Gibbs sampling is a MCMC method to draw samples from a complex distribution (usually a posterior in Bayesian inference). In this post I aim to show how to do Gibbs sampling using Edward, "a Python library for probabilistic modeling". If you are new to Edward, you can install the package by following up these steps. In above code x0 and x1 are two place holders for samples of X0X 0X0 and X1X 1X1 from previous iteration. Edward helped us to write Gibbs sampling with less than 10 line of codes.

We consider a particular instance of a common problem in recommender systems: using a database of book reviews to inform user-targeted recommendations. In our dataset, books are categorized into genres and sub-genres. To exploit this nested taxonomy, we use a hierarchical model that enables information pooling across across similar items at many levels within the genre hierarchy. The main challenge in deploying this model is computational: the data sizes are large, and fitting the model at scale using off-the-shelf maximum likelihood procedures is prohibitive. To get around this computational bottleneck, we extend a moment-based fitting procedure proposed for fitting single-level hierarchical models to the general case of arbitrarily deep hierarchies. This extension is an order of magnetite faster than standard maximum likelihood procedures. The fitting method can be deployed beyond recommender systems to general contexts with deeply-nested hierarchical generalized linear mixed models.

In statistics and machine learning, approximation of an intractable integration is often achieved by using the unbiased Monte Carlo estimator, but the variances of the estimation are generally high in many applications. Control variates approaches are well-known to reduce the variance of the estimation. These control variates are typically constructed by employing predefined parametric functions or polynomials, determined by using those samples drawn from the relevant distributions. Instead, we propose to construct those control variates by learning neural networks to handle the cases when test functions are complex. In many applications, obtaining a large number of samples for Monte Carlo estimation is expensive, which may result in overfitting when training a neural network. We thus further propose to employ auxiliary random variables induced by the original ones to extend data samples for training the neural networks. We apply the proposed control variates with augmented variables to thermodynamic integration and reinforcement learning. Experimental results demonstrate that our method can achieve significant variance reduction compared with other alternatives.

This paper analyzes asymptotic performance of a regularized multi-task learning model where task parameters are optimized jointly. If tasks are closely related, empirical work suggests multi-task learning models to outperform single-task ones in finite sample cases. As data size grows indefinitely, we show the learned multi-classifier to optimize an average misclassification error function which depicts the risk of applying multi-task learning algorithm to making decisions. This technique conclusion demonstrates the regularized multi-task learning model to be able to produce reliable decision rule for each task in the sense that it will asymptotically converge to the corresponding Bayes rule. Also, we find the interaction effect between tasks vanishes as data size growing indefinitely, which is quite different from the behavior in finite sample cases.

According to Dennett, the same system may be described using a `physical' (mechanical) explanatory stance, or using an `intentional' (belief- and goal-based) explanatory stance. Humans tend to find the physical stance more helpful for certain systems, such as planets orbiting a star, and the intentional stance for others, such as living animals. We define a formal counterpart of physical and intentional stances within computational theory: a description of a system as either a device, or an agent, with the key difference being that `devices' are directly described in terms of an input-output mapping, while `agents' are described in terms of the function they optimise. Bayes' rule can then be applied to calculate the subjective probability of a system being a device or an agent, based only on its behaviour. We illustrate this using the trajectories of an object in a toy grid-world domain.

Hidden tree Markov models allow learning distributions for tree structured data while being interpretable as nondeterministic automata. We provide a concise summary of the main approaches in literature, focusing in particular on the causality assumptions introduced by the choice of a specific tree visit direction. We will then sketch a novel non-parametric generalization of the bottom-up hidden tree Markov model with its interpretation as a nondeterministic tree automaton with infinite states.

It is no doubt that the sub-field of machine learning / artificial intelligence has increasingly gained more popularity in the past couple of years. As Big Data is the hottest trend in the tech industry at the moment, machine learning is incredibly powerful to make predictions or calculated suggestions based on large amounts of data. Some of the most common examples of machine learning are Netflix's algorithms to make movie suggestions based on movies you have watched in the past or Amazon's algorithms that recommend books based on books you have bought before. So if you want to learn more about machine learning, how do you start? For me, my first introduction is when I took an Artificial Intelligence class when I was studying abroad in Copenhagen. My lecturer is a full-time Applied Math and CS professor at the Technical University of Denmark, in which his research areas are logic and artificial, focusing primarily on the use of logic to model human-like planning, reasoning and problem solving.

The demand of metals by modern technology has been shifting from common base metals to a variety of minor metals, such as cobalt or indium. The industrial importance and limited geological availability of some minor metals have led to them being considered more "critical," and there is a growing interest in such critical metals and their producing companies. In this research, we create a novel framework, Dynamic Advisor-Based Ensemble (dynABE), to predict the stock trend of major critical metal producers. Specifically, dynABE first utilizes domain knowledge to group the features into different "advisors," each advisor dealing with a particular economic sector. Then through ensembles of weak classifiers, each advisor produces a prediction result, and all the advisors are combined again in a biased online update fashion to dynamically make the final prediction. Based on a misclassification error of 32% for Jinchuan Group's stock (HKG: 2362), we further test a simple stock trading strategy, which leads to a back-tested return of 296%, or an excess return of 130% within one year. In addition, the feature set selected by dynABE also suggests potentially influential factors to metal criticality, because stock prices of major producers influence metal production. Therefore, not only does this research propose a novel framework for specialized stock trend prediction, it also provides domain insights into dynamic features that potentially influence metal criticality.

Causally insufficient structures (models with latent or hidden variables, or with confounding etc.) of joint probability distributions have been subject of intense study not only in statistics, but also in various AI systems. In AI, belief networks, being representations of joint probability distribution with an underlying directed acyclic graph structure, are paid special attention due to the fact that efficient reasoning (uncertainty propagation) methods have been developed for belief network structures. Algorithms have been therefore developed to acquire the belief network structure from data. As artifacts due to variable hiding negatively influence the performance of derived belief networks, models with latent variables have been studied and several algorithms for learning belief network structure under causal insufficiency have also been developed. Regrettably, some of them are known already to be erroneous (e.g. IC algorithm of [Pearl:Verma:91]. This paper is devoted to another algorithm, the Fast Causal Inference (FCI) Algorithm of [Spirtes:93]. It is proven by a specially constructed example that this algorithm, as it stands in [Spirtes:93], is also erroneous. Fundamental reason for failure of this algorithm is the temporary introduction of non-real links between nodes of the network with the intention of later removal. While for trivial dependency structures these non-real links may be actually removed, this may not be the case for complex ones, e.g. for the case described in this paper. A remedy of this failure is proposed.

We consider the problem of how to improve automatic target recognition by fusing the naive sensor-level classification decisions with "intuition," or context, in a mathematically principled way. This is a general approach that is compatible with many definitions of context, but for specificity, we consider context as co-occurrence in imagery. In particular, we consider images that contain multiple objects identified at various confidence levels. We learn the patterns of co-occurrence in each context, then use these patterns as hyper-parameters for a Hierarchical Bayesian Model. The result is that low-confidence sensor classification decisions can be dramatically improved by fusing those readings with context. We further use hyperpriors to address the case where multiple contexts may be appropriate. We also consider the Bayesian Network, an alternative to the Hierarchical Bayesian Model, which is computationally more efficient but assumes that context and sensor readings are uncorrelated.