Both frequentist and Bayesian probability have a role to play in machine learning. For example, if dealing with truly random and discrete variables, such as landing a six in a die roll, the traditional approach of simply calculating the odds (frequency) is the fastest way to model a likely outcome. However, if the six keeps coming up far more often than the predicated 1/6 odds, only Bayesian probability would take that new observation into account and increase the confidence level that someone is playing with loaded dice.

Of these, probably the most common is the use of supervised classification to construct low-dimensional event summaries, which are informative to carry out statistical inference for a given set of parameters of interest. The learned summary statistics -functions of the data that are informative on their relevant properties-can efficiently combine high-dimensional information from each event into one or a few variables which can be used as the basis of statistical inference. The informational source for this compression are simulated observations produced by a complex generative model; the latter reproduces the chain of physical processes occurring in subatomic collisions and the subsequent interaction of the produced final state particles with the detection elements.

Cells regulate themselves via dizzyingly complex biochemical processes called signaling pathways. These are usually depicted as a network, where nodes represent proteins and edges indicate their influence on each other. In order to understand diseases and therapies at the cellular level, it is crucial to have an accurate understanding of the signaling pathways at work. Since signaling pathways can be modified by disease, the ability to infer signaling pathways from condition- or patient-specific data is highly valuable. A variety of techniques exist for inferring signaling pathways. We build on past works that formulate signaling pathway inference as a Dynamic Bayesian Network structure estimation problem on phosphoproteomic time course data. We take a Bayesian approach, using Markov Chain Monte Carlo to estimate a posterior distribution over possible Dynamic Bayesian Network structures. Our primary contributions are (i) a novel proposal distribution that efficiently samples sparse graphs and (ii) the relaxation of common restrictive modeling assumptions. We implement our method, named Sparse Signaling Pathway Sampling, in Julia using the Gen probabilistic programming language. Probabilistic programming is a powerful methodology for building statistical models. The resulting code is modular, extensible, and legible. The Gen language, in particular, allows us to customize our inference procedure for biological graphs and ensure efficient sampling. We evaluate our algorithm on simulated data and the HPN-DREAM pathway reconstruction challenge, comparing our performance against a variety of baseline methods. Our results demonstrate the vast potential for probabilistic programming, and Gen specifically, for biological network inference. Find the full codebase at https://github.com/gitter-lab/ssps

Even before the recent George Floyd case, there has been much debate about the extent to which claims of systemic racism are supported by statistical evidence. For example (Ross 2015) claims that unarmed blacks are 3.5 times more likely to be shot by police than unarmed whites when adjusting for relative differences in population size. However, (Fryer 2016) - formally published later as (Fryer 2019) - found that there was no such racial disparity when the data were conditioned on people being stopped by police, and there was a similar conclusion in (Patty and Hanson 2020) that was produced in direct response to public concerns about the Floyd case. In response to Fryer, (Ross, Winterhalder, and McElreath 2018) argued that Fryer's analysis was compromised because it was essentially an example of Simpson's paradox (Simpson 1951; Bickel, Hammel, and O'Connell 1975; Fenton, Neil, and Constantinou 2019) whereby conclusions based on pooled statistics are reversed when drilling down into relevant subcategories. A new paper (Knox, Lowe, and Mummolo 2020) explains why Simpson's paradox is not the only statistical explanation for the apparently contradictory conclusions of Ross and Fryer.

Stochastic block models (SBM) are widely used in network science due to their interpretable structure that allows inference on groups of nodes having common connectivity patterns. Although providing a well established model-based approach for community detection, such formulations are still the object of intense research to address the key problem of inferring the unknown number of communities. This has motivated the development of several probabilistic mechanisms to characterize the node partition process, covering solutions with fixed, random and infinite number of communities. In this article we provide a unified view of all these formulations within a single extended stochastic block model (ESBM), that relies on Gibbs-type processes and encompasses most existing representations as special cases. Connections with Bayesian nonparametric literature open up new avenues that allow the natural inclusion of several unexplored options to model the nodes partition process and to incorporate node attributes in a principled manner. Among these new alternatives, we focus on the Gnedin process as an example of a probabilistic mechanism with desirable theoretical properties and nice empirical performance. A collapsed Gibbs sampler that can be applied to the whole ESBM class is proposed, and refined methods for estimation, uncertainty quantification and model assessment are outlined. The performance of ESBM is assessed in simulations and an application to bill co-sponsorship networks in the Italian parliament, where we find key hidden block structures and core-periphery patterns.

There has been increased interest in missing sensor data imputation, which is ubiquitous in the field of structural health monitoring (SHM) due to discontinuous sensing caused by sensor malfunction. To address this fundamental issue, this paper presents an incremental Bayesian tensor learning method for reconstruction of spatiotemporal missing data in SHM and forecasting of structural response. In particular, a spatiotemporal tensor is first constructed followed by Bayesian tensor factorization that extracts latent features for missing data imputation. To enable structural response forecasting based on incomplete sensing data, the tensor decomposition is further integrated with vector autoregression in an incremental learning scheme. The performance of the proposed approach is validated on continuous field-sensing data (including strain and temperature records) of a concrete bridge, based on the assumption that strain time histories are highly correlated to temperature recordings. The results indicate that the proposed probabilistic tensor learning approach is accurate and robust even in the presence of large rates of random missing, structured missing and their combination. The effect of rank selection on the imputation and prediction performance is also investigated. The results show that a better estimation accuracy can be achieved with a higher rank for random missing whereas a lower rank for structured missing.

Anomaly estimation, or the problem of finding a subset of a dataset that differs from the rest of the dataset, is a classic problem in machine learning and data mining. In both theoretical work and in applications, the anomaly is assumed to have a specific structure defined by membership in an $\textit{anomaly family}$. For example, in temporal data the anomaly family may be time intervals, while in network data the anomaly family may be connected subgraphs. The most prominent approach for anomaly estimation is to compute the Maximum Likelihood Estimator (MLE) of the anomaly. However, it was recently observed that for some anomaly families, the MLE is an asymptotically $\textit{biased}$ estimator of the anomaly. Here, we demonstrate that the bias of the MLE depends on the size of the anomaly family. We prove that if the number of sets in the anomaly family that contain the anomaly is sub-exponential, then the MLE is asymptotically unbiased. At the same time, we provide empirical evidence that the converse is also true: if the number of such sets is exponential, then the MLE is asymptotically biased. Our analysis unifies a number of earlier results on the bias of the MLE for specific anomaly families, including intervals, submatrices, and connected subgraphs. Next, we derive a new anomaly estimator using a mixture model, and we empirically demonstrate that our estimator is asymptotically unbiased regardless of the size of the anomaly family. We illustrate the benefits of our estimator on both simulated disease outbreak data and a real-world highway traffic dataset.

We introduce in this work a new method for online approximate Bayesian learning, whose main idea is to approximate the sequence $(\pi_t)_{t\geq 1}$ of posterior distributions by a sequence $(\tilde{\pi}_t)_{t\geq 1}$ which (i) can be estimated in an online fashion using sequential Monte Carlo methods and (ii) is shown to converge to the same distribution as the sequence $(\pi_t)_{t\geq 1}$, under weak assumptions on the statistical model at hand. In its simplest version, the proposed approach amounts to take for $(\tilde{\pi}_t)_{t\geq 1}$ the sequence of filtering distributions associated to a particular state-space model, and to approximate this sequence using a standard particle filter algorithm. We illustrate on several challenging examples the benefits of this procedure for online approximate Bayesian parameter inference, and with one real data example we show that its online predictive performance can significantly outperform that of stochastic gradient descent and of streaming variational Bayes.

Fraudulent activities are an expensive problem for many financial institutions, costing billions of dollars to corporations annually. More commonly occurring activities in this regard are credit card frauds. In this context, the credit card fraud detection concept has been developed over the lines of incorporating the uncertainty in our prediction system to ensure better judgment in such a crucial task. We propose to use a sparse Gaussian classification method to work with the large data-set and use the concept of pseudo or inducing inputs. We perform the same with different sets of kernels and the different number of inducing data points to show the best accuracy was obtained with the selection of RBF kernel with a higher number of inducing points. Our approach was able to work over large financial data given the stochastic nature of our method employed and also good test accuracy with low variance over the prediction suggesting confidence and robustness in our model. Using the methodologies of Bayesian learning techniques with the incorporated inducing points phenomenon, are successfully able to obtain a healthy accuracy and a high confidence score.

Latent confounders---unobserved variables that influence both treatment and outcome---can bias estimates of causal effects. In some cases, these confounders are shared across observations, e.g. all students taking a course are influenced by the course's difficulty in addition to any educational interventions they receive individually. This paper shows how to semiparametrically model latent confounders that have this structure and thereby improve estimates of causal effects. The key innovations are a hierarchical Bayesian model, Gaussian processes with structured latent confounders (GP-SLC), and a Monte Carlo inference algorithm for this model based on elliptical slice sampling. GP-SLC provides principled Bayesian uncertainty estimates of individual treatment effect with minimal assumptions about the functional forms relating confounders, covariates, treatment, and outcome. Finally, this paper shows GP-SLC is competitive with or more accurate than widely used causal inference techniques on three benchmark datasets, including the Infant Health and Development Program and a dataset showing the effect of changing temperatures on state-wide energy consumption across New England.