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.

Sign languages aren't easy to learn and are even harder to teach. They use not just hand gestures but also mouthings, facial expressions and body posture to communicate meaning. This complexity means professional teaching programmes are still rare and often expensive. But this could all change soon, with a little help from artificial intelligence (AI). My colleagues and I are working on software for teaching yourself sign languages in an automated, intuitive way.

We propose a principle for exploring context in machine learning models. Starting with a simple assumption that each observation (random variables) may or may not depend on its context (conditional variables), a conditional probability distribution is decomposed into two parts: context-free and context-sensitive. Then by employing the log-linear word production model for relating random variables to their embedding space representation and making use of the convexity of natural exponential function, we show that the embedding of an observation can also be decomposed into a weighted sum of two vectors, representing its context-free and context-sensitive parts, respectively. This simple treatment of context provides a unified view of many existing deep learning models, leading to revisions of these models able to achieve significant performance boost. Specifically, our upgraded version of a recent sentence embedding model (Arora et al., 2017) not only outperforms the original one by a large margin, but also leads to a new, principled approach for compositing the embeddings of bag-of-words features, as well as a new architecture for modeling attention in deep neural networks. More surprisingly, our new principle provides a novel understanding of the gates and equations defined by the long short term memory (LSTM) model, which also leads to a new model that is able to converge significantly faster and achieve much lower prediction errors. Furthermore, our principle also inspires a new type of generic neural network layer that better resembles real biological neurons than the traditional linear mapping plus nonlinear activation based architecture. Its multi-layer extension provides a new principle for deep neural networks which subsumes residual network (ResNet) as its special case, and its extension to convolutional neutral network model accounts for irrelevant input (e.g., background in an image) in addition to filtering. Our models are validated through a series of benchmark datasets and we show that in many cases, simply replacing existing layers with our context-aware counterparts is sufficient to significantly improve the results.

Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with data-driven models while obeying the constraints of the problem at hand. This work employs a convolutional encoder-decoder neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the Boltzmann-Gibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to out-of-distribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.

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.

That means the pooling layer computes a feature vector of size 128 which is passed into dense layers of the feedforward network as we mentioned above. The overall structure of the DNN can be understood as a preprocessor defined in the first part that is being trained to map text sequences into feature vectors in such a way that the weights of the second part can be trained to obtain optimal classification results from the overall network. More details on the implementation and text preprocessing can be found in my GitHub repository for this project. I trained this network for 10 epochs with a batch size of 128 using an 80-20 training/hold-out set. A couple of notes on additional parameters: The vast majority of documents in this collection is of length 5000 or less. So for the maximum input sequence length for the DNN I chose 5000 words. There are roughly 100,000 unique words in this collection of documents. I arbitrarily limited the dictionary that the DNN can learn to 25% of that: 25,000 words. Finally, for the embedding dimension, I chose 300 simply because that is the default embedding dimension for both word2vec and GloVe.

In an effort to improve the performance of deep neural networks in data-scarce, non-i.i.d., or unsupervised settings, much recent research has been devoted to encoding invariance under symmetry transformations into neural network architectures. We treat the neural network input and output as random variables, and consider group invariance from the perspective of probabilistic symmetry. Drawing on tools from probability and statistics, we establish a link between functional and probabilistic symmetry, and obtain generative functional representations of joint and conditional probability distributions that are invariant or equivariant under the action of a compact group. Those representations completely characterize the structure of neural networks that can be used to model such distributions and yield a general program for constructing invariant stochastic or deterministic neural networks. We develop the details of the general program for exchangeable sequences and arrays, recovering a number of recent examples as special cases.

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.