This paper is a step-by-step tutorial for fitting a mixture distribution to data. It merely assumes the reader has the background of calculus and linear algebra. Other required background is briefly reviewed before explaining the main algorithm. In explaining the main algorithm, first, fitting a mixture of two distributions is detailed and examples of fitting two Gaussians and Poissons, respectively for continuous and discrete cases, are introduced. Thereafter, fitting several distributions in general case is explained and examples of several Gaussians (Gaussian Mixture Model) and Poissons are again provided. Model-based clustering, as one of the applications of mixture distributions, is also introduced. Numerical simulations are also provided for both Gaussian and Poisson examples for the sake of better clarification.

Machine learning is using data we have (known as training data) to learn a function that we can use to make predictions about data we don't have yet. The simplest example of this is linear regression, where we learn the slope and intercept of a line so we can predict the vertical position of points from their horizontal position. This is shown below, the training data are the blue points and the learnt function is the red line. Machine learning is an extension of linear regression in a few ways. Secondly, modern ML uses much more powerful methods for extracting patterns of which deep learning is only one of many.

Generative models, either by simple clustering algorithms or deep neural network architecture, have been developed as a probabilistic estimation method for dimension reduction or to model the underlying properties of data structures. Although their apparent use has largely been limited to image recognition and classification, generative machine learning algorithms can be a powerful tool for travel behaviour research. In this paper, we examine the generative machine learning approach for analyzing multiple discrete-continuous (MDC) travel behaviour data to understand the underlying heterogeneity and correlation, increasing the representational power of such travel behaviour models. We show that generative models are conceptually similar to choice selection behaviour process through information entropy and variational Bayesian inference. Specifically, we consider a restricted Boltzmann machine (RBM) based algorithm with multiple discrete-continuous layer, formulated as a variational Bayesian inference optimization problem. We systematically describe the proposed machine learning algorithm and develop a process of analyzing travel behaviour data from a generative learning perspective. We show parameter stability from model analysis and simulation tests on an open dataset with multiple discrete-continuous dimensions and a size of 293,330 observations. For interpretability, we derive analytical methods for conditional probabilities as well as elasticities. Our results indicate that latent variables in generative models can accurately represent joint distribution consistently w.r.t multiple discrete-continuous variables. Lastly, we show that our model can generate statistically similar data distributions for travel forecasting and prediction.

Bayesian networks in their Factor Graph Reduced Normal Form (FGrn) are a powerful paradigm for implementing inference graphs. Unfortunately, the computational and memory costs of these networks may be considerable, even for relatively small networks, and this is one of the main reasons why these structures have often been underused in practice. In this work, through a detailed algorithmic and structural analysis, various solutions for cost reduction are proposed. An online version of the classic batch learning algorithm is also analyzed, showing very similar results (in an unsupervised context); which is essential even if multilevel structures are to be built. The solutions proposed, together with the possible online learning algorithm, are included in a C++ library that is quite efficient, especially if compared to the direct use of the well-known sum-product and Maximum Likelihood (ML) algorithms. The results are discussed with particular reference to a Latent Variable Model (LVM) structure.

A regression-based BNN model is proposed to predict spatiotemporal quantities like hourly rider demand with calibrated uncertainties. The main contributions of this paper are (i) A feed-forward deterministic neural network (DetNN) architecture that predicts cyclical time series data with sensitivity to anomalous forecasting events; (ii) A Bayesian framework applying SVGD to train large neural networks for such tasks, capable of producing time series predictions as well as measures of uncertainty surrounding the predictions. Experiments show that the proposed BNN reduces average estimation error by 10% across 8 U.S. cities compared to a fine-tuned multilayer perceptron (MLP), and 4% better than the same network architecture trained without SVGD.

Human social behavior is structured by relationships. We form teams, groups, tribes, and alliances at all scales of human life. These structures guide multi-agent cooperation and competition, but when we observe others these underlying relationships are typically unobservable and hence must be inferred. Humans make these inferences intuitively and flexibly, often making rapid generalizations about the latent relationships that underlie behavior from just sparse and noisy observations. Rapid and accurate inferences are important for determining who to cooperate with, who to compete with, and how to cooperate in order to compete. Towards the goal of building machine-learning algorithms with human-like social intelligence, we develop a generative model of multi-agent action understanding based on a novel representation for these latent relationships called Composable Team Hierarchies (CTH). This representation is grounded in the formalism of stochastic games and multi-agent reinforcement learning. We use CTH as a target for Bayesian inference yielding a new algorithm for understanding behavior in groups that can both infer hidden relationships as well as predict future actions for multiple agents interacting together. Our algorithm rapidly recovers an underlying causal model of how agents relate in spatial stochastic games from just a few observations. The patterns of inference made by this algorithm closely correspond with human judgments and the algorithm makes the same rapid generalizations that people do.

Generalized Bayesian learning algorithms are increasingly popular in machine learning, due to their PAC generalization properties and flexibility. The present paper aims at providing a self-contained survey on the resulting PAC-Bayes framework and some of its main theoretical and algorithmic developments.

Domain adaptation has become a prominent problem setting in machine learning and related fields. This review asks the questions: when and how a classifier can learn from a source domain and generalize to a target domain. As for when, we review conditions that allow for cross-domain generalization error bounds. As for how, we present a categorization of approaches, divided into, what we refer to as, sample-based, feature-based and inference-based methods. Sample-based methods focus on weighting individual observations during training based on their importance to the target domain. Feature-based methods focus on mapping, projecting and representing features such that a source classifier performs well on the target domain and inference-based methods focus on alternative estimators, such as robust, minimax or Bayesian. Our categorization highlights recurring ideas and raises a number of questions important to further research.

In binary classification framework, we are interested in making cost sensitive label predictions in the presence of uniform/symmetric label noise. We first observe that $0$-$1$ Bayes classifiers are not (uniform) noise robust in cost sensitive setting. To circumvent this impossibility result, we present two schemes; unlike the existing methods, our schemes do not require noise rate. The first one uses $\alpha$-weighted $\gamma$-uneven margin squared loss function, $l_{\alpha, usq}$, which can handle cost sensitivity arising due to domain requirement (using user given $\alpha$) or class imbalance (by tuning $\gamma$) or both. However, we observe that $l_{\alpha, usq}$ Bayes classifiers are also not cost sensitive and noise robust. We show that regularized ERM of this loss function over the class of linear classifiers yields a cost sensitive uniform noise robust classifier as a solution of a system of linear equations. We also provide a performance bound for this classifier. The second scheme that we propose is a re-sampling based scheme that exploits the special structure of the uniform noise models and uses in-class probability estimates. Our computational experiments on some UCI datasets with class imbalance show that classifiers of our two schemes are on par with the existing methods and in fact better in some cases w.r.t. Accuracy and Arithmetic Mean, without using/tuning noise rate. We also consider other cost sensitive performance measures viz., F measure and Weighted Cost for evaluation.

The great challenge was to generalize the reconstruction of the arm for any background without using segmentation. For that purpose, several background images were synthetically generated and were overlaid by automated labelled masks (i.e., boolean mask of the arm in the visual field) by means of background subtraction (Figure 1(b)). An example of the results of the generated arm given the a joint angle configuration is shown in Figure 1(c). The most right generated image shows difficulties of the model to properly reconstruct the robot arm when the majority of it is outside the field of view. Anyhow, the statistical evaluation of the network, over all experiments, showed an accuracy of 84.4% when comparing the matching between the original versus the generated image mask.