Probabilistic Models in Machine Learning is the use of the codes of statistics to data examination. It was one of the initial methods of machine learning. It's quite extensively used to this day. Individual of the best-known algorithms in this group is the Naive Bayes algorithm. Probabilistic modelling delivers a framework for accepting what learning is.

For the binary prevalence quantification problem under prior probability shift, we determine the asymptotic variance of the maximum likelihood estimator. We find that it is a function of the Brier score for the regression of the class label against the features under the test data set distribution. This observation suggests that optimising the accuracy of a base classifier on the training data set helps to reduce the variance of the related quantifier on the test data set. Therefore, we also point out training criteria for the base classifier that imply optimisation of both of the Brier scores on the training and the test data sets.

This paper presents Bayesian parameter estimation for first order Grey system models' parameters (or sometimes referred to as hyperparameters). There are different forms of first-order Grey System Models. These include $GM(1,1)$, $GM(1,1| \cos(\omega t)$, $GM(1,1| \sin(\omega t)$, and $GM(1,1| \cos(\omega t), \sin(\omega t)$. The whitenization equation of these models is a first-order linear differential equation of the form \[ \frac{dx}{dt} + a x = f(t) \] where $a$ is a parameter and $f(t) = b$ in $GM(1,1|)$ , $f(t) = b_1\cos(\omega t) + b_2$ in $GM(1,1| cos(\omega t)$, $f(t) = b_1\sin(\omega t)+b_2$ in $GM(1,1| \sin(\omega t)$, $f(t) = b_1\sin(\omega t) + b_2\cos(\omega t) + b_3$ in $GM(1,1| \cos(\omega t), \sin(\omega t)$, $f(t) = b x^2$ in Grey Verhulst model (GVM), and where $b, b_1, b_2$, and $b_3$ are parameters. The results from Bayesian estimations are compared to the least square estimated models with fixed $\omega$. We found that using rolling Bayesian estimations for GM parameters can allow us to estimate the parameters in all possible forms. Based on the data used for the comparison, the numerical results showed that models with Bayesian parameter estimations are up to 45\% more accurate in mean squared errors.

When faced with severely imbalanced binary classification problems, we often train models on bootstrapped data in which the number of instances of each class occur in a more favorable ratio, e.g., one. We view algorithmic inequity through the lens of imbalanced classification: in order to balance the performance of a classifier across groups, we can bootstrap to achieve training sets that are balanced with respect to both labels and group identity. For an example problem with severe class imbalance---prediction of suicide death from administrative patient records---we illustrate how an equity-directed bootstrap can bring test set sensitivities and specificities much closer to satisfying the equal odds criterion. In the context of na\"ive Bayes and logistic regression, we analyze the equity-directed bootstrap, demonstrating that it works by bringing odds ratios close to one, and linking it to methods involving intercept adjustment, thresholding, and weighting.

We propose a very fast approximate Markov Chain Monte Carlo (MCMC) sampling framework that is applicable to a large class of sparse Bayesian inference problems, where the computational cost per iteration in several models is of order $O(ns)$, where $n$ is the sample size, and $s$ the underlying sparsity of the model. This cost can be further reduced by data sub-sampling when stochastic gradient Langevin dynamics are employed. The algorithm is an extension of the asynchronous Gibbs sampler of Johnson et al. (2013), but can be viewed from a statistical perspective as a form of Bayesian iterated sure independent screening (Fan et al. (2009)). We show that in high-dimensional linear regression problems, the Markov chain generated by the proposed algorithm admits an invariant distribution that recovers correctly the main signal with high probability under some statistical assumptions. Furthermore we show that its mixing time is at most linear in the number of regressors. We illustrate the algorithm with several models.

We present a new subspace-based method to construct probabilistic models for high-dimensional data and highlight its use in anomaly detection. The approach is based on a statistical estimation of probability density using densities of random subspaces combined with geometric averaging. In selecting random subspaces, equal representation of each attribute is used to ensure correct statistical limits. Gaussian mixture models (GMMs) are used to create the probability densities for each subspace with techniques included to mitigate singularities allowing for the ability to handle both numerical and categorial attributes. The number of components for each GMM is determined automatically through Bayesian information criterion to prevent overfitting. The proposed algorithm attains competitive AUC scores compared with prominent algorithms against benchmark anomaly detection datasets with the added benefits of being simple, scalable, and interpretable.

In machine learning and statistics, computing the accuracy, or loss, of a model is crucial for understanding the quality of the model and what improvements can be made to increase accuracy. Typically, researchers choose a loss function de- pending on their task, and this loss function runs over their test set of data, after training. However, in many cases, researchers want an estimation of their loss either before they test it or in cases when testing data is not yet available. This estimation is known as expected loss, or risk, and is usually utilized in order to assess how precarious an action or event will be. The foundations of Bayesian statistics are rooted in Bayes' Theorem, a theorem developed by Thomas Bayes who was an English mathematician and theologian during the 1700s.

Stacking is a widely used model averaging technique that asymptotically yields optimal predictions among linear averages. We show that stacking is most effective when model predictive performance is heterogeneous in inputs, and we can further improve the stacked mixture with a hierarchical model. We generalize stacking to Bayesian hierarchical stacking. The model weights are varying as a function of data, partially-pooled, and inferred using Bayesian inference. We further incorporate discrete and continuous inputs, other structured priors, and time series and longitudinal data.

Bayesian optimization (BO) is a flexible and powerful framework that is suitable for computationally expensive simulation-based applications and guarantees statistical convergence to the global optimum. While remaining as one of the most popular optimization methods, its capability is hindered by the size of data, the dimensionality of the considered problem, and the nature of sequential optimization. These scalability issues are intertwined with each other and must be tackled simultaneously. In this work, we propose the Scalable$^3$-BO framework, which employs sparse GP as the underlying surrogate model to scope with Big Data and is equipped with a random embedding to efficiently optimize high-dimensional problems with low effective dimensionality. The Scalable$^3$-BO framework is further leveraged with asynchronous parallelization feature, which fully exploits the computational resource on HPC within a computational budget. As a result, the proposed Scalable$^3$-BO framework is scalable in three independent perspectives: with respect to data size, dimensionality, and computational resource on HPC. The goal of this work is to push the frontiers of BO beyond its well-known scalability issues and minimize the wall-clock waiting time for optimizing high-dimensional computationally expensive applications. We demonstrate the capability of Scalable$^3$-BO with 1 million data points, 10,000-dimensional problems, with 20 concurrent workers in an HPC environment.

We study query and computationally efficient planning algorithms with linear function approximation and a simulator. We assume that the agent only has local access to the simulator, meaning that the agent can only query the simulator at states that have been visited before. This setting is more practical than many prior works on reinforcement learning with a generative model. We propose an algorithm named confident Monte Carlo least square policy iteration (Confident MC-LSPI) for this setting. Under the assumption that the Q-functions of all deterministic policies are linear in known features of the state-action pairs, we show that our algorithm has polynomial query and computational complexities in the dimension of the features, the effective planning horizon and the targeted sub-optimality, while these complexities are independent of the size of the state space. One technical contribution of our work is the introduction of a novel proof technique that makes use of a virtual policy iteration algorithm. We use this method to leverage existing results on $\ell_\infty$-bounded approximate policy iteration to show that our algorithm can learn the optimal policy for the given initial state even only with local access to the simulator. We believe that this technique can be extended to broader settings beyond this work.