Single-site Markov Chain Monte Carlo (MCMC) is a variant of MCMC in which a single coordinate in the state space is modified in each step. Structured relational models are a good candidate for this style of inference. In the single-site context, second order methods become feasible because the typical cubic costs associated with these methods is now restricted to the dimension of each coordinate. Our work, which we call Newtonian Monte Carlo (NMC), is a method to improve MCMC convergence by analyzing the first and second order gradients of the target density to determine a suitable proposal density at each point. Existing first order gradient-based methods suffer from the problem of determining an appropriate step size. Too small a step size and it will take a large number of steps to converge, while a very large step size will cause it to overshoot the high density region. NMC is similar to the Newton-Raphson update in optimization where the second order gradient is used to automatically scale the step size in each dimension. However, our objective is to find a parameterized proposal density rather than the maxima. As a further improvement on existing first and second order methods, we show that random variables with constrained supports don't need to be transformed before taking a gradient step. We demonstrate the efficiency of NMC on a number of different domains. For statistical models where the prior is conjugate to the likelihood, our method recovers the posterior quite trivially in one step. However, we also show results on fairly large non-conjugate models, where NMC performs better than adaptive first order methods such as NUTS or other inexact scalable inference methods such as Stochastic Variational Inference or bootstrapping.

Mutual independence is a key concept in statistics that characterizes the structural relationships between variables. Existing methods to investigate mutual independence rely on the definition of two competing models, one being nested into the other and used to generate a null distribution for a statistic of interest, usually under the asymptotic assumption of large sample size. As such, these methods have a very restricted scope of application. In the present manuscript, we propose to change the investigation of mutual independence from a hypothesis-driven task that can only be applied in very specific cases to a blind and automated search within patterns of mutual independence. To this end, we treat the issue as one of model comparison that we solve in a Bayesian framework. We show the relationship between such an approach and existing methods in the case of multivariate normal distributions as well as cross-classified multinomial distributions. We propose a general Markov chain Monte Carlo (MCMC) algorithm to numerically approximate the posterior distribution on the space of all patterns of mutual independence. The relevance of the method is demonstrated on synthetic data as well as two real datasets, showing the unique insight provided by this approach.

As systems are getting more autonomous with the development of artificial intelligence, it is important to discover the causal knowledge from observational sensory inputs. By encoding a series of cause-effect relations between events, causal networks can facilitate the prediction of effects from a given action and analyze their underlying data generation mechanism. However, missing data are ubiquitous in practical scenarios. Directly performing existing casual discovery algorithms on partially observed data may lead to the incorrect inference. To alleviate this issue, we proposed a deep learning framework, dubbed Imputated Causal Learning (ICL), to perform iterative missing data imputation and causal structure discovery. Through extensive simulations on both synthetic and real data, we show that ICL can outperform state-of-the-art methods under different missing data mechanisms.

Let's face reality, mathematics is far from being enjoyable. To learn it, we often lack time, and most importantly, motivation. Why do we need all these symbols and a bunch of figures? It turns out, a lot of sense. Especially if you have something to do with machine learning. The point here is not to acquire knowledge, but to be able to use it.

This paper introduces a new data-driven methodology for estimating sparse covariance matrices of the random coefficients in logit mixture models. Researchers typically specify covariance matrices in logit mixture models under one of two extreme assumptions: either an unrestricted full covariance matrix (allowing correlations between all random coefficients), or a restricted diagonal matrix (allowing no correlations at all). Our objective is to find optimal subsets of correlated coefficients for which we estimate covariances. We propose a new estimator, called MISC, that uses a mixed-integer optimization (MIO) program to find an optimal block diagonal structure specification for the covariance matrix, corresponding to subsets of correlated coefficients, for any desired sparsity level using Markov Chain Monte Carlo (MCMC) posterior draws from the unrestricted full covariance matrix. The optimal sparsity level of the covariance matrix is determined using out-of-sample validation. We demonstrate the ability of MISC to correctly recover the true covariance structure from synthetic data. In an empirical illustration using a stated preference survey on modes of transportation, we use MISC to obtain a sparse covariance matrix indicating how preferences for attributes are related to one another.

Variational Bayes is a method to find a good approximation of the posterior probability distribution of latent variables from a parametric family of distributions. The evidence lower bound (ELBO), which is nothing but the model evidence minus the Kullback-Leibler divergence, has been commonly used as a quality measure in the optimization process. However, the model evidence itself has been considered computationally intractable since it is expressed as a nested expectation with an outer expectation with respect to the training dataset and an inner conditional expectation with respect to latent variables. Similarly, if the Kullback-Leibler divergence is replaced with another divergence metric, the corresponding lower bound on the model evidence is often given by such a nested expectation. The standard (nested) Monte Carlo method can be used to estimate such quantities, whereas the resulting estimate is biased and the variance is often quite large. Recently the authors provided an unbiased estimator of the model evidence with small variance by applying the idea from multilevel Monte Carlo (MLMC) methods. In this article, we give more examples involving nested expectations in the context of variational Bayes where MLMC methods can help construct low-variance unbiased estimators, and provide numerical results which demonstrate the effectiveness of our proposed estimators.

This paper presents a new approach to a robust Gaussian process (GP) regression. Most existing approaches replace an outlier-prone Gaussian likelihood with a non-Gaussian likelihood induced from a heavy tail distribution, such as the Laplace distribution and Student-t distribution. However, the use of a non-Gaussian likelihood would incur the need for a computationally expensive Bayesian approximate computation in the posterior inferences. The proposed approach models an outlier as a noisy and biased observation of an unknown regression function, and accordingly, the likelihood contains bias terms to explain the degree of deviations from the regression function. We entail how the biases can be estimated accurately with other hyperparameters by a regularized maximum likelihood estimation. Conditioned on the bias estimates, the robust GP regression can be reduced to a standard GP regression problem with analytical forms of the predictive mean and variance estimates. Therefore, the proposed approach is simple and very computationally attractive. It also gives a very robust and accurate GP estimate for many tested scenarios. For the numerical evaluation, we perform a comprehensive simulation study to evaluate the proposed approach with the comparison to the existing robust GP approaches under various simulated scenarios of different outlier proportions and different noise levels. The approach is applied to data from two measurement systems, where the predictors are based on robust environmental parameter measurements and the response variables utilize more complex chemical sensing methods that contain a certain percentage of outliers. The utility of the measurement systems and value of the environmental data are improved through the computationally efficient GP regression and bias model.

With the rapid development of online education system, knowledge tracing which aims at predicting students' knowledge state is becoming a critical and fundamental task in personalized education. Traditionally, existing methods are domain-specified. However, there are a larger number of domains (e.g., subjects, schools) in the real world and the lacking of data in some domains, how to utilize the knowledge and information in other domains to help train a knowledge tracing model for target domains is increasingly important. We refer to this problem as domain adaptation for knowledge tracing (DAKT) which contains two aspects: (1) how to achieve great knowledge tracing performance in each domain. (2) how to transfer good performed knowledge tracing model between domains. To this end, in this paper, we propose a novel adaptable framework, namely adaptable knowledge tracing (AKT) to address the DAKT problem. Specifically, for the first aspect, we incorporate the educational characteristics (e.g., slip, guess, question texts) based on the deep knowledge tracing (DKT) to obtain a good performed knowledge tracing model. For the second aspect, we propose and adopt three domain adaptation processes. First, we pre-train an auto-encoder to select useful source instances for target model training. Second, we minimize the domain-specific knowledge state distribution discrepancy under maximum mean discrepancy (MMD) measurement to achieve domain adaptation. Third, we adopt fine-tuning to deal with the problem that the output dimension of source and target domain are different to make the model suitable for target domains. Extensive experimental results on two private datasets and seven public datasets clearly prove the effectiveness of AKT for great knowledge tracing performance and its superior transferable ability.

An open question in systems and computational neuroscience is how neural circuits accumulate evidence towards a decision. Fitting models of decision-making theory to neural activity helps answer this question, but current approaches limit the number of these models that we can fit to neural data. Here we propose a unifying framework for modeling neural activity during decision-making tasks. The framework includes the canonical drift-diffusion model and enables extensions such as multi-dimensional accumulators, variable and collapsing boundaries, and discrete jumps. Our framework is based on constraining the parameters of recurrent state-space models, for which we introduce a scalable variational Laplace-EM inference algorithm. We applied the modeling approach to spiking responses recorded from monkey parietal cortex during two decision-making tasks. We found that a two-dimensional accumulator better captured the trial-averaged responses of a set of parietal neurons than a single accumulator model. Next, we identified a variable lower boundary in the responses of an LIP neuron during a random dot motion task.

The scalable inference for Bayesian nonparametric models with big data is still challenging. Current variational inference methods fail to characterise the correlation structure among latent variables due to the mean-field setting and cannot infer the true posterior dimension because of the universal truncation. To overcome these limitations, we build a general framework to infer Bayesian nonparametric models by maximising the proposed nonparametric evidence lower bound, and then develop a novel approach by combining Monte Carlo sampling and stochastic variational inference framework. Our method has several advantages over the traditional online variational inference method. First, it achieves a smaller divergence between variational distributions and the true posterior by factorising variational distributions under the conditional setting instead of the mean-field setting to capture the correlation pattern. Second, it reduces the risk of underfitting or overfitting by truncating the dimension adaptively rather than using a prespecified truncated dimension for all latent variables. Third, it reduces the computational complexity by approximating the posterior functionally instead of updating the stick-breaking parameters individually. We apply the proposed method on hierarchical Dirichlet process and gamma--Dirichlet process models, two essential Bayesian nonparametric models in topic analysis. The empirical study on three large datasets including arXiv, New York Times and Wikipedia reveals that our proposed method substantially outperforms its competitor in terms of lower perplexity and much clearer topic-words clustering.