DOCSLIB.ORG

Tutorial on Probabilistic Programming with Pymc3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Tutorial on Probabilistic Programming with Pymc3 185.A83 Machine Learning for Health Informatics 2017S, VU, 2.0 h, 3.0 ECTS Tutorial 02 - 04.04.2017 Tutorial on Probabilistic Programming with PyMC3 [email protected] http://hci-kdd.org/machine-learning-for-health-informatics-course Holzinger Group hci-kdd.org 1 MAKE Health T01 Schedule ▪ 01. Introduction to Probabilistic Programming ▪ 02. PyMC3 ▪ 03. linear regression – the Bayesian way ▪ 04. generalized linear models with PyMC3 Holzinger Group hci-kdd.org 2 MAKE Health T01 01. Introduction and Overview ▪ Probabilistic Programming (PP) ▪ allows automatic Bayesian inference ▪ on complex, user-defined probabilistic models ▪ utilizing “Markov chain Monte Carlo” (MCMC) sampling ▪ PyMC3 ▪ a PP framework ▪ compiles probabilistic programs on-the-fly to C ▪ allows model specification in Python code Salvatier J, Wiecki TV, Fonnesbeck C. (2016) Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55 https://doi.org/10.7717/peerj-cs.55 Holzinger Group hci-kdd.org 3 MAKE Health T01 Properties of Probabilistic Programs ▪ IS NOT ▪ Software that behaves probabilistically ▪ General programming language ▪ IS ▪ Toolset for statistical / Bayesian modeling ▪ Framework to describe probabilistic models ▪ Tool to perform (automatic) inference ▪ Closely related to graphical models and Bayesian networks ▪ Extension to basic language (e.g. PyMC3 for Python) “does in 50 lines of code what used to take thousands” Kulkarni, T. D., Kohli, P., Tenenbaum, J. B. & Mansinghka, V. Picture: A probabilistic programming language for scene perception. in Proceedings of the ieee conference on computer vision and pattern recognition 4390–4399 (2015). Holzinger Group hci-kdd.org 4 MAKE Health T01 Probabilistic Programs ▪ Machine learning algorithms / models often a black box PP “open box” ▪ Simple approach 1. Define and build model 2. Automatic inference 3. Interpretation of results not much equations anymore! ▪ “inference”: guess latent variables based on observations, using e.g. MCMC Holzinger Group hci-kdd.org 5 MAKE Health T01 Markov chain Monte Carlo (MCMC) ▪ Markov chain ▪ Stochastic process ▪ “memoryless” (Markov property) ▪ Conditional probability distribution of future states depends only upon the present state ▪ Sampling from probability distributions ▪ State of chain sample of distribution ▪ Quality improves with number of steps ▪ Class of algorithms / methods ▪ Numerical approximation of complex integrals Holzinger Group hci-kdd.org 6 MAKE Health T01 Markov chain Monte Carlo (MCMC) (animated) Holzinger Group hci-kdd.org 7 MAKE Health T01 Markov chain Monte Carlo (MCMC) ▪ Metropolis-Hastings: random walk ▪ Gibbs-sampling: popular, complex, no tuning ▪ PyMC3 ▪ No-U-Turn Sampler (NUTS) ▪ Hamiltonian Monte Carlo (HMC) ▪ Metropolis ▪ Slice ▪ BinaryMetropolis Holzinger Group hci-kdd.org 8 MAKE Health T01 Prior & posterior distributions ▪ Quantity of interest: 휃 (theta) ▪ Prior = probability distribution ▪ Uncertainty before observation: p(휃) ▪ Belief in absence of data ▪ Posterior = probability distribution ▪ Uncertainty after observation X: p(휃|X) ▪ Likelihood: p(푋|휃) 푃표푠푡푒푟표푟 ∝ 퐿푘푒푙ℎ표표푑 × 푃푟표푟 푝 푥 휃 푝(휃) 푝 휃 푥 = 푝(푥) ▪ Calculating posterior from observation and prior = updating beliefs Coin toss example Holzinger Group hci-kdd.org 9 MAKE Health T01 PyMC3 ▪ Python 3 package / framework ▪ Probabilistic machine learning ▪ specification and fitting of Bayesian models ▪ Inference by MCMC & variational fitting algorithms ▪ Performance enhancements: cross-compilation to C (Python numerical computation package “Theano”) ▪ Accessible, natural syntax ▪ Various capabilities: GPU computing, sampling backends, object-oriented, extendable design Holzinger Group hci-kdd.org 10 MAKE Health T01 Live presentation ▪ PyMC3 syntax introduction ▪ linear regression – the Bayesian way ▪ generalized linear models with PyMC3 Holzinger Group hci-kdd.org 11 MAKE Health T01 Thank you! Holzinger Group hci-kdd.org 12 MAKE Health T01 Sample Questions ▪ What is probabilistic programming? What type of problems can be solved? ▪ What are the typical steps in a probabilistic program? ▪ What is inference? ▪ What is PyMC3? ▪ What is the posterior distribution? Holzinger Group hci-kdd.org 13 MAKE Health T01 Appendix ▪ Main sources ▪ Salvatier J, Wiecki TV, Fonnesbeck C. (2016) Probabilistic programming in Python using PyMC3. PeerJ Computer Science 2:e55 https://doi.org/10.7717/peerj-cs.55 ▪ Davidson-Pilon, C. Bayesian Methods for Hackers: Probabilistic Programming and Bayesian Inference. (Addison-Wesley Professional, 2015). ▪ PyMC3’s documentation http://pymc-devs.github.io/pymc3/index.html Holzinger Group hci-kdd.org 14 MAKE Health T01.
Load more

Recommended publications
  • University of North Carolina at Charlotte Belk College of Business
    University of North Carolina at Charlotte Belk College of Business BPHD 8220 Financial Bayesian Analysis Fall 2019 Course Time: Tuesday 12:20 - 3:05 pm Location: Friday Building 207 Professor: Dr. Yufeng Han Office Location: Friday Building 340A Telephone: (704) 687-8773 E-mail: [email protected] Office Hours: by appointment Textbook: Introduction to Bayesian Econometrics, 2nd Edition, by Edward Greenberg (required) An Introduction to Bayesian Inference in Econometrics, 1st Edition, by Arnold Zellner Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan, 2nd Edition, by Joseph M. Hilbe, de Souza, Rafael S., Emille E. O. Ishida Bayesian Analysis with Python: Introduction to statistical modeling and probabilistic programming using PyMC3 and ArviZ, 2nd Edition, by Osvaldo Martin The Oxford Handbook of Bayesian Econometrics, 1st Edition, by John Geweke Topics: 1. Principles of Bayesian Analysis 2. Simulation 3. Linear Regression Models 4. Multivariate Regression Models 5. Time-Series Models 6. State-Space Models 7. Volatility Models Page | 1 8. Endogeneity Models Software: 1. Stan 2. Edward 3. JAGS 4. BUGS & MultiBUGS 5. Python Modules: PyMC & ArviZ 6. R, STATA, SAS, Matlab, etc. Course Assessment: Homework assignments, paper presentation, and replication (or project). Grading: Homework: 30% Paper presentation: 40% Replication (project): 30% Selected Papers: Vasicek, O. A. (1973), A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas. Journal of Finance, 28: 1233-1239. doi:10.1111/j.1540- 6261.1973.tb01452.x Shanken, J. (1987). A Bayesian approach to testing portfolio efficiency. Journal of Financial Economics, 19(2), 195–215. https://doi.org/10.1016/0304-405X(87)90002-X Guofu, C., & Harvey, R.
    [Show full text]
  • The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo
    Journal of Machine Learning Research 15 (2014) 1593-1623 Submitted 11/11; Revised 10/13; Published 4/14 The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo Matthew D. Hoffman [email protected] Adobe Research 601 Townsend St. San Francisco, CA 94110, USA Andrew Gelman [email protected] Departments of Statistics and Political Science Columbia University New York, NY 10027, USA Editor: Anthanasios Kottas Abstract Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior and sensitivity to correlated parameters that plague many MCMC methods by taking a series of steps informed by first-order gradient information. These features allow it to converge to high-dimensional target distributions much more quickly than simpler methods such as random walk Metropolis or Gibbs sampling. However, HMC's performance is highly sensitive to two user-specified parameters: a step size and a desired number of steps L. In particular, if L is too small then the algorithm exhibits undesirable random walk behavior, while if L is too large the algorithm wastes computation. We introduce the No-U-Turn Sampler (NUTS), an extension to HMC that eliminates the need to set a number of steps L. NUTS uses a recursive algorithm to build a set of likely candidate points that spans a wide swath of the target distribution, stopping automatically when it starts to double back and retrace its steps. Empirically, NUTS performs at least as efficiently as (and sometimes more efficiently than) a well tuned standard HMC method, without requiring user intervention or costly tuning runs.
    [Show full text]
  • Arxiv:1801.02309V4 [Stat.ML] 11 Dec 2019
    Journal of Machine Learning Research 20 (2019) 1-42 Submitted 4/19; Revised 11/19; Published 12/19 Log-concave sampling: Metropolis-Hastings algorithms are fast Raaz Dwivedi∗;y [email protected] Yuansi Chen∗;} [email protected] Martin J. Wainwright};y;z [email protected] Bin Yu};y [email protected] Department of Statistics} Department of Electrical Engineering and Computer Sciencesy University of California, Berkeley Voleon Groupz, Berkeley Editor: Suvrit Sra Abstract We study the problem of sampling from a strongly log-concave density supported on Rd, and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin diffusion, combined with an accept-reject step. Relative to known guarantees for the unadjusted Langevin algorithm (ULA), our bounds show that the use of an accept-reject step in MALA leads to an ex- ponentially improved dependence on the error-tolerance. Concretely, in order to obtain samples with TV error at most δ for a density with condition number κ, we show that MALA requires κd log(1/δ) steps from a warm start, as compared to the κ2d/δ2 steps establishedO in past work on ULA. We also demonstrate the gains of a modifiedO version of MALA over ULA for weakly log-concave densities. Furthermore, we derive mixing time bounds for the Metropolized random walk (MRW) and obtain (κ) mixing time slower than MALA. We provide numerical examples that support ourO theoretical findings, and demonstrate the benefits of Metropolis-Hastings adjustment for Langevin-type sampling algorithms.
    [Show full text]
  • An Introduction to Data Analysis Using the Pymc3 Probabilistic Programming Framework: a Case Study with Gaussian Mixture Modeling
    An introduction to data analysis using the PyMC3 probabilistic programming framework: A case study with Gaussian Mixture Modeling Shi Xian Liewa, Mohsen Afrasiabia, and Joseph L. Austerweila aDepartment of Psychology, University of Wisconsin-Madison, Madison, WI, USA August 28, 2019 Author Note Correspondence concerning this article should be addressed to: Shi Xian Liew, 1202 West Johnson Street, Madison, WI 53706. E-mail: [email protected]. This work was funded by a Vilas Life Cycle Professorship and the VCRGE at University of Wisconsin-Madison with funding from the WARF 1 RUNNING HEAD: Introduction to PyMC3 with Gaussian Mixture Models 2 Abstract Recent developments in modern probabilistic programming have offered users many practical tools of Bayesian data analysis. However, the adoption of such techniques by the general psychology community is still fairly limited. This tutorial aims to provide non-technicians with an accessible guide to PyMC3, a robust probabilistic programming language that allows for straightforward Bayesian data analysis. We focus on a series of increasingly complex Gaussian mixture models – building up from fitting basic univariate models to more complex multivariate models fit to real-world data. We also explore how PyMC3 can be configured to obtain significant increases in computational speed by taking advantage of a machine’s GPU, in addition to the conditions under which such acceleration can be expected. All example analyses are detailed with step-by-step instructions and corresponding Python code. Keywords: probabilistic programming; Bayesian data analysis; Markov chain Monte Carlo; computational modeling; Gaussian mixture modeling RUNNING HEAD: Introduction to PyMC3 with Gaussian Mixture Models 3 1 Introduction Over the last decade, there has been a shift in the norms of what counts as rigorous research in psychological science.
    [Show full text]
  • Hamiltonian Monte Carlo Swindles
    Hamiltonian Monte Carlo Swindles Dan Piponi Matthew D. Hoffman Pavel Sountsov Google Research Google Research Google Research Abstract become indistinguishable (Mangoubi and Smith, 2017; Bou-Rabee et al., 2018). These results were proven in Hamiltonian Monte Carlo (HMC) is a power- service of proofs of convergence and mixing speed for ful Markov chain Monte Carlo (MCMC) algo- HMC, but have been used by Heng and Jacob (2019) rithm for estimating expectations with respect to derive unbiased estimators from finite-length HMC to continuous un-normalized probability dis- chains. tributions. MCMC estimators typically have Empirically, we observe that two HMC chains targeting higher variance than classical Monte Carlo slightly different stationary distributions still couple with i.i.d. samples due to autocorrelations; approximately when driven by the same randomness. most MCMC research tries to reduce these In this paper, we propose methods that exploit this autocorrelations. In this work, we explore approximate-coupling phenomenon to reduce the vari- a complementary approach to variance re- ance of HMC-based estimators. These methods are duction based on two classical Monte Carlo based on two classical Monte Carlo variance-reduction “swindles”: first, running an auxiliary coupled techniques (sometimes called “swindles”): antithetic chain targeting a tractable approximation to sampling and control variates. the target distribution, and using the aux- iliary samples as control variates; and sec- In the antithetic scheme, we induce anti-correlation ond, generating anti-correlated ("antithetic") between two chains by driving the second chain with samples by running two chains with flipped momentum variables that are negated versions of the randomness. Both ideas have been explored momenta from the first chain.
    [Show full text]
  • Is Structure Based Drug Design Ready for Selectivity Optimization?
    bioRxiv preprint doi: https://doi.org/10.1101/2020.07.02.185132; this version posted July 3, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. PrEPRINT AHEAD OF SUBMISSION — July 2, 2020 1 IS STRUCTURE BASED DRUG DESIGN READY FOR 2 SELECTIVITY optimization? 1,2,† 2 3 4 4 3 SteVEN K. Albanese , John D. ChoderA , AndrEA VOLKAMER , Simon Keng , Robert Abel , 4* 4 Lingle WANG Louis V. Gerstner, Jr. GrADUATE School OF Biomedical Sciences, Memorial Sloan Kettering Cancer 5 1 Center, NeW York, NY 10065; Computational AND Systems Biology PrOGRam, Sloan Kettering 6 2 Institute, Memorial Sloan Kettering Cancer Center, NeW York, NY 10065; Charité – 7 3 Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin; Schrödinger, NeW York, NY 10036 8 4 [email protected] (LW) 9 *For CORRespondence: †Schrödinger, NeW York, NY 10036 10 PrESENT ADDRess: 11 12 AbstrACT 13 Alchemical FREE ENERGY CALCULATIONS ARE NOW WIDELY USED TO DRIVE OR MAINTAIN POTENCY IN SMALL MOLECULE LEAD 14 OPTIMIZATION WITH A ROUGHLY 1 Kcal/mol ACCURACY. Despite this, THE POTENTIAL TO USE FREE ENERGY CALCULATIONS 15 TO DRIVE OPTIMIZATION OF COMPOUND SELECTIVITY AMONG TWO SIMILAR TARGETS HAS BEEN RELATIVELY UNEXPLORED IN 16 PUBLISHED studies. IN THE MOST OPTIMISTIC scenario, THE SIMILARITY OF BINDING SITES MIGHT LEAD TO A FORTUITOUS 17 CANCELLATION OF ERRORS AND ALLOW SELECTIVITY TO BE PREDICTED MORE ACCURATELY THAN AffiNITY. Here, WE ASSESS 18 THE ACCURACY WITH WHICH SELECTIVITY CAN BE PREDICTED IN THE CONTEXT OF SMALL MOLECULE KINASE inhibitors, 19 CONSIDERING THE VERY SIMILAR BINDING SITES OF HUMAN KINASES CDK2 AND CDK9, AS WELL AS ANOTHER SERIES OF 20 LIGANDS ATTEMPTING TO ACHIEVE SELECTIVITY BETWEEN THE MORE DISTANTLY RELATED KINASES CDK2 AND ERK2.
    [Show full text]
  • Hamiltonian Monte Carlo Within Stan
    Hamiltonian Monte Carlo within Stan Daniel Lee Columbia University, Statistics Department [email protected] BayesComp mc-stan.org 1 Why MCMC? • Have data. • Have a rich statistical model. • No analytic solution. • (Point estimate not adequate.) 2 Review: MCMC • Markov Chain Monte Carlo. The samples form a Markov Chain. • Markov property: Pr(θn+1 j θ1; : : : ; θn) = Pr(θn+1 j θn) • Invariant distribution: π × Pr = π • Detailed balance: sufficient condition: R Pr(θn+1; A) = A q(θn+1; θn)dy π(θn+1)q(θn+1; θn) = π(θn)q(θn; θn+1) 3 Review: RWMH • Want: samples from posterior distribution: Pr(θjx) • Need: some function proportional to joint model. f (x, θ) / Pr(x, θ) • Algorithm: Given f (x, θ), x, N, Pr(θn+1 j θn) For n = 1 to N do Sample θˆ ∼ q(θˆ j θn−1) f (x,θ)ˆ With probability α = min 1; , set θn θˆ, else θn θn−1 f (x,θn−1) 4 Review: Hamiltonian Dynamics • (Implicit: d = dimension) • q = position (d-vector) • p = momentum (d-vector) • U(q) = potential energy • K(p) = kinectic energy • Hamiltonian system: H(q; p) = U(q) + K(p) 5 Review: Hamiltonian Dynamics • for i = 1; : : : ; d dqi = @H dt @pi dpi = − @H dt @qi • kinectic energy usually defined as K(p) = pT M−1p=2 • for i = 1; : : : ; d dqi −1 dt = [M p]i dpi = − @U dt @qi 6 Connection to MCMC • q, position, is the vector of parameters • U(q), potential energy, is (proportional to) the minus the log probability density of the parameters • p, momentum, are augmented variables • K(p), kinectic energy, is calculated • Hamiltonian dynamics used to update q.
    [Show full text]
  • Mixed Hamiltonian Monte Carlo for Mixed Discrete and Continuous Variables
    Mixed Hamiltonian Monte Carlo for Mixed Discrete and Continuous Variables Guangyao Zhou Vicarious AI Union City, CA 94587, USA [email protected] Abstract Hamiltonian Monte Carlo (HMC) has emerged as a powerful Markov Chain Monte Carlo (MCMC) method to sample from complex continuous distributions. However, a fundamental limitation of HMC is that it can not be applied to distributions with mixed discrete and continuous variables. In this paper, we propose mixed HMC (M- HMC) as a general framework to address this limitation. M-HMC is a novel family of MCMC algorithms that evolves the discrete and continuous variables in tandem, allowing more frequent updates of discrete variables while maintaining HMC’s ability to suppress random-walk behavior. We establish M-HMC’s theoretical properties, and present an efficient implementation with Laplace momentum that introduces minimal overhead compared to existing HMC methods. The superior performances of M-HMC over existing methods are demonstrated with numerical experiments on Gaussian mixture models (GMMs), variable selection in Bayesian logistic regression (BLR), and correlated topic models (CTMs). 1 Introduction Markov chain Monte Carlo (MCMC) is one of the most powerful methods for sampling from probability distributions. The Metropolis-Hastings (MH) algorithm is a commonly used general- purpose MCMC method, yet is inefficient for complex, high-dimensional distributions because of the random walk nature of its movements. Recently, Hamiltonian Monte Carlo (HMC) [13, 22, 2] has emerged as a powerful alternative to MH for complex continuous distributions due to its ability to follow the curvature of target distributions using gradients information and make distant proposals with high acceptance probabilities.
    [Show full text]
  • Migrating MATLAB®To Python
    Migrating MATLAB® to Python Strategies, Comparisons and a Guide to Converting for Experts Migrating MATLAB® to Python Strategies, Comparisons and a Guide to Converting for Experts Alexandre Chabot-Leclerc Enthought, Inc. ©2020 Enthought, Inc. Written by Enthought, Inc. All Rights Reserved. Use only permitted under license. Copying, sharing, redistributing, or other unauthorized use strictly prohibited. All trademarks and registered trademarks are the property of their respective owners. MATLAB and Simulink are registered trademark of The MathWorks, Inc. Enthought, Inc. 200 W Cesar Chavez St Suite 202 Austin, TX 78701 United States www.enthought.com Version 1.2.0 Migrating MATLAB® to Python B Introduction 1 Why Python 2 Diff erences Between Python and MATLAB® 4 Fundamental Data Types 4 Organizing Code in Packages, not Toolboxes 6 Syntax 6 Indexing and Slicing: Why Zero-Based Indexing 8 NumPy Arrays Are Not Matrices 10 Programming Paradigm: Object-Oriented vs. Procedural 13 Anti-Patterns 15 How Do I? 17 Load Data 17 Signal Processing 19 Linear Algebra 19 Machine Learning 20 Statistical Analysis 21 Image Processing and Computer Vision 21 Optimization 22 Natural Language Processing 22 Data Visualization 22 Save Data 25 What Else? 26 Strategies for Converting to Python 27 From the Bottom Up: Converting One Function at a Time 27 From the Top Down: Calling Python from MATLAB® 33 What Next? 37 Acknowledgments 37 Appendix 37 Code Example: Profiling Contiguous Array Operations 37 Complete Version of main.py, in Chapter 4 38 References 39 Accelerate your Python migration with Enthought’s Python for Scientists and Engineers training course! Migrating MATLAB® to Python C Introduction This document will guide you through the transition from MATLAB® to Python.
    [Show full text]
  • Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations
    risks Article Markov Chain Monte Carlo Methods for Estimating Systemic Risk Allocations Takaaki Koike * and Marius Hofert Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON N2L 3G1, Canada; [email protected] * Correspondence: [email protected] Received: 24 September 2019; Accepted: 7 January 2020; Published: 15 January 2020 Abstract: In this paper, we propose a novel framework for estimating systemic risk measures and risk allocations based on Markov Chain Monte Carlo (MCMC) methods. We consider a class of allocations whose jth component can be written as some risk measure of the jth conditional marginal loss distribution given the so-called crisis event. By considering a crisis event as an intersection of linear constraints, this class of allocations covers, for example, conditional Value-at-Risk (CoVaR), conditional expected shortfall (CoES), VaR contributions, and range VaR (RVaR) contributions as special cases. For this class of allocations, analytical calculations are rarely available, and numerical computations based on Monte Carlo (MC) methods often provide inefficient estimates due to the rare-event character of the crisis events. We propose an MCMC estimator constructed from a sample path of a Markov chain whose stationary distribution is the conditional distribution given the crisis event. Efficient constructions of Markov chains, such as the Hamiltonian Monte Carlo and Gibbs sampler, are suggested and studied depending on the crisis event and the underlying loss distribution. The efficiency of the MCMC estimators is demonstrated in a series of numerical experiments. Keywords: systemic risk measures; conditional Value-at-Risk (CoVaR); capital allocation; copula models; quantitative risk management 1.
    [Show full text]
  • The Geometric Foundations of Hamiltonian Monte Carlo 3
    Submitted to Statistical Science The Geometric Foundations of Hamiltonian Monte Carlo Michael Betancourt, Simon Byrne, Sam Livingstone, and Mark Girolami Abstract. Although Hamiltonian Monte Carlo has proven an empirical success, the lack of a rigorous theoretical understanding of the algo- rithm has in many ways impeded both principled developments of the method and use of the algorithm in practice. In this paper we develop the formal foundations of the algorithm through the construction of measures on smooth manifolds, and demonstrate how the theory natu- rally identifies efficient implementations and motivates promising gen- eralizations. Key words and phrases: Markov Chain Monte Carlo, Hamiltonian Monte Carlo, Disintegration, Differential Geometry, Smooth Manifold, Fiber Bundle, Riemannian Geometry, Symplectic Geometry. The frontier of Bayesian inference requires algorithms capable of fitting complex mod- els with hundreds, if not thousands of parameters, intricately bound together with non- linear and often hierarchical correlations. Hamiltonian Monte Carlo (Duane et al., 1987; Neal, 2011) has proven tremendously successful at extracting inferences from these mod- els, with applications spanning computer science (Sutherland, P´oczos and Schneider, 2013; Tang, Srivastava and Salakhutdinov, 2013), ecology (Schofield et al., 2014; Terada, Inoue and Nishihara, 2013), epidemiology (Cowling et al., 2012), linguistics (Husain, Vasishth and Srinivasan, 2014), pharmacokinetics (Weber et al., 2014), physics (Jasche et al., 2010; Porter and Carr´e,
    [Show full text]
  • Probabilistic Programming in Python Using Pymc3
    Probabilistic programming in Python using PyMC3 John Salvatier1, Thomas V. Wiecki2 and Christopher Fonnesbeck3 1 AI Impacts, Berkeley, CA, United States 2 Quantopian Inc, Boston, MA, United States 3 Department of Biostatistics, Vanderbilt University, Nashville, TN, United States ABSTRACT Probabilistic programming allows for automatic Bayesian inference on user-defined probabilistic models. Recent advances in Markov chain Monte Carlo (MCMC) sampling allow inference on increasingly complex models. This class of MCMC, known as Hamiltonian Monte Carlo, requires gradient information which is often not readily available. PyMC3 is a new open source probabilistic programming framework written in Python that uses Theano to compute gradients via automatic differentiation as well as compile probabilistic programs on-the-fly to C for increased speed. Contrary to other probabilistic programming languages, PyMC3 allows model specification directly in Python code. The lack of a domain specific language allows for great flexibility and direct interaction with the model. This paper is a tutorial-style introduction to this software package. Subjects Data Mining and Machine Learning, Data Science, Scientific Computing and Simulation Keywords Bayesian statistic, Probabilistic Programming, Python, Markov chain Monte Carlo, Statistical modeling INTRODUCTION Probabilistic programming (PP) allows for flexible specification and fitting of Bayesian Submitted 9 September 2015 statistical models. PyMC3 is a new, open-source PP framework with an intuitive and Accepted 8 March 2016 readable, yet powerful, syntax that is close to the natural syntax statisticians use to Published 6 April 2016 describe models. It features next-generation Markov chain Monte Carlo (MCMC) Corresponding author Thomas V. Wiecki, sampling algorithms such as the No-U-Turn Sampler (NUTS) (Hoffman & Gelman, [email protected] 2014), a self-tuning variant of Hamiltonian Monte Carlo (HMC) (Duane et al., 1987).
    [Show full text]