field has been restricted to simple and unrealistic scenarios. In this paper, we propose a new hybrid approach to SaT that allows us to successfully address large-scale and complex SaT missions. The probabilistic structure of SaT is compiled into a deterministic planning model and Bayesian inference is directly incorporated in the planning mechanism. Thanks to this tight integration between automated planning and probabilistic reasoning, we are able to exploit the power of both approaches. Planning provides the tools to efficiently explore big search spaces, while Bayesian inference, by readily combining prior knowledge with observable data, allows the planner to make more informed and effective decisions. We offer experimental evidence of the potential of our approach.

We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} \sim P$ if $i, j$ are both in the community and $A_{ij} \sim Q$ otherwise, for two known probability distributions $P$ and $Q$. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case ($P={{\rm Bern}}(p)$ and $Q={{\rm Bern}}(q)$ with $p>q$) and the Gaussian case ($P=\mathcal{N}(\mu,1)$ and $Q=\mathcal{N}(0,1)$ with $\mu>0$), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If $K=\omega( n /\log n)$, SDP attains the information-theoretic recovery limits with sharp constants; (2) If $K=\Theta(n/\log n)$, SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If $K=o(n/\log n)$ and $K \to \infty$, SDP is order-wise suboptimal. The same critical scaling for $K$ is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of $n$ vertices partitioned into multiple communities of equal size $K$. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.

Driving styles have a great influence on vehicle fuel economy, active safety, and drivability. To recognize driving styles of path-tracking behaviors for different divers, a statistical pattern-recognition method is developed to deal with the uncertainty of driving styles or characteristics based on probability density estimation. First, to describe driver path-tracking styles, vehicle speed and throttle opening are selected as the discriminative parameters, and a conditional kernel density function of vehicle speed and throttle opening is built, respectively, to describe the uncertainty and probability of two representative driving styles, e.g., aggressive and normal. Meanwhile, a posterior probability of each element in feature vector is obtained using full Bayesian theory. Second, a Euclidean distance method is involved to decide to which class the driver should be subject instead of calculating the complex covariance between every two elements of feature vectors. By comparing the Euclidean distance between every elements in feature vector, driving styles are classified into seven levels ranging from low normal to high aggressive. Subsequently, to show benefits of the proposed pattern-recognition method, a cross-validated method is used, compared with a fuzzy logic-based pattern-recognition method. The experiment results show that the proposed statistical pattern-recognition method for driving styles based on kernel density estimation is more efficient and stable than the fuzzy logic-based method.

There are three different types of topics in machine learning, the first ones are like NLP, Computer vision, Robotics etc. and other ones are algorithms in machine learning like genetic algorithms, neural networks, bayesian networks etc and thirdly there are concepts like decision trees, random forest, PCA etc. So, how are all these topics related when I say PhD in Bayesian Networks or PhD in NLP or PhD in boosting etc?

Kernel methods are one of the mainstays of machine learning, but the problem of kernel learning remains challenging, with only a few heuristics and very little theory. This is of particular importance in methods based on estimation of kernel mean embeddings of probability measures. For characteristic kernels, which include most commonly used ones, the kernel mean embedding uniquely determines its probability measure, so it can be used to design a powerful statistical testing framework, which includes nonparametric two-sample and independence tests. In practice, however, the performance of these tests can be very sensitive to the choice of kernel and its lengthscale parameters. To address this central issue, we propose a new probabilistic model for kernel mean embeddings, the Bayesian Kernel Embedding model, combining a Gaussian process prior over the Reproducing Kernel Hilbert Space containing the mean embedding with a conjugate likelihood function, thus yielding a closed form posterior over the mean embedding. The posterior mean of our model is closely related to recently proposed shrinkage estimators for kernel mean embeddings, while the posterior uncertainty is a new, interesting feature with various possible applications. Critically for the purposes of kernel learning, our model gives a simple, closed form marginal pseudolikelihood of the observed data given the kernel hyperparameters. This marginal pseudolikelihood can either be optimized to inform the hyperparameter choice or fully Bayesian inference can be used.