Approximate Bayesian computation (ABC) is now an established technique for statistical inference used in cases where the likelihood function is computationally expensive or not available. It relies on the use of a model that is specified in the form of a simulator, and approximates the likelihood at a parameter $\theta$ by simulating auxiliary data sets $x$ and evaluating the distance of $x$ from the true data $y$. However, ABC is not computationally feasible in cases where using the simulator for each $\theta$ is very expensive. This paper investigates this situation in cases where a cheap, but approximate, simulator is available. The approach is to employ delayed acceptance Markov chain Monte Carlo (MCMC) within an ABC sequential Monte Carlo (SMC) sampler in order to, in a first stage of the kernel, use the cheap simulator to rule out parts of the parameter space that are not worth exploring, so that the "true" simulator is only run (in the second stage of the kernel) where there is a reasonable chance of accepting proposed values of $\theta$. We show that this approach can be used quite automatically, with the only tuning parameter choice additional to ABC-SMC being the number of particles we wish to carry through to the second stage of the kernel. Applications to stochastic differential equation models and latent doubly intractable distributions are presented.

Machine Learning, the branch of Artificial Intelligence is based on the idea that machines should be able to learn and adapt through experience. It is increasingly gaining popularity over the last couple of years. Machine learning is one approach to achieve Artificial Intelligence by using algorithms. It is predicted that Machine Learning Algorithms may replace a wealth of jobs in the coming years. Logistic Regression is a powerful statistical way of estimating discrete values (usually binary values) from a set of independent variables.

Describing the dimension reduction (DR) techniques by means of probabilistic models has recently been given special attention. Probabilistic models, in addition to a better interpretability of the DR methods, provide a framework for further extensions of such algorithms. One of the new approaches to the probabilistic DR methods is to preserving the internal structure of data. It is meant that it is not necessary that the data first be converted from the matrix or tensor format to the vector format in the process of dimensionality reduction. In this paper, a latent variable model for matrix-variate data for canonical correlation analysis (CCA) is proposed. Since in general there is not any analytical maximum likelihood solution for this model, we present two approaches for learning the parameters. The proposed methods are evaluated using the synthetic data in terms of convergence and quality of mappings. Also, real data set is employed for assessing the proposed methods with several probabilistic and none-probabilistic CCA based approaches. The results confirm the superiority of the proposed methods with respect to the competing algorithms. Moreover, this model can be considered as a framework for further extensions.

Informative Bayesian priors are often difficult to elicit, and when this is the case, modelers usually turn to noninformative or objective priors. However, objective priors such as the Jeffreys and reference priors are not tractable to derive for many models of interest. We address this issue by proposing techniques for learning reference prior approximations: we select a parametric family and optimize a black-box lower bound on the reference prior objective to find the member of the family that serves as a good approximation. We experimentally demonstrate the method's effectiveness by recovering Jeffreys priors and learning the Variational Autoencoder's reference prior.

Learning from demonstration (LfD) is the process of building behavioral models of a task from demonstrations provided by an expert. These models can be used e.g. for system control by generalizing the expert demonstrations to previously unencountered situations. Most LfD methods, however, make strong assumptions about the expert behavior, e.g. they assume the existence of a deterministic optimal ground truth policy or require direct monitoring of the expert's controls, which limits their practical use as part of a general system identification framework. In this work, we consider the LfD problem in a more general setting where we allow for arbitrary stochastic expert policies, without reasoning about the optimality of the demonstrations. Following a Bayesian methodology, we model the full posterior distribution of possible expert controllers that explain the provided demonstration data. Moreover, we show that our methodology can be applied in a nonparametric context to infer the complexity of the state representation used by the expert, and to learn task-appropriate partitionings of the system state space.

The goal of Machine Learning to automatically learn from data, extract knowledge and to make decisions without any human intervention. Such automatic (aML) approaches show impressive success. Recent results even demonstrate intriguingly that deep learning applied for automatic classification of skin lesions is on par with the performance of dermatologists, yet outperforms the average. As human perception is inherently limited, such approaches can discover patterns, e.g. that two objects are similar, in arbitrarily high-dimensional spaces what no human is able to do. Humans can deal only with limited amounts of data, whilst big data is beneficial for aML; however, in health informatics, we are often confronted with a small number of data sets, where aML suffer of insufficient training samples and many problems are computationally hard. Here, interactive machine learning (iML) may be of help, where a human-in-the-loop contributes to reduce the complexity of NP-hard problems. A further motivation for iML is that standard black-box approaches lack transparency, hence do not foster trust and acceptance of ML among end-users. Rising legal and privacy aspects, e.g. with the new European General Data Protection Regulations, make black-box approaches difficult to use, because they often are not able to explain why a decision has been made. In this paper, we present some experiments to demonstrate the effectiveness of the human-in-the-loop approach, particularly in opening the black-box to a glass-box and thus enabling a human directly to interact with an learning algorithm. We selected the Ant Colony Optimization framework, and applied it on the Traveling Salesman Problem, which is a good example, due to its relevance for health informatics, e.g. for the study of protein folding. From studies of how humans extract so much from so little data, fundamental ML-research also may benefit.

We consider structure discovery of undirected graphical models from observational data. Inferring likely structures from few examples is a complex task often requiring the formulation of priors and sophisticated inference procedures. Popular methods rely on estimating a penalized maximum likelihood of the precision matrix. However, in these approaches structure recovery is an indirect consequence of the data-fit term, the penalty can be difficult to adapt for domain-specific knowledge, and the inference is computationally demanding. By contrast, it may be easier to generate training samples of data that arise from graphs with the desired structure properties. We propose here to leverage this latter source of information as training data to learn a function, parametrized by a neural network that maps empirical covariance matrices to estimated graph structures. Learning this function brings two benefits: it implicitly models the desired structure or sparsity properties to form suitable priors, and it can be tailored to the specific problem of edge structure discovery, rather than maximizing data likelihood. Applying this framework, we find our learnable graph-discovery method trained on synthetic data generalizes well: identifying relevant edges in both synthetic and real data, completely unknown at training time. We find that on genetics, brain imaging, and simulation data we obtain performance generally superior to analytical methods.

In Bayesian statistics, probabilities are interpreted as degrees of belief. For any set of mutually exclusive and exhaustive events, one expresses the state of knowledge as a probability distribution over that set. The probability of an event then describes the personal confidence that this event will happen or has happened. As a consequence, probabilities are subjective properties reflecting the amount of knowledge an observer has about the events; a different observer might know which event happened and assign different probabilities. If an observer gains information, she updates the probabilities she had assigned before. If the set of possible mutually exclusive and exhaustive events is infinite, it is generally impossible to store all entries of the corresponding probability distribution on a computer or communicate it through a channel with finite bandwidth. One therefore needs to approximate the probability distribution which describes one's belief. Given a limited set X of approximative beliefs q(s) on a quantity s, what is the best belief to approximate the actual belief as expressed by the probability p(s)? In the literature, it is sometimes claimed that the best approximation is given by the q X that minimizes the Kullback-Leibler divergence ("approximation" KL) [1] KL(p, q) () p(s) p(s) ln (1) q(s)

In the last few months, I have had several people contact me about their enthusiasm for venturing into the world of data science and using Machine Learning (ML) techniques to probe statistical regularities and build impeccable data-driven products. However, I've observed that some actually lack the necessary mathematical intuition and framework to get useful results. This is the main reason I decided to write this blog post. Recently, there has been an upsurge in the availability of many easy-to-use machine and deep learning packages such as scikit-learn, Weka, Tensorflow etc. Machine Learning theory is a field that intersects statistical, probabilistic, computer science and algorithmic aspects arising from learning iteratively from data and finding hidden insights which can be used to build intelligent applications. Despite the immense possibilities of Machine and Deep Learning, a thorough mathematical understanding of many of these techniques is necessary for a good grasp of the inner workings of the algorithms and getting good results. There are many reasons why the mathematics of Machine Learning is important and I'll highlight some of them below: The main question when trying to understand an interdisciplinary field such as Machine Learning is the amount of maths necessary and the level of maths needed to understand these techniques.

The simplest solutions are usually the most powerful ones, and Naive Bayes is a good proof of that. In spite of the great advances of the Machine Learning in the last years, it has proven to not only be simple but also fast, accurate and reliable. It has been successfully used for many purposes, but it works particularly well with natural language processing (NLP) problems. Naive Bayes is a family of probabilistic algorithms that take advantage of probability theory and Bayes' Theorem to predict the category of a sample (like a piece of news or a customer review). They are probabilistic, which means that they calculate the probability of each category for a given sample, and then output the category with the highest one.