Nonuniform subsampling methods are effective to reduce computational burden and maintain estimation efficiency for massive data. Existing methods mostly focus on subsampling with replacement due to its high computational efficiency. If the data volume is so large that nonuniform subsampling probabilities cannot be calculated all at once, then subsampling with replacement is infeasible to implement. This paper solves this problem using Poisson subsampling. We first derive optimal Poisson subsampling probabilities in the context of quasi-likelihood estimation under the A- and L-optimality criteria. For a practically implementable algorithm with approximated optimal subsampling probabilities, we establish the consistency and asymptotic normality of the resultant estimators. To deal with the situation that the full data are stored in different blocks or at multiple locations, we develop a distributed subsampling framework, in which statistics are computed simultaneously on smaller partitions of the full data. Asymptotic properties of the resultant aggregated estimator are investigated. We illustrate and evaluate the proposed strategies through numerical experiments on simulated and real data sets.

Bayesian statistics is a statistical theory based on the Bayesian interpretation of probability. To understand Bayesian Statistics, we need to first understand conditional probability and Bayes' theorem. Conditional probability measures the probability of an event occurring based on the fact that another event has already occurred. Just as the formula below shows, event A occurs given that event B occurred, is the division of the joint probability of event A and B and the probability of event B occurring. There are two notes regarding this formula.

The missing data problem has been broadly studied in the last few decades and has various applications in different areas such as statistics or bioinformatics. Even though many methods have been developed to tackle this challenge, most of those are imputation techniques that require multiple iterations through the data before yielding convergence. In addition, such approaches may introduce extra biases and noises to the estimated parameters. In this work, we propose novel algorithms to find the maximum likelihood estimates (MLEs) for a one-class/multiple-class randomly missing data set under some mild assumptions. As the computation is direct without any imputation, our algorithms do not require multiple iterations through the data, thus promising to be less time-consuming than other methods while maintaining superior estimation performance. We validate these claims by empirical results on various data sets of different sizes and release all codes in a GitHub repository to contribute to the research community related to this problem.

The shortcomings of maximum likelihood estimation in the context of model-based reinforcement learning have been highlighted by an increasing number of papers. When the model class is misspecified or has a limited representational capacity, model parameters with high likelihood might not necessarily result in high performance of the agent on a downstream control task. To alleviate this problem, we propose an end-to-end approach for model learning which directly optimizes the expected returns using implicit differentiation. We treat a value function that satisfies the Bellman optimality operator induced by the model as an implicit function of model parameters and show how to differentiate the function. We provide theoretical and empirical evidence highlighting the benefits of our approach in the model misspecification regime compared to likelihood-based methods.

Meta-Learning promises to enable more data-efficient inference by harnessing previous experience from related learning tasks. While existing meta-learning methods help us to improve the accuracy of our predictions in face of data scarcity, they fail to supply reliable uncertainty estimates, often being grossly overconfident in their predictions. Addressing these shortcomings, we introduce a novel meta-learning framework, called F-PACOH, that treats meta-learned priors as stochastic processes and performs meta-level regularization directly in the function space. This allows us to directly steer the probabilistic predictions of the meta-learner towards high epistemic uncertainty in regions of insufficient meta-training data and, thus, obtain well-calibrated uncertainty estimates. Finally, we showcase how our approach can be integrated with sequential decision making, where reliable uncertainty quantification is imperative. In our benchmark study on meta-learning for Bayesian Optimization (BO), F-PACOH significantly outperforms all other meta-learners and standard baselines. Even in a challenging lifelong BO setting, where optimization tasks arrive one at a time and the meta-learner needs to build up informative prior knowledge incrementally, our proposed method demonstrates strong positive transfer.

In this article, we'll talk about Microsoft AI, the pathway to learn for beginners who are curious to explore the Microsoft AI Platforms, various functionalities and features supported by Machine Learning Studio in Azure, and the processes to train and better the Machine Learning Models with Azure. We also learn about different algorithms and thus gain the overall knowledge to get started and work with Microsoft Azure AI. Check out the official website of the summit to register as an attendee or to be a speaker and share your knowledge with the community. Microsoft AI is a powerful framework that enables organizations, researchers, and non-profits to use AI technologies with its powerful framework which offers services and features across domains of Machine Learning, Robotics, Data Science, IoT, and many more. One of the advantages of Azure can be realized with this example of how Machine Learning becomes more scalable in the Cloud even while working on Notebooks.

A general Bayesian framework is introduced for mixture modelling and inference with real-valued time series. At the top level, the state space is partitioned via the choice of a discrete context tree, so that the resulting partition depends on the values of some of the most recent samples. At the bottom level, a different model is associated with each region of the partition. This defines a very rich and flexible class of mixture models, for which we provide algorithms that allow for efficient, exact Bayesian inference. In particular, we show that the maximum a posteriori probability (MAP) model (including the relevant MAP context tree partition) can be precisely identified, along with its exact posterior probability. The utility of this general framework is illustrated in detail when a different autoregressive (AR) model is used in each state-space region, resulting in a mixture-of-AR model class. The performance of the associated algorithmic tools is demonstrated in the problems of model selection and forecasting on both simulated and real-world data, where they are found to provide results as good or better than state-of-the-art methods.

This paper investigates the theoretical and empirical performance of Fisher-Pitman-type permutation tests for assessing the equality of unknown Poisson mixture distributions. Building on nonparametric maximum likelihood estimators (NPMLEs) of the mixing distribution, these tests are theoretically shown to be able to adapt to complicated unspecified structures of count data and also consistent against their corresponding ANOVA-type alternatives; the latter is a result in parallel to classic claims made by Robinson (Robinson, 1973). The studied methods are then applied to a single-cell RNA-seq data obtained from different cell types from brain samples of autism subjects and healthy controls; empirically, they unveil genes that are differentially expressed between autism and control subjects yet are missed using common tests. For justifying their use, rate optimality of NPMLEs is also established in settings similar to nonparametric Gaussian (Wu and Yang, 2020a) and binomial mixtures (Tian et al., 2017; Vinayak et al., 2019).

Interacting particle or agent systems that display a rich variety of collection motions are ubiquitous in science and engineering. A fundamental and challenging goal is to understand the link between individual interaction rules and collective behaviors. In this paper, we study the data-driven discovery of distance-based interaction laws in second-order interacting particle systems. We propose a learning approach that models the latent interaction kernel functions as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of interaction kernel function with the pointwise uncertainty quantification, and the other one is the inference of unknown parameters in the non-collective forces of the system. We formulate learning interaction kernel functions as a statistical inverse problem and provide a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. We provide a finite-sample analysis, showing that our posterior mean estimator converges at an optimal rate equal to the one in the classical 1-dimensional Kernel Ridge regression. Numerical results on systems that exhibit different collective behaviors demonstrate efficient learning of our approach from scarce noisy trajectory data.

Integrating discrete probability distributions and combinatorial optimization problems into neural networks has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable: it only requires the ability to compute the most probable states; and does not rely on smooth relaxations. The framework encompasses several approaches, such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP. Moreover, we show that I-MLE simplifies to maximum likelihood estimation when used in some recently studied learning settings that involve combinatorial solvers. Experiments on several datasets suggest that I-MLE is competitive with and often outperforms existing approaches which rely on problem-specific relaxations.