Probabilistic Programming in Python Using Pymc3
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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. -
Stan: a Probabilistic Programming Language
JSS Journal of Statistical Software January 2017, Volume 76, Issue 1. doi: 10.18637/jss.v076.i01 Stan: A Probabilistic Programming Language Bob Carpenter Andrew Gelman Matthew D. Hoffman Columbia University Columbia University Adobe Creative Technologies Lab Daniel Lee Ben Goodrich Michael Betancourt Columbia University Columbia University Columbia University Marcus A. Brubaker Jiqiang Guo Peter Li York University NPD Group Columbia University Allen Riddell Indiana University Abstract Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.14.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propa- gation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line using the cmdstan package, through R using the rstan package, and through Python using the pystan package. All three interfaces sup- port sampling and optimization-based inference with diagnostics and posterior analysis. rstan and pystan also provide access to log probabilities, gradients, Hessians, parameter transforms, and specialized plotting. -
Stan: a Probabilistic Programming Language
JSS Journal of Statistical Software MMMMMM YYYY, Volume VV, Issue II. http://www.jstatsoft.org/ Stan: A Probabilistic Programming Language Bob Carpenter Andrew Gelman Matt Hoffman Columbia University Columbia University Adobe Research Daniel Lee Ben Goodrich Michael Betancourt Columbia University Columbia University University of Warwick Marcus A. Brubaker Jiqiang Guo Peter Li University of Toronto, NPD Group Columbia University Scarborough Allen Riddell Dartmouth College Abstract Stan is a probabilistic programming language for specifying statistical models. A Stan program imperatively defines a log probability function over parameters conditioned on specified data and constants. As of version 2.2.0, Stan provides full Bayesian inference for continuous-variable models through Markov chain Monte Carlo methods such as the No-U-Turn sampler, an adaptive form of Hamiltonian Monte Carlo sampling. Penalized maximum likelihood estimates are calculated using optimization methods such as the Broyden-Fletcher-Goldfarb-Shanno algorithm. Stan is also a platform for computing log densities and their gradients and Hessians, which can be used in alternative algorithms such as variational Bayes, expectation propa- gation, and marginal inference using approximate integration. To this end, Stan is set up so that the densities, gradients, and Hessians, along with intermediate quantities of the algorithm such as acceptance probabilities, are easily accessible. Stan can be called from the command line, through R using the RStan package, or through Python using the PyStan package. All three interfaces support sampling and optimization-based inference. RStan and PyStan also provide access to log probabilities, gradients, Hessians, and data I/O. Keywords: probabilistic program, Bayesian inference, algorithmic differentiation, Stan. -
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. -
Advancedhmc.Jl: a Robust, Modular and Efficient Implementation of Advanced HMC Algorithms
2nd Symposium on Advances in Approximate Bayesian Inference, 20191{10 AdvancedHMC.jl: A robust, modular and efficient implementation of advanced HMC algorithms Kai Xu [email protected] University of Edinburgh Hong Ge [email protected] University of Cambridge Will Tebbutt [email protected] University of Cambridge Mohamed Tarek [email protected] UNSW Canberra Martin Trapp [email protected] Graz University of Technology Zoubin Ghahramani [email protected] University of Cambridge & Uber AI Labs Abstract Stan's Hamilton Monte Carlo (HMC) has demonstrated remarkable sampling robustness and efficiency in a wide range of Bayesian inference problems through carefully crafted adaption schemes to the celebrated No-U-Turn sampler (NUTS) algorithm. It is challeng- ing to implement these adaption schemes robustly in practice, hindering wider adoption amongst practitioners who are not directly working with the Stan modelling language. AdvancedHMC.jl (AHMC) contributes a modular, well-tested, standalone implementation of NUTS that recovers and extends Stan's NUTS algorithm. AHMC is written in Julia, a modern high-level language for scientific computing, benefiting from optional hardware acceleration and interoperability with a wealth of existing software written in both Julia and other languages, such as Python. Efficacy is demonstrated empirically by comparison with Stan through a third-party Markov chain Monte Carlo benchmarking suite. 1. Introduction Hamiltonian Monte Carlo (HMC) is an efficient Markov chain Monte Carlo (MCMC) algo- rithm which avoids random walks by simulating Hamiltonian dynamics to make proposals (Duane et al., 1987; Neal et al., 2011). Due to the statistical efficiency of HMC, it has been widely applied to fields including physics (Duane et al., 1987), differential equations (Kramer et al., 2014), social science (Jackman, 2009) and Bayesian inference (e.g. -
Maple Advanced Programming Guide
Maple 9 Advanced Programming Guide M. B. Monagan K. O. Geddes K. M. Heal G. Labahn S. M. Vorkoetter J. McCarron P. DeMarco c Maplesoft, a division of Waterloo Maple Inc. 2003. ii ¯ Maple, Maplesoft, Maplet, and OpenMaple are trademarks of Water- loo Maple Inc. c Maplesoft, a division of Waterloo Maple Inc. 2003. All rights re- served. The electronic version (PDF) of this book may be downloaded and printed for personal use or stored as a copy on a personal machine. The electronic version (PDF) of this book may not be distributed. Information in this document is subject to change without notice and does not rep- resent a commitment on the part of the vendor. The software described in this document is furnished under a license agreement and may be used or copied only in accordance with the agreement. It is against the law to copy the software on any medium as specifically allowed in the agreement. Windows is a registered trademark of Microsoft Corporation. Java and all Java based marks are trademarks or registered trade- marks of Sun Microsystems, Inc. in the United States and other countries. Maplesoft is independent of Sun Microsystems, Inc. All other trademarks are the property of their respective owners. This document was produced using a special version of Maple that reads and updates LATEX files. Printed in Canada ISBN 1-894511-44-1 Contents Preface 1 Audience . 1 Worksheet Graphical Interface . 2 Manual Set . 2 Conventions . 3 Customer Feedback . 3 1 Procedures, Variables, and Extending Maple 5 Prerequisite Knowledge . 5 In This Chapter . -
End-User Probabilistic Programming
End-User Probabilistic Programming Judith Borghouts, Andrew D. Gordon, Advait Sarkar, and Neil Toronto Microsoft Research Abstract. Probabilistic programming aims to help users make deci- sions under uncertainty. The user writes code representing a probabilistic model, and receives outcomes as distributions or summary statistics. We consider probabilistic programming for end-users, in particular spread- sheet users, estimated to number in tens to hundreds of millions. We examine the sources of uncertainty actually encountered by spreadsheet users, and their coping mechanisms, via an interview study. We examine spreadsheet-based interfaces and technology to help reason under uncer- tainty, via probabilistic and other means. We show how uncertain values can propagate uncertainty through spreadsheets, and how sheet-defined functions can be applied to handle uncertainty. Hence, we draw conclu- sions about the promise and limitations of probabilistic programming for end-users. 1 Introduction In this paper, we discuss the potential of bringing together two rather distinct approaches to decision making under uncertainty: spreadsheets and probabilistic programming. We start by introducing these two approaches. 1.1 Background: Spreadsheets and End-User Programming The spreadsheet is the first \killer app" of the personal computer era, starting in 1979 with Dan Bricklin and Bob Frankston's VisiCalc for the Apple II [15]. The primary interface of a spreadsheet|then and now, four decades later|is the grid, a two-dimensional array of cells. Each cell may hold a literal data value, or a formula that computes a data value (and may depend on the data in other cells, which may themselves be computed by formulas). -
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. -
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. -
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. -
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. -
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.