Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.

Machine Learning : A Probabilistic Perspective

Statistics for Spatial Data.

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Convex Analysisの二,三の進展について

Most common human traits and diseases have a polygenic pattern of inheritance: DNA sequence variants at many genetic loci influence the phenotype. Genome-wide association (GWA) studies have identified more than 600 variants associated with human traits, but these typically explain small fractions of phenotypic variation, raising questions about the use of further studies. Here, using 183,727 individuals, we show that hundreds of genetic variants, in at least 180 loci, influence adult height, a highly heritable and classic polygenic trait. The large number of loci reveals patterns with important implications for genetic studies of common human diseases and traits. First, the 180 loci are not random, but instead are enriched for genes that are connected in biological pathways (P = 0.016) and that underlie skeletal growth defects (P < 0.001). Second, the likely causal gene is often located near the most strongly associated variant: in 13 of 21 loci containing a known skeletal growth gene, that gene was closest to the associated variant. Third, at least 19 loci have multiple independently associated variants, suggesting that allelic heterogeneity is a frequent feature of polygenic traits, that comprehensive explorations of already-discovered loci should discover additional variants and that an appreciable fraction of associated loci may have been identified. Fourth, associated variants are enriched for likely functional effects on genes, being over-represented among variants that alter amino-acid structure of proteins and expression levels of nearby genes. Our data explain approximately 10% of the phenotypic variation in height, and we estimate that unidentified common variants of similar effect sizes would increase this figure to approximately 16% of phenotypic variation (approximately 20% of heritable variation). Although additional approaches are needed to dissect the genetic architecture of polygenic human traits fully, our findings indicate that GWA studies can identify large numbers of loci that implicate biologically relevant genes and pathways.

Hundreds of variants clustered in genomic loci and biological pathways affect human height

We report on gravitational wave discoveries from compact binary coalescences detected by Advanced LIGO and Advanced Virgo in the first half of the third observing run (O3a) between 1 April 2019 15:00 UTC and 1 October 2019 15:00. By imposing a false-alarm-rate threshold of two per year in each of the four search pipelines that constitute our search, we present 39 candidate gravitational wave events. At this threshold, we expect a contamination fraction of less than 10%. Of these, 26 candidate events were reported previously in near real-time through GCN Notices and Circulars; 13 are reported here for the first time. The catalog contains events whose sources are black hole binary mergers up to a redshift of ~0.8, as well as events whose components could not be unambiguously identified as black holes or neutron stars. For the latter group, we are unable to determine the nature based on estimates of the component masses and spins from gravitational wave data alone. The range of candidate events which are unambiguously identified as binary black holes (both objects $\geq 3~M_\odot$) is increased compared to GWTC-1, with total masses from $\sim 14~M_\odot$ for GW190924_021846 to $\sim 150~M_\odot$ for GW190521. For the first time, this catalog includes binary systems with significantly asymmetric mass ratios, which had not been observed in data taken before April 2019. We also find that 11 of the 39 events detected since April 2019 have positive effective inspiral spins under our default prior (at 90% credibility), while none exhibit negative effective inspiral spin. Given the increased sensitivity of Advanced LIGO and Advanced Virgo, the detection of 39 candidate events in ~26 weeks of data (~1.5 per week) is consistent with GWTC-1.

GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run

Modern data science applications often involve complex relational data with dynamic structures. An abrupt change in such dynamic relational data is typically observed in systems that undergo regime changes due to interventions. In such a case, we consider a factorized fusion shrinkage model in which all decomposed factors are dynamically shrunk towards group-wise fusion structures, where the shrinkage is obtained by applying global-local shrinkage priors to the successive differences of the row vectors of the factorized matrices. The proposed priors enjoy many favorable properties in comparison and clustering of the estimated dynamic latent factors. Comparing estimated latent factors involves both adjacent and long-term comparisons, with the time range of comparison considered as a variable. Under certain conditions, we demonstrate that the posterior distribution attains the minimax optimal rate up to logarithmic factors. In terms of computation, we present a structured mean-field variational inference framework that balances optimal posterior inference with computational scalability, exploiting both the dependence among components and across time. The framework can accommodate a wide variety of models, including dynamic matrix factorization, latent space models for networks and low-rank tensors. The effectiveness of our methodology is demonstrated through extensive simulations and real-world data analysis.

Factorized Fusion Shrinkage for Dynamic Relational Data

We propose BMLE, a new family of bandit algorithms, that are formulated in a general way based on the Biased Maximum Likelihood Estimation method originally appearing in the adaptive control literature. We design the cost-bias term to tackle the exploration and exploitation tradeoff for stochastic bandit problems. We provide an explicit closed form expression for the index of an arm for Bernoulli bandits, which is trivial to compute. We also provide a general recipe for extending the BMLE algorithm to other families of reward distributions. We prove that for Bernoulli bandits, the BMLE algorithm achieves a logarithmic finite-time regret bound and hence attains order-optimality. Through extensive simulations, we demonstrate that the proposed algorithms achieve regret performance comparable to the best of several state-of-the-art baseline methods, while having a significant computational advantage in comparison to other best performing methods. The generality of the proposed approach makes it possible to address more complex models, including general adaptive control of Markovian systems.

Bandit Learning Through Biased Maximum Likelihood Estimation

A pseudo-spectral based efficient volume penalization scheme for Cahn-Hilliard equation in complex geometries

The advent of ML-driven decision-making and policy formation has led to an increasing focus on algorithmic fairness. As clustering is one of the most commonly used unsupervised machine learning approaches, there has naturally been a proliferation of literature on {\em fair clustering}. A popular notion of fairness in clustering mandates the clusters to be {\em balanced}, i.e., each level of a protected attribute must be approximately equally represented in each cluster. Building upon the original framework, this literature has rapidly expanded in various aspects. In this article, we offer a novel model-based formulation of fair clustering, complementing the existing literature which is almost exclusively based on optimizing appropriate objective functions.

Fair Clustering via Hierarchical Fair-Dirichlet Process

We study Bayesian histograms for distribution estimation on $[0,1]^d$ under the Wasserstein $W_v, 1 \leq v<\infty$ distance in the i.i.d sampling regime. We newly show that when $d<2v$, histograms possess a special \textit{memory efficiency} property, whereby in reference to the sample size $n$, order $n^{d/2v}$ bins are needed to obtain minimax rate optimality. This result holds for the posterior mean histogram and with respect to posterior contraction: under the class of Borel probability measures and some classes of smooth densities. The attained memory footprint overcomes existing minimax optimal procedures by a polynomial factor in $n$; for example an $n^{1 - d/2v}$ factor reduction in the footprint when compared to the empirical measure, a minimax estimator in the Borel probability measure class. Additionally constructing both the posterior mean histogram and the posterior itself can be done super--linearly in $n$. Due to the popularity of the $W_1,W_2$ metrics and the coverage provided by the $d<2v$ case, our results are of most practical interest in the $(d=1,v =1,2), (d=2,v=2), (d=3,v=2)$ settings and we provide simulations demonstrating the theory in several of these instances.

Anirban Bhattacharya

Papers

Factorized Fusion Shrinkage for Dynamic Relational Data

Bandit Learning Through Biased Maximum Likelihood Estimation

A pseudo-spectral based efficient volume penalization scheme for Cahn-Hilliard equation in complex geometries

Fair Clustering via Hierarchical Fair-Dirichlet Process

Memory Efficient And Minimax Distribution Estimation Under Wasserstein Distance Using Bayesian Histograms