The EM algorithm for HMMs
|
|
- Brent Morrison
- 5 years ago
- Views:
Transcription
1 The EM algorithm for HMMs Michael Collins February 22, 2012
2 Maximum-Likelihood Estimation for Fully Observed Data (Recap from earlier) We have fully observed data, x i,1... x i,m, s i,1... s i,m for i = 1... n. The likelihood function is n L(θ) = log p(x i,1... x i,m, s i,1... s i,m ; θ) i=1 Maximum-likelihood estimates of transition probabilities are n t(s i=1 s) = count(i, s s ) n i=1 s count(i, s s ) Maximum-likelihood estimates of emission probabilities are e(x s) = n i=1 count(i, s x) n i=1 x count(i, s x)
3 Maximum-Likelihood Estimation for Partially Observed Data We have partially observed data, x i,1... x i,m for i = 1... n. Note we do not have state sequences. The likelihood function is n L(θ) = log p(x i,1... x i,m, s 1... s m ; θ) s 1...s m i=1 We can maximize this function using EM... (the algorithm will converge to a local maximum of the likelihood function)
4 An Example Suppose we have an HMM with two states (k = 2) and 4 possible emissions (a, b, x, y) and our (partially observed) training data consists of the following counts of 4 different sequences (no other sequences are seen): a x (100 times) a y (100 times) b x (100 times) b y (100 times) What are the maximum-likelihood estimates for the HMM?
5 Forward and Backward Probabilities Define α[j, s] to be the sum of probabilities of all paths ending in state s at position j in the sequence, for j = 1... m and s {1... k}. More formally: α[j, s] = s 1...s j 1 [ t(s 1 )e(x 1 s 1 ) ( j 1 k=2 t(s k s k 1 )e(x k s k ) ) t(s s j 1 )e(x j s) Define β[j, s] for s {1... k} and j {1... (m 1)} to be the sum of probabilities of all paths starting with state s at position j and going to the end of the sequence. More formally: β[j, s] = s j+1...s m t(s j+1 s)e(x j+1 s j+1 ) m k=j+2 t(s k s k 1 )e(x k s k ) ]
6 Recursive Definitions of the Forward Probabilities Initialization: for s = 1... k α[1, s] = t(s)e(x 1 s) For j = 2... m: α[j, s] = (α[j 1, s ] t(s s ) e(x j s)) s {1...k}
7 Recursive Definitions of the Backward Probabilities Initialization: for s = 1... k β[m, s] = 1 For j = m : β[j, s] = (β[j + 1, s ] t(s s) e(x j+1 s )) s {1...k}
8 The Forward-Backward Algorithm Given these definitions: p(x 1... x m, S j = s; θ) = s 1...s m:s j =s p(x 1... x m, s 1... s m ; θ) = α[j, s] β[j, s] Note: we ll assume the special definition that β[m, s] = 1 for all s
9 The Forward-Backward Algorithm Given these definitions: p(x 1... x m, S j = s, S j+1 = s ; θ) = s 1...s m:s j =s,s j+1 =s p(x 1... x m, s 1... s m ; θ) = α[j, s] t(s s) e(x j+1 s ) β[j + 1, s ] Note: we ll assume the special definition that β[m, s] = 1 for all s
10 Things we can Compute Using Forward-Backward Probabilities The probability of any sequence: p(x 1... x m ; θ) = = s s 1...s m p(x 1... x m, s 1... s m ; θ) α[m, s] The probability of any state transition: p(x 1... x m, S j = s, S j+1 = s ; θ) = p(x 1... x m, s 1... s m ; θ) s 1...s m:s j =s,s j+1 =s = α[j, s] t(s s) e(x j+1 s ) β[j + 1, s ]
11 Things we can Compute Using Forward-Backward Probabilities (continued) The conditional probability of any state transition: p(s j = s, S j+1 = s x 1... x m ; θ) = α[j, s] t(s s) e(x j+1 s ) β[j + 1, s ] α[m, s] s The conditional probability of any state at any position: p(s j = s x 1... x m ; θ) = α[j, s] β[j, s] α[m, s] s
12 Things we can Compute Using Forward-Backward Probabilities (continued) Define count(i, s s ; θ) to be the expected number of times the transition s s is seen in the training example x i,1, x i,2,..., x i,m, for parameters θ. Then count(i, s s ; θ) = m 1 j=1 p(s j = s, S j+1 = s x i,1... x i,m ; θ) (We can compute p(s j = s, S j+1 = s x i,1... x i,m ; θ) using the forward-backward probabilities, see previous slide)
13 Things we can Compute Using Forward-Backward Probabilities (continued) For completeness, a formal definition of count(i, s s ; θ): count(i, s s ; θ) = s 1...s m p(s 1... s m x i,1... x i,m ; θ)count(s s, s 1... s m ) where count(s s, s 1... s m ) is the number of times the transition s s is seen in the sequence s 1... s m
14 Things we can Compute Using Forward-Backward Probabilities (continued) Define count(i, s z; θ) to be the expected number of times the state s is paired with the emission z in the training sequence x i,1, x i,2,..., x i,m, for parameters θ. Then count(i, s z; θ) = m p(s j = s x i,1... x i,m ; θ)[[x i,j = z]] j=1 (We can compute p(s j = s x i,1... x i,m ; θ) using the forward-backward probabilities, see previous slides)
15 The EM Algorithm for HMMs Initialization: set initial parameters θ 0 to some value For t = 1... T : Use the forward-backward algorithm to compute all expected counts of the form count(i, s s ; θ t 1 ) or count(i, s z; θ t 1 ) Update the parameters based on the expected counts: n t t (s i=1 s) = count(i, s s ; θ t 1 ) n i=1 s count(i, s s ; θ t 1 ) n e t i=1 (x s) = count(i, s x; θt 1 ) n i=1 x count(i, s x; θt 1 )
16 The Initial State Probabilities For simplicity I ve omitted the estimates for the initial state parameters t(s), but these are simple to derive in a similar way to the transition and the emission parameters For completeness, the expected counts are: count(i, s; θ t 1 ) = α[1, s] β[1, s] α[m, s] s (the expected number of times state s is seen as the initial state) The parameter updates are then t t (s) = n i=1 count(i, s; θt 1 ) n
Notes on the EM Algorithm Michael Collins, September 24th 2005
Notes on the EM Algorithm Michael Collins, September 24th 2005 1 Hidden Markov Models A hidden Markov model (N, Σ, Θ) consists of the following elements: N is a positive integer specifying the number of
More informationA potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples
1.3 Regime switching models A potentially useful approach to model nonlinearities in time series is to assume different behavior (structural break) in different subsamples (or regimes). If the dates, the
More informationComputer Vision Group Prof. Daniel Cremers. 7. Sequential Data
Group Prof. Daniel Cremers 7. Sequential Data Bayes Filter (Rep.) We can describe the overall process using a Dynamic Bayes Network: This incorporates the following Markov assumptions: (measurement) (state)!2
More informationExact Inference (9/30/13) 2 A brief review of Forward-Backward and EM for HMMs
STA561: Probabilistic machine learning Exact Inference (9/30/13) Lecturer: Barbara Engelhardt Scribes: Jiawei Liang, He Jiang, Brittany Cohen 1 Validation for Clustering If we have two centroids, η 1 and
More informationEstimation of the Markov-switching GARCH model by a Monte Carlo EM algorithm
Estimation of the Markov-switching GARCH model by a Monte Carlo EM algorithm Maciej Augustyniak Fields Institute February 3, 0 Stylized facts of financial data GARCH Regime-switching MS-GARCH Agenda Available
More informationChapter 7: Estimation Sections
Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions Frequentist Methods: 7.5 Maximum Likelihood Estimators
More informationDecision Theory: Value Iteration
Decision Theory: Value Iteration CPSC 322 Decision Theory 4 Textbook 9.5 Decision Theory: Value Iteration CPSC 322 Decision Theory 4, Slide 1 Lecture Overview 1 Recap 2 Policies 3 Value Iteration Decision
More informationHidden Markov Models. Selecting model parameters or training
idden Markov Models Selecting model parameters or training idden Markov Models Motivation: The n'th observation in a chain of observations is influenced by a corresponding latent variable... Observations
More informationa 13 Notes on Hidden Markov Models Michael I. Jordan University of California at Berkeley Hidden Markov Models The model
Notes on Hidden Markov Models Michael I. Jordan University of California at Berkeley Hidden Markov Models This is a lightly edited version of a chapter in a book being written by Jordan. Since this is
More informationBCJR Algorithm. Veterbi Algorithm (revisted) Consider covolutional encoder with. And information sequences of length h = 5
Chapter 2 BCJR Algorithm Ammar Abh-Hhdrohss Islamic University -Gaza ١ Veterbi Algorithm (revisted) Consider covolutional encoder with And information sequences of length h = 5 The trellis diagram has
More informationLecture 17: More on Markov Decision Processes. Reinforcement learning
Lecture 17: More on Markov Decision Processes. Reinforcement learning Learning a model: maximum likelihood Learning a value function directly Monte Carlo Temporal-difference (TD) learning COMP-424, Lecture
More informationProbability Distributions: Discrete
Probability Distributions: Discrete Introduction to Data Science Algorithms Jordan Boyd-Graber and Michael Paul SEPTEMBER 27, 2016 Introduction to Data Science Algorithms Boyd-Graber and Paul Probability
More informationLecture Notes: November 29, 2012 TIME AND UNCERTAINTY: FUTURES MARKETS
Lecture Notes: November 29, 2012 TIME AND UNCERTAINTY: FUTURES MARKETS Gerard says: theory's in the math. The rest is interpretation. (See Debreu quote in textbook, p. 204) make the markets for goods over
More informationLecture Notes 6. Assume F belongs to a family of distributions, (e.g. F is Normal), indexed by some parameter θ.
Sufficient Statistics Lecture Notes 6 Sufficiency Data reduction in terms of a particular statistic can be thought of as a partition of the sample space X. Definition T is sufficient for θ if the conditional
More informationEstimating Mixed Logit Models with Large Choice Sets. Roger H. von Haefen, NC State & NBER Adam Domanski, NOAA July 2013
Estimating Mixed Logit Models with Large Choice Sets Roger H. von Haefen, NC State & NBER Adam Domanski, NOAA July 2013 Motivation Bayer et al. (JPE, 2007) Sorting modeling / housing choice 250,000 individuals
More informationLogistics. CS 473: Artificial Intelligence. Markov Decision Processes. PS 2 due today Midterm in one week
CS 473: Artificial Intelligence Markov Decision Processes Dan Weld University of Washington [Slides originally created by Dan Klein & Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188 materials
More informationNotes on Syllabus Section VI: TIME AND UNCERTAINTY, FUTURES MARKETS
Economics 200B UCSD; Prof. R. Starr, Ms. Kaitlyn Lewis, Winter 2017; Syllabus Section VI Notes1 Notes on Syllabus Section VI: TIME AND UNCERTAINTY, FUTURES MARKETS Overview: The mathematical abstraction
More informationHandout 4: Deterministic Systems and the Shortest Path Problem
SEEM 3470: Dynamic Optimization and Applications 2013 14 Second Term Handout 4: Deterministic Systems and the Shortest Path Problem Instructor: Shiqian Ma January 27, 2014 Suggested Reading: Bertsekas
More informationPakes (1986): Patents as Options: Some Estimates of the Value of Holding European Patent Stocks
Pakes (1986): Patents as Options: Some Estimates of the Value of Holding European Patent Stocks Spring 2009 Main question: How much are patents worth? Answering this question is important, because it helps
More informationHidden Markov Model for High Frequency Data
Hidden Markov Model for High Frequency Data Department of Mathematics, Florida State University Joint Math Meeting, Baltimore, MD, January 15 What are HMMs? A Hidden Markov model (HMM) is a stochastic
More informationReasoning with Uncertainty
Reasoning with Uncertainty Markov Decision Models Manfred Huber 2015 1 Markov Decision Process Models Markov models represent the behavior of a random process, including its internal state and the externally
More informationEstimating a Dynamic Oligopolistic Game with Serially Correlated Unobserved Production Costs. SS223B-Empirical IO
Estimating a Dynamic Oligopolistic Game with Serially Correlated Unobserved Production Costs SS223B-Empirical IO Motivation There have been substantial recent developments in the empirical literature on
More information4 Reinforcement Learning Basic Algorithms
Learning in Complex Systems Spring 2011 Lecture Notes Nahum Shimkin 4 Reinforcement Learning Basic Algorithms 4.1 Introduction RL methods essentially deal with the solution of (optimal) control problems
More information6. Genetics examples: Hardy-Weinberg Equilibrium
PBCB 206 (Fall 2006) Instructor: Fei Zou email: fzou@bios.unc.edu office: 3107D McGavran-Greenberg Hall Lecture 4 Topics for Lecture 4 1. Parametric models and estimating parameters from data 2. Method
More informationAlgorithms and Networking for Computer Games
Algorithms and Networking for Computer Games Chapter 4: Game Trees http://www.wiley.com/go/smed Game types perfect information games no hidden information two-player, perfect information games Noughts
More informationChapter 4: Asymptotic Properties of MLE (Part 3)
Chapter 4: Asymptotic Properties of MLE (Part 3) Daniel O. Scharfstein 09/30/13 1 / 1 Breakdown of Assumptions Non-Existence of the MLE Multiple Solutions to Maximization Problem Multiple Solutions to
More informationAsymptotic results discrete time martingales and stochastic algorithms
Asymptotic results discrete time martingales and stochastic algorithms Bernard Bercu Bordeaux University, France IFCAM Summer School Bangalore, India, July 2015 Bernard Bercu Asymptotic results for discrete
More informationUnobserved Heterogeneity Revisited
Unobserved Heterogeneity Revisited Robert A. Miller Dynamic Discrete Choice March 2018 Miller (Dynamic Discrete Choice) cemmap 7 March 2018 1 / 24 Distributional Assumptions about the Unobserved Variables
More informationCS340 Machine learning Bayesian model selection
CS340 Machine learning Bayesian model selection Bayesian model selection Suppose we have several models, each with potentially different numbers of parameters. Example: M0 = constant, M1 = straight line,
More informationReinforcement Learning. Slides based on those used in Berkeley's AI class taught by Dan Klein
Reinforcement Learning Slides based on those used in Berkeley's AI class taught by Dan Klein Reinforcement Learning Basic idea: Receive feedback in the form of rewards Agent s utility is defined by the
More informationReinforcement Learning
Reinforcement Learning Basic idea: Receive feedback in the form of rewards Agent s utility is defined by the reward function Must (learn to) act so as to maximize expected rewards Grid World The agent
More informationTwo hours. To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER
Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions.
More informationStatistical estimation
Statistical estimation Statistical modelling: theory and practice Gilles Guillot gigu@dtu.dk September 3, 2013 Gilles Guillot (gigu@dtu.dk) Estimation September 3, 2013 1 / 27 1 Introductory example 2
More informationNon-Deterministic Search
Non-Deterministic Search MDP s 1 Non-Deterministic Search How do you plan (search) when your actions might fail? In general case, how do you plan, when the actions have multiple possible outcomes? 2 Example:
More informationBack to estimators...
Back to estimators... So far, we have: Identified estimators for common parameters Discussed the sampling distributions of estimators Introduced ways to judge the goodness of an estimator (bias, MSE, etc.)
More informationSome Discrete Distribution Families
Some Discrete Distribution Families ST 370 Many families of discrete distributions have been studied; we shall discuss the ones that are most commonly found in applications. In each family, we need a formula
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532l Lecture 10 Stochastic Games and Bayesian Games CPSC 532l Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games 4 Analyzing Bayesian
More informationThe change of correlation structure across industries: an analysis in the regime-switching framework
Kyoto University, Graduate School of Economics Research Project Center Discussion Paper Series The change of correlation structure across industries: an analysis in the regime-switching framework Masahiko
More informationCS 188: Artificial Intelligence
CS 188: Artificial Intelligence Markov Decision Processes Dan Klein, Pieter Abbeel University of California, Berkeley Non Deterministic Search Example: Grid World A maze like problem The agent lives in
More informationHidden Markov Models for Financial Market Predictions
Hidden Markov Models for Financial Market Predictions Department of Mathematics and Statistics Youngstown State University Central Spring Sectional Meeting, Michigan State University, March 15 1 Introduction
More informationChapter 7: Estimation Sections
1 / 40 Chapter 7: Estimation Sections 7.1 Statistical Inference Bayesian Methods: Chapter 7 7.2 Prior and Posterior Distributions 7.3 Conjugate Prior Distributions 7.4 Bayes Estimators Frequentist Methods:
More informationSTAT 111 Recitation 2
STAT 111 Recitation 2 Linjun Zhang October 10, 2017 Misc. Please collect homework 1 (graded). 1 Misc. Please collect homework 1 (graded). Office hours: 4:30-5:30pm every Monday, JMHH F86. 1 Misc. Please
More informationOccasional Paper. Dynamic Methods for Analyzing Hedge-Fund Performance: A Note Using Texas Energy-Related Funds. Jiaqi Chen and Michael L.
DALLASFED Occasional Paper Dynamic Methods for Analyzing Hedge-Fund Performance: A Note Using Texas Energy-Related Funds Jiaqi Chen and Michael L. Tindall Federal Reserve Bank of Dallas Financial Industry
More informationSequential Decision Making
Sequential Decision Making Dynamic programming Christos Dimitrakakis Intelligent Autonomous Systems, IvI, University of Amsterdam, The Netherlands March 18, 2008 Introduction Some examples Dynamic programming
More informationSTATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics. Ph. D. Comprehensive Examination: Macroeconomics Fall, 2010
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Comprehensive Examination: Macroeconomics Fall, 2010 Section 1. (Suggested Time: 45 Minutes) For 3 of the following 6 statements, state
More informationStat 260/CS Learning in Sequential Decision Problems. Peter Bartlett
Stat 260/CS 294-102. Learning in Sequential Decision Problems. Peter Bartlett 1. Gittins Index: Discounted, Bayesian (hence Markov arms). Reduces to stopping problem for each arm. Interpretation as (scaled)
More informationIEOR E4004: Introduction to OR: Deterministic Models
IEOR E4004: Introduction to OR: Deterministic Models 1 Dynamic Programming Following is a summary of the problems we discussed in class. (We do not include the discussion on the container problem or the
More informationPolitical Lobbying in a Recurring Environment
Political Lobbying in a Recurring Environment Avihai Lifschitz Tel Aviv University This Draft: October 2015 Abstract This paper develops a dynamic model of the labor market, in which the employed workers,
More informationModelling financial data with stochastic processes
Modelling financial data with stochastic processes Vlad Ardelean, Fabian Tinkl 01.08.2012 Chair of statistics and econometrics FAU Erlangen-Nuremberg Outline Introduction Stochastic processes Volatility
More informationCourse information FN3142 Quantitative finance
Course information 015 16 FN314 Quantitative finance This course is aimed at students interested in obtaining a thorough grounding in market finance and related empirical methods. Prerequisite If taken
More informationTo earn the extra credit, one of the following has to hold true. Please circle and sign.
CS 188 Fall 2018 Introduction to rtificial Intelligence Practice Midterm 2 To earn the extra credit, one of the following has to hold true. Please circle and sign. I spent 2 or more hours on the practice
More informationIntroduction to Political Economy Problem Set 3
Introduction to Political Economy 14.770 Problem Set 3 Due date: Question 1: Consider an alternative model of lobbying (compared to the Grossman and Helpman model with enforceable contracts), where lobbies
More informationDouble Chain Ladder and Bornhutter-Ferguson
Double Chain Ladder and Bornhutter-Ferguson María Dolores Martínez Miranda University of Granada, Spain mmiranda@ugr.es Jens Perch Nielsen Cass Business School, City University, London, U.K. Jens.Nielsen.1@city.ac.uk,
More informationSum-Product: Message Passing Belief Propagation
Sum-Product: Message Passing Belief Propagation 40-956 Advanced Topics in AI: Probabilistic Graphical Models Sharif University of Technology Soleymani Spring 2015 All single-node marginals If we need the
More informationSum-Product: Message Passing Belief Propagation
Sum-Product: Message Passing Belief Propagation Probabilistic Graphical Models Sharif University of Technology Spring 2017 Soleymani All single-node marginals If we need the full set of marginals, repeating
More informationEE641 Digital Image Processing II: Purdue University VISE - October 29,
EE64 Digital Image Processing II: Purdue University VISE - October 9, 004 The EM Algorithm. Suffient Statistics and Exponential Distributions Let p(y θ) be a family of density functions parameterized by
More information1 Overview. 2 The Gradient Descent Algorithm. AM 221: Advanced Optimization Spring 2016
AM 22: Advanced Optimization Spring 206 Prof. Yaron Singer Lecture 9 February 24th Overview In the previous lecture we reviewed results from multivariate calculus in preparation for our journey into convex
More informationBayesian course - problem set 3 (lecture 4)
Bayesian course - problem set 3 (lecture 4) Ben Lambert November 14, 2016 1 Ticked off Imagine once again that you are investigating the occurrence of Lyme disease in the UK. This is a vector-borne disease
More informationHidden Markov Models. Slides by Carl Kingsford. Based on Chapter 11 of Jones & Pevzner, An Introduction to Bioinformatics Algorithms
Hidden Markov Models Slides by Carl Kingsford Based on Chapter 11 of Jones & Pevzner, An Introduction to Bioinformatics Algorithms Eukaryotic Genes & Exon Splicing Prokaryotic (bacterial) genes look like
More informationBMI/CS 776 Lecture #15: Multiple Alignment - ProbCons. Colin Dewey
BMI/CS 776 Lecture #15: Multiple Alignment - ProbCons Colin Dewey 2007.03.13 1 Probabilistic multiple alignment Like Needleman-Wunsch, pair HMMs can be generalized to n > 2 sequences Unfortunately, the
More informationFor every job, the start time on machine j+1 is greater than or equal to the completion time on machine j.
Flow Shop Scheduling - makespan A flow shop is one where all the jobs visit all the machine for processing in the given order. If we consider a flow shop with n jobs and two machines (M1 and M2), all the
More informationPredicting Electricity Pool Prices Using Hidden Markov Models
Preprints of the 9th International Symposium on Advanced Control of Chemical Processes The International Federation of Automatic Control June 7-1, 215, Whistler, British Columbia, Canada MoPoster2.7 Predicting
More informationCMPSCI 311: Introduction to Algorithms Second Midterm Practice Exam SOLUTIONS
CMPSCI 311: Introduction to Algorithms Second Midterm Practice Exam SOLUTIONS November 17, 2016. Name: ID: Instructions: Answer the questions directly on the exam pages. Show all your work for each question.
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532L Lecture 10 Stochastic Games and Bayesian Games CPSC 532L Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games Stochastic Games
More informationProject exam for STK Computational statistics
Project exam for STK4051 - Computational statistics Fall 2017 Part 1 (of 2) This is the first part of the exam project set for STK4051/9051, fall semester 2017. It is made available on the course website
More informationAlgorithmic Trading using Reinforcement Learning augmented with Hidden Markov Model
Algorithmic Trading using Reinforcement Learning augmented with Hidden Markov Model Simerjot Kaur (sk3391) Stanford University Abstract This work presents a novel algorithmic trading system based on reinforcement
More informationCapital Allocation Principles
Capital Allocation Principles Maochao Xu Department of Mathematics Illinois State University mxu2@ilstu.edu Capital Dhaene, et al., 2011, Journal of Risk and Insurance The level of the capital held by
More informationA start of Variational Methods for ERGM Ranran Wang, UW
A start of Variational Methods for ERGM Ranran Wang, UW MURI-UCI April 24, 2009 Outline A start of Variational Methods for ERGM [1] Introduction to ERGM Current methods of parameter estimation: MCMCMLE:
More information1 A tax on capital income in a neoclassical growth model
1 A tax on capital income in a neoclassical growth model We look at a standard neoclassical growth model. The representative consumer maximizes U = β t u(c t ) (1) t=0 where c t is consumption in period
More informationArrow Debreu Equilibrium. October 31, 2015
Arrow Debreu Equilibrium October 31, 2015 Θ 0 = {s 1,...s S } - the set of (unknown) states of the world assuming there are S unknown states. information is complete but imperfect n - number of consumers
More information6.825 Homework 3: Solutions
6.825 Homework 3: Solutions 1 Easy EM You are given the network structure shown in Figure 1 and the data in the following table, with actual observed values for A, B, and C, and expected counts for D.
More informationDefinition 9.1 A point estimate is any function T (X 1,..., X n ) of a random sample. We often write an estimator of the parameter θ as ˆθ.
9 Point estimation 9.1 Rationale behind point estimation When sampling from a population described by a pdf f(x θ) or probability function P [X = x θ] knowledge of θ gives knowledge of the entire population.
More informationLecture 10: Point Estimation
Lecture 10: Point Estimation MSU-STT-351-Sum-17B (P. Vellaisamy: MSU-STT-351-Sum-17B) Probability & Statistics for Engineers 1 / 31 Basic Concepts of Point Estimation A point estimate of a parameter θ,
More informationAcademic Research Review. Classifying Market Conditions Using Hidden Markov Model
Academic Research Review Classifying Market Conditions Using Hidden Markov Model INTRODUCTION Best known for their applications in speech recognition, Hidden Markov Models (HMMs) are able to discern and
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationA Production-Based Model for the Term Structure
A Production-Based Model for the Term Structure U Wharton School of the University of Pennsylvania U Term Structure Wharton School of the University 1 / 19 Production-based asset pricing in the literature
More informationInference in Bayesian Networks
Andrea Passerini passerini@disi.unitn.it Machine Learning Inference in graphical models Description Assume we have evidence e on the state of a subset of variables E in the model (i.e. Bayesian Network)
More informationDecision Theory: Sequential Decisions
Decision Theory: CPSC 322 Decision Theory 2 Textbook 9.3 Decision Theory: CPSC 322 Decision Theory 2, Slide 1 Lecture Overview 1 Recap 2 Decision Theory: CPSC 322 Decision Theory 2, Slide 2 Decision Variables
More informationCS 188: Artificial Intelligence. Outline
C 188: Artificial Intelligence Markov Decision Processes (MDPs) Pieter Abbeel UC Berkeley ome slides adapted from Dan Klein 1 Outline Markov Decision Processes (MDPs) Formalism Value iteration In essence
More informationCSE 473: Artificial Intelligence
CSE 473: Artificial Intelligence Markov Decision Processes (MDPs) Luke Zettlemoyer Many slides over the course adapted from Dan Klein, Stuart Russell or Andrew Moore 1 Announcements PS2 online now Due
More informationA note on the nested Logit model
Erik Biørn Version of September 17 2008 ECON5115 - ECONOMETRICS: MICROECONOMETRICS AND DISCRETE CHOICE AUTUMN 2008 A note on the nested Logit model In this note we present the basic idea of the nested
More informationSTP Problem Set 3 Solutions
STP 425 - Problem Set 3 Solutions 4.4) Consider the separable sequential allocation problem introduced in Sections 3.3.3 and 4.6.3, where the goal is to maximize the sum subject to the constraints f(x
More informationDavid A. Robalino (World Bank) Eduardo Zylberstajn (Fundacao Getulio Vargas, Brazil) Extended Abstract
Incentive Effects of Risk Pooling, Redistributive and Savings Arrangements in Unemployment Benefit Systems: Evidence from a Structural Model for Brazil David A. Robalino (World Bank) Eduardo Zylberstajn
More informationSTATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics. Ph. D. Comprehensive Examination: Macroeconomics Fall, 2016
STATE UNIVERSITY OF NEW YORK AT ALBANY Department of Economics Ph. D. Comprehensive Examination: Macroeconomics Fall, 2016 Section 1. (Suggested Time: 45 Minutes) For 3 of the following 6 statements, state
More informationDRAFT. 1 exercise in state (S, t), π(s, t) = 0 do not exercise in state (S, t) Review of the Risk Neutral Stock Dynamics
Chapter 12 American Put Option Recall that the American option has strike K and maturity T and gives the holder the right to exercise at any time in [0, T ]. The American option is not straightforward
More informationModelling, Estimation and Hedging of Longevity Risk
IA BE Summer School 2016, K. Antonio, UvA 1 / 50 Modelling, Estimation and Hedging of Longevity Risk Katrien Antonio KU Leuven and University of Amsterdam IA BE Summer School 2016, Leuven Module II: Fitting
More informationExercise. Show the corrected sample variance is an unbiased estimator of population variance. S 2 = n i=1 (X i X ) 2 n 1. Exercise Estimation
Exercise Show the corrected sample variance is an unbiased estimator of population variance. S 2 = n i=1 (X i X ) 2 n 1 Exercise S 2 = = = = n i=1 (X i x) 2 n i=1 = (X i µ + µ X ) 2 = n 1 n 1 n i=1 ((X
More informationCSE 473: Ar+ficial Intelligence
CSE 473: Ar+ficial Intelligence Hidden Markov Models Luke Ze@lemoyer - University of Washington [These slides were created by Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188
More information91.420/543: Artificial Intelligence UMass Lowell CS Fall 2010
91.420/543: Artificial Intelligence UMass Lowell CS Fall 2010 Lecture 17 & 18: Markov Decision Processes Oct 12 13, 2010 A subset of Lecture 9 slides from Dan Klein UC Berkeley Many slides over the course
More informationMA 1125 Lecture 14 - Expected Values. Wednesday, October 4, Objectives: Introduce expected values.
MA 5 Lecture 4 - Expected Values Wednesday, October 4, 27 Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationOn the Minimum Description Length Complexity of Multinomial Processing Tree Models
On the Minimum Description Length Complexity of Multinomial Processing Tree Models Hao Wu and Jay I. Myung The Ohio State University William H. Batchelder University of California, Irvine Abstract Multinomial
More informationBetting Against Beta: A State-Space Approach
Betting Against Beta: A State-Space Approach An Alternative to Frazzini and Pederson (2014) David Puelz and Long Zhao UT McCombs April 20, 2015 Overview Background Frazzini and Pederson (2014) A State-Space
More informationMarkov Decision Processes
Markov Decision Processes Robert Platt Northeastern University Some images and slides are used from: 1. CS188 UC Berkeley 2. RN, AIMA Stochastic domains Image: Berkeley CS188 course notes (downloaded Summer
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics May 29, 2015 Instructions This exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationPoint Estimation. Some General Concepts of Point Estimation. Example. Estimator quality
Point Estimation Some General Concepts of Point Estimation Statistical inference = conclusions about parameters Parameters == population characteristics A point estimate of a parameter is a value (based
More informationValuing American Options by Simulation
Valuing American Options by Simulation Hansjörg Furrer Market-consistent Actuarial Valuation ETH Zürich, Frühjahrssemester 2008 Valuing American Options Course material Slides Longstaff, F. A. and Schwartz,
More informationOptimal Stopping. Nick Hay (presentation follows Thomas Ferguson s Optimal Stopping and Applications) November 6, 2008
(presentation follows Thomas Ferguson s and Applications) November 6, 2008 1 / 35 Contents: Introduction Problems Markov Models Monotone Stopping Problems Summary 2 / 35 The Secretary problem You have
More informationCalibration of Interest Rates
WDS'12 Proceedings of Contributed Papers, Part I, 25 30, 2012. ISBN 978-80-7378-224-5 MATFYZPRESS Calibration of Interest Rates J. Černý Charles University, Faculty of Mathematics and Physics, Prague,
More information2D penalized spline (continuous-by-continuous interaction)
2D penalized spline (continuous-by-continuous interaction) Two examples (RWC, Section 13.1): Number of scallops caught off Long Island Counts are made at specific coordinates. Incidence of AIDS in Italian
More informationSUPPLEMENT TO EQUILIBRIA IN HEALTH EXCHANGES: ADVERSE SELECTION VERSUS RECLASSIFICATION RISK (Econometrica, Vol. 83, No. 4, July 2015, )
Econometrica Supplementary Material SUPPLEMENT TO EQUILIBRIA IN HEALTH EXCHANGES: ADVERSE SELECTION VERSUS RECLASSIFICATION RISK (Econometrica, Vol. 83, No. 4, July 2015, 1261 1313) BY BEN HANDEL, IGAL
More information