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Bayesian and frequentist results are not the same, ever


I often hear people say that the results from Bayesian methods are the same as the results from frequentist methods, at least under certain conditions. And sometimes it even comes from people who understand Bayesian methods. Today I saw this tweet from Julia Rohrer: "Running a Bayesian multi-membership multi-level probit model with a custom function to generate average marginal effects only to find that the estimate is precisely the same as the one generated by linear regression with dummy-coded group membership." Which elicited what I interpret as good-natured teasing, like this tweet from Daniël Lakens: "I always love it when people realize that the main difference between a frequentist and Bayesian analysis is that for the latter approach you first need to wait 24 hours for the results." Ok, that's funny, but there is a serious point here I want to respond to because both of these comments are based on the premise that we can compare the results from Bayesian and frequentist methods.

10 Must-Know Statistical Concepts for Data Scientists - KDnuggets


Data science is an interdisciplinary field. One of the building blocks of data science is statistics. Without a decent level of statistics knowledge, it would be highly difficult to understand or interpret the data. Statistics helps us explain the data. We use statistics to infer results about a population based on a sample drawn from that population.

Robustness Meets Algorithms

Communications of the ACM

In every corner of machine learning and statistics, there is a need for estimators that work not just in an idealized model, but even when their assumptions are violated. Unfortunately, in high dimensions, being provably robust and being efficiently computable are often at odds with each other. We give the first efficient algorithm for estimating the parameters of a high-dimensional Gaussian that is able to tolerate a constant fraction of corruptions that is independent of the dimension. Prior to our work, all known estimators either needed time exponential in the dimension to compute or could tolerate only an inverse-polynomial fraction of corruptions. Not only does our algorithm bridge the gap between robustness and algorithms, but also it turns out to be highly practical in a variety of settings. Machine learning is filled with examples of estimators that work well in idealized settings but fail when their assumptions are violated. In fact, these are examples of a more general paradigm within statistics called maximum likelihood estimation: When we know the distribution comes from some parametric family, we choose the parameters that are the most likely to have generated the observed data. In 1922, Ronald Fisher12 formulated the maximum likelihood principle. It has many wonderful properties (under various technical conditions), such as converging to the true parameters as the number of samples goes to infinity, a property called consistency. Moreover, it has asymptotically the smallest possible variance among all unbiased estimators, a property called asymptotic consistency. In 1960, John Tukey24 challenged the conventional wisdom in parametric estimation by asking a simple question: Are there provably robust methods to estimate the parameters of a one-dimensional Gaussian?

A Primer on the EM Algorithm


The Expectation-Maximization (EM) algorithm is one of the main algorithms in machine learning for estimation of model parameters [2][3][4]. For example, it is used to estimate mixing coefficients, means, and covariances in mixture models as shown in Figure 1. Its objective is to maximize the likelihood p(X θ) where X is a matrix of observed data and θ is a vector of model parameters. This is maximum likelihood estimation and in practice the log-likelihood ln p(X θ) is maximized. The model parameters that maximize this function are deemed to be the correct model parameters.

Active Tree Search in Large POMDPs Artificial Intelligence

Model-based planning and prospection are widely studied in both cognitive neuroscience and artificial intelligence (AI), but from different perspectives - and with different desiderata in mind (biological realism versus scalability) that are difficult to reconcile. Here, we introduce a novel method to plan in large POMDPs - Active Tree Search - that combines the normative character and biological realism of a leading planning theory in neuroscience (Active Inference) and the scalability of Monte-Carlo methods in AI. This unification is beneficial for both approaches. On the one hand, using Monte-Carlo planning permits scaling up the biologically grounded approach of Active Inference to large-scale problems. On the other hand, the theory of Active Inference provides a principled solution to the balance of exploration and exploitation, which is often addressed heuristically in Monte-Carlo methods. Our simulations show that Active Tree Search successfully navigates binary trees that are challenging for sampling-based methods, problems that require adaptive exploration, and the large POMDP problem Rocksample. Furthermore, we illustrate how Active Tree Search can be used to simulate neurophysiological responses (e.g., in the hippocampus and prefrontal cortex) of humans and other animals that contain large planning problems. These simulations show that Active Tree Search is a principled realisation of neuroscientific and AI theories of planning, which offers both biological realism and scalability.

The Inescapable Duality of Data and Knowledge Artificial Intelligence

We will discuss how over the last 30 to 50 years, systems that focused only on data have been handicapped with success focused on narrowly focused tasks, and knowledge has been critical in developing smarter, intelligent, more effective systems. We will draw a parallel with the role of knowledge and experience in human intelligence based on cognitive science. And we will end with the recent interest in neuro-symbolic or hybrid AI systems in which knowledge is the critical enabler for combining data-intensive statistical AI systems with symbolic AI systems which results in more capable AI systems that support more human-like intelligence.

Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior Machine Learning

In this work we address the problem of solving ill-posed inverse problems in imaging where the prior is a variational autoencoder (VAE). Specifically we consider the decoupled case where the prior is trained once and can be reused for many different log-concave degradation models without retraining. Whereas previous MAP-based approaches to this problem lead to highly non-convex optimization algorithms, our approach computes the joint (space-latent) MAP that naturally leads to alternate optimization algorithms and to the use of a stochastic encoder to accelerate computations. The resulting technique (JPMAP) performs Joint Posterior Maximization using an Autoencoding Prior. We show theoretical and experimental evidence that the proposed objective function is quite close to bi-convex. Indeed it satisfies a weak bi-convexity property which is sufficient to guarantee that our optimization scheme converges to a stationary point. We also highlight the importance of correctly training the VAE using a denoising criterion, in order to ensure that the encoder generalizes well to out-of-distribution images, without affecting the quality of the generative model. This simple modification is key to providing robustness to the whole procedure. Finally we show how our joint MAP methodology relates to more common MAP approaches, and we propose a continuation scheme that makes use of our JPMAP algorithm to provide more robust MAP estimates. Experimental results also show the higher quality of the solutions obtained by our JPMAP approach with respect to other non-convex MAP approaches which more often get stuck in spurious local optima.

Recovery of Joint Probability Distribution from one-way marginals: Low rank Tensors and Random Projections Machine Learning

Joint probability mass function (PMF) estimation is a fundamental machine learning problem. The number of free parameters scales exponentially with respect to the number of random variables. Hence, most work on nonparametric PMF estimation is based on some structural assumptions such as clique factorization adopted by probabilistic graphical models, imposition of low rank on the joint probability tensor and reconstruction from 3-way or 2-way marginals, etc. In the present work, we link random projections of data to the problem of PMF estimation using ideas from tomography. We integrate this idea with the idea of low-rank tensor decomposition to show that we can estimate the joint density from just one-way marginals in a transformed space. We provide a novel algorithm for recovering factors of the tensor from one-way marginals, test it across a variety of synthetic and real-world datasets, and also perform MAP inference on the estimated model for classification.

An evidential classifier based on Dempster-Shafer theory and deep learning Artificial Intelligence

We propose a new classifier based on Dempster-Shafer (DS) theory and a convolutional neural network (CNN) architecture for set-valued classification. In this classifier, called the evidential deep-learning classifier, convolutional and pooling layers first extract high-dimensional features from input data. The features are then converted into mass functions and aggregated by Dempster's rule in a DS layer. Finally, an expected utility layer performs set-valued classification based on mass functions. We propose an end-to-end learning strategy for jointly updating the network parameters. Additionally, an approach for selecting partial multi-class acts is proposed. Experiments on image recognition, signal processing, and semantic-relationship classification tasks demonstrate that the proposed combination of deep CNN, DS layer, and expected utility layer makes it possible to improve classification accuracy and to make cautious decisions by assigning confusing patterns to multi-class sets.