#### c

Minnesota, STAT 8931
Excerpt: ... Markov Chain Monte Carlo Maximum Likelihood Charles J. Geyer School of Statistics University of Minnesota Minneapolis, MN 55455 Markov chain Monte Carlo (e. g., the Metropolis algorithm and Gibbs sampler) is a general tool for simulation of complex stochastic processes useful in many types of statistical inference. The basics of Markov chain Monte Carlo are reviewed, including choice of algorithms and variance estimation, and some new methods are introduced. The use of Markov chain Monte Carlo for maximum likelihood estimation is explained, and its performance is compared with maximum pseudo likelihood estimation. Key Words: Markov chain, Monte Carlo, Maximum likelihood, Metropolis algorithm, Gibbs sampler, Variance estimation. For many complex stochastic processes very little can be accomplished by analytic calculations, but simulation of the process is possible using Markov chain Monte Carlo (Metropolis, et al., 1953; Hastings, 1970; Geman and Geman, 1984). The simulation can be used to calculate integral ...

#### c

Minnesota, STAT 8931
Excerpt: ... Markov Chain Monte Carlo Maximum Likelihood Charles J. Geyer School of Statistics University of Minnesota Minneapolis, MN 55455 Markov chain Monte Carlo (e. g., the Metropolis algorithm and Gibbs sampler) is a general tool for simulation of complex stochastic processes useful in many types of statistical inference. The basics of Markov chain Monte Carlo are reviewed, including choice of algorithms and variance estimation, and some new methods are introduced. The use of Markov chain Monte Carlo for maximum likelihood estimation is explained, and its performance is compared with maximum pseudo likelihood estimation. Key Words: Markov chain, Monte Carlo, Maximum likelihood, Metropolis algorithm, Gibbs sampler, Variance estimation. For many complex stochastic processes very little can be accomplished by analytic calculations, but simulation of the process is possible using Markov chain Monte Carlo (Metropolis, et al., 1953; Hastings, 1970; Geman and Geman, 1984). The simulation can be used to calculate integral ...

#### topics

Wisconsin, CS 809
Excerpt: ... gorithms, bin packing, etc. 4. Random graphs. The Gn.p model and random graph evolution. Geometric processes for random networks. Average-behavior analysis of algorithms for minimum spanning trees, matching, traveling salesman and other NP-complete problems. 5. Other algorithmically interesting processes. Martingales. Markov chain Monte Carlo . Percolation. Branching random walks. Random bisection. Queues. Applications to probabilistic counting, approximation of hard counting problems, asynchronous computation, greedy optimization, random variate generation, etc. ...

#### flyer

UCLA, MATH 473
Excerpt: ... pical problem instances drawn from an ensemble. Possible topics include: Markov Chain Monte Carlo ; simulated annealing; spin models; average-case complexity; phase transitions in combinatorial optimization; solution clustering; message-passing algorithms. Students will be exposed to potential thesis topics at the junction of applied mathematics, computational complexity and statistical physics. Instructor: Prof. Allon Percus Background and prerequisites: The seminar is intended for a broad audience, including graduate students as well as advanced undergraduates in mathematics, computer science, electrical engineering and physics. Knowledge of probability and combinatorics is assumed. Other helpful (but not required) background includes analysis of algorithms, computational complexity, and statistical mechanics. Format: A combination of lecture, discussions, and student presentations. ...

#### M9_Slides_Cognition

Acton School of Business, CAAM 415
Excerpt: ... ons The beliefs, goals and plans of other people Social structures, conventions, and rules Probabilistic view Probabilistic view People have prior knowledge People have prior knowledge Prior knowledge is often highly structured Learning approximates optimal statistical inference Learning approximates optimal statistical inference Ultimately this is about how to learn as much as possible from the statistics of your environment possible from the statistics of your environment Work by Tenenbaum, Griffiths, Kemp Work by Tenenbaum, Griffiths, Kemp 1. The discovery of structural form (Kemp and 1 The discovery of structural form (Kemp and Tenenbaum, 2008) 2. Optimal predictions in everyday cognition 2 Optimal predictions in everyday cognition (Griffiths and Tenenbaum, 2006) 3. Markov Chain Monte Carlo with people 3 M k Ch i M C l ih l (Sanborn and Griffiths, 2008) Learning structures Learning structures Scientists discover structure in their data: Scientists discover structure in their data: ...

#### lecture01

Fayetteville State University, MCMC 06
Excerpt: ... Markov Chain Monte Carlo Simulations and Their Statistical Analysis An Overview Bernd Berg FSU, August 29, 2006 Content 1. Statistics as needed 2. Markov Chain Monte Carlo (MC) 3. Statistical Analysis of MC Data and Advanced MC. 1 Probability Distributions and Sampling In N experiments we may find an event A to occur n times. The frequency definition of the probability of the event is n . P (A) = lim N N Let P (a, b) be the probability that xr [a, b] where xr is a random variable drawn in the interval (-, +) with a probability density f (x) > 0. Then, b P (a, b) = a dx f (x) and f (x) = lim P (y, x) . yx x - y The (cumulative) distribution function of the random variable xr is defined as x F (x) = P (xr x) = - f (x ) dx . 2 For uniform probability distribution between [0, 1), u(x) = 1 for 0 x < 1; 0 elsewhere. The corresponding distribution function is x U (x) = - u(x ) dx = 0 for x < 0; x for 0 x 1; 1 for x > 1. It allows for the construction of general probability distributions. ...

#### mcmc1

Kentucky, STA 705
Excerpt: ... Markov Chain Monte Carlo STA705 Fall 2003 1 Introduction Recall we have been placing computer algorithms into two camps, those based on maximizing a function f (x), aimed toward computing maximum likelihood estimators and posterior modes, those based on computing an expectation EF [h(X)]. Recall one common set of methods for computing EF [h(X)] was to compute an iid sample X1 , . . . , Xn F and then use the sample average hn (X) = (1/n) h(Xi ) as an estimator of EF [h(X)]. Rejection sampling was one method for drawing the iid sample. Importance sampling was based on a similar idea, but instead the sample was drawn from a separate distribution G and a weighted average was used. Markov Chain Monte Carlo is another method for sampling from F . Like importance sampling, there is a twist. In rejection sampling there is an iid sample from F and you can use straightforward central limit theorem results. In importance sampling, the sample is independent from another distribution G and requires a weighted av ...

#### lec4

Harvard, CS 225
Excerpt: ... raph, then (G) = (1/(dn)2 ) Combining Theorem 12 with Corollary 9, we deduce Theorem 6. (The nonbipartite assumption in Theorem 12 can achieved by adding a self-loop to each vertex, which only increases the hitting time.) We note that the bounds presented in this lecture are not tight. 4 Markov Chain Monte Carlo Random walks are a very widely used tool in the design of randomized algorithms. In particular, they are the heart of the " Markov Chain Monte Carlo " method, which is widely used in statistical 5 physics and for solving approximate counting problems. In these applications, the goal is generate a random sample from an exponentially large space, such as an (almost) uniformly random perfect matching for a given bipartite graph G. (It turns out that this is equivalent to approximately counting the number of perfect matchings in G.) The approach is to do a random walk on an ^ appropriate (regular) graph G defined on the state space (e.g. by doing random local changes ^ on the current perfect matching). ...

#### Stat531

UPenn, STAT 531
Excerpt: ... Statistics 531 (Spring 2004) Markov Models, State Space Models, and Markov Chain Monte Carlo . Professor J. Michael Steele Prerequisites: This course serves as a rst graduate course in stochastic processes with a focus on model building and computational methods. It is often taken after Statistics 530, but it is designed to be independent of Statistics 530. It is neither more advanced nor less advance than Statistics 530, but it does focus on material that is more immediately applicable. Mathematical prerequisites are limited to knowledge of undergraduate probability (say at the level of Statistics 430), a knowledge of advanced calculus, and some experience with scientic computing. Special This Year: This year the course will be suciently dierent from earlier years that students who have already received credit for Statistics 531 can still take this course, do a project, and get credit for a reading course. Such a reading course might be called Markov Models and Financial Applications, but st ...

#### lec7

Berkeley, CS 294
Excerpt: ... l assumes: P(F = f , L = l ) = g = f + Our solution: e -U ( f ,l ) T Z P(X = G = g ) = P(X = , G = g ) P(G = g ) P(G = g X = w)P(X = ) P(G = g ) prior distribution = normalizing constant Interesting term: P(G = g X = ) = P(n = f - g ) 1 - - (g - f ) exp 2 2 2 = ( 2 ) n Where we assume every pixel has independent noise ~ N ( , ) e.g. Poisson process noise in CCDs Result: P (X = G = g ) = e -U ( f ,l )+term T Z' e -U p ( f ,l ) T = Z' is the posterior probability of a particular f, l given g. Note this is also a Gibbs distribution! MAP (maximum a posteriori) Estimate If you insist on a single answer then return f*, l* that maximizes - U p ( f *, l *) T 1 P(X = * G = g ) = exp Z Z' or equivalently, minimizes the energy function U p ( f *,l *) Problem: Solution: Technique: distribution f , l space is very large! Construct samples of f , l in this space with high probablity Markov Chain Monte Carlo (MCMC) lets you sample the posterior Sampling a Distribution Q: A: ...

#### poster_aptak

Excerpt: ... Bayesian Analysis of X-ray Luminosity Functions A. Ptak (JHU) Abstract Often only a relatively small number of sources of a given class are detected in X-ray surveys, requiring careful handling of the statistics. We previously addressed this issue i ...

#### garren

JMU, MATH 0001
Excerpt: ... James Madison University Statistics Colloquium Introduction to Sampling Methods: Markov Chain Monte Carlo Steven Garren JMU Wednesday, March 14, 2001 4:30pm (Refreshments served at 4:20) Room 141, Burruss Hall Abstract Markov chain Monte Carlo is used to generate pseudo-random numbers on a computer in both Bayesian and frequentist settings, when the probability distribution of interest is intractable (i.e., too complicated for approaches such as numerical quadrature). Special cases include Gibbs sampling and the Metropolis-Hastings algorithms. ...

#### pas20521

East Los Angeles College, PAS 205
Excerpt: ... e of occurrence, and the decisions whether to retain or delete are made using further pseudo-random variates. (iv) Spatial Poisson process There is a method here analogous to that in (ii). Suppose, for instance, that we are in two dimensions and wish to simulate a Poisson process within a rectangle. We may simulate the total number of points of the process using a single variate from the appropriate Poisson distribution, and then simulate x and y co-ordinates of these points using independent variates from two uniform distributions, giving a two-dimensional uniform distribution over the rectangle. 21.3 Markov Chain Monte Carlo (MCMC) methods These are methods used widely in Bayesian Statistics to simulate samples from an otherwise rather intractable posterior distribution. These samples are then used to get an idea of what the distribution looks like. Only a bare outline is given here. The idea is to construct a Markov chain (actually with continuous state space) which has the distribution of interest as its ...

#### outline09

Berkeley, EE 291
Excerpt: ... s, collective intelligence); hybrid system simulation; control and optimization of hybrid systems; observability of hybrid systems; model identication. Security of Network Embedded Systems: attacks on network embedded systems can be modeled as games between the adversary and the controller. With the ubiquitous use of network embedded systems in physical infrastructure in so-called SCADA (Supervisory Control And Data Acquisition) systems, it is important to derive provably correct defenses to certain classes of attacks. Applications: groups of coordinating vehicles; identication of modes in ATC observed data; gait modeling, stability and control; engine control; guidance of a UAV; biological modeling and control; embedded control and real time scheduling. Open Problems. Examples include: Observers and State Estimation for Hybrid Systems, approaches such as Generalized Principal Component Analysis or Markov Chain Monte Carlo methods have been proposed; Model Predictive Control or Finite Horizon Control an ...

#### top

SUNY Albany, LECTURE 6
Excerpt: ... <!DOCTYPE HTML PUBLIC "-/W3C/DTD HTML 4.0 Transitional/EN" "http:/www.w3.org/TR/REC-html40/loose.dtd"> <HTML> <META NAME="GENERATOR" CONTENT="TtH 2.32"> <title> Lecture VI</title> <H1 align=center>Lecture VI </H1> <body BGCOLOR="#FFFFFF" TEXT="#000000" LINK="#CC0000" ALINK="#FF3300" VLINK="#330099"> <BASE HREF="http:/omega.stat.psu.edu:8008/summer99/lecture6/"> <CENTER><A HREF="carlos"><IMG SRC="/urhere.gif" ALT="Another service from Omega" border=0></A></CENTER> <H1 align=center>An Introduction to Markov Chain Monte Carlo </H1> <img alt="*" src="/dot_green.gif" align="left" width="550" height="10"> <br> ...

#### top

SUNY Albany, LECTURE 5
Excerpt: ... <!DOCTYPE HTML PUBLIC "-/W3C/DTD HTML 4.0 Transitional/EN" "http:/www.w3.org/TR/REC-html40/loose.dtd"> <HTML> <META NAME="GENERATOR" CONTENT="TtH 2.32"> <title> Lecture V</title> <H1 align=center>Lecture V </H1> <body BGCOLOR="#FFFFFF" TEXT="#000000" LINK="#CC0000" ALINK="#FF3300" VLINK="#330099"> <BASE HREF="http:/omega.stat.psu.edu:8008/summer99/lecture5/"> <CENTER><A HREF="carlos"><IMG SRC="/urhere.gif" ALT="Another service from Omega" border=0></A></CENTER> <H1 align=center>An Introduction to Markov Chain Monte Carlo </H1> <img alt="*" src="/dot_green.gif" align="left" width="550" height="10"> <br> ...

#### top

SUNY Albany, LECTURE 8
Excerpt: ... <!DOCTYPE HTML PUBLIC "-/W3C/DTD HTML 4.0 Transitional/EN" "http:/www.w3.org/TR/REC-html40/loose.dtd"> <HTML> <META NAME="GENERATOR" CONTENT="TtH 2.32"> <title> Lecture VIII</title> <H1 align=center>Lecture VIII </H1> <body BGCOLOR="#FFFFFF" TEXT="#000000" LINK="#CC0000" ALINK="#FF3300" VLINK="#330099"> <BASE HREF="http:/omega.stat.psu.edu:8008/summer99/lecture8/"> <CENTER><A HREF="carlos"><IMG SRC="/urhere.gif" ALT="Another service from Omega" border=0></A></CENTER> <H1 align=center>An Introduction to Markov Chain Monte Carlo </H1> <img alt="*" src="/dot_green.gif" align="left" width="550" height="10"> <br> ...

#### top

SUNY Albany, LECTURE 4
Excerpt: ... <!DOCTYPE HTML PUBLIC "-/W3C/DTD HTML 4.0 Transitional/EN" "http:/www.w3.org/TR/REC-html40/loose.dtd"> <HTML> <META NAME="GENERATOR" CONTENT="TtH 2.32"> <title> Lecture IV</title> <H1 align=center>Lecture IV </H1> <body BGCOLOR="#FFFFFF" TEXT="#000000" LINK="#CC0000" ALINK="#FF3300" VLINK="#330099"> <BASE HREF="http:/omega.stat.psu.edu:8008/summer99/lecture4/"> <CENTER><A HREF="carlos"><IMG SRC="/urhere.gif" ALT="Another service from Omega" border=0></A></CENTER> <H1 align=center>An Introduction to Markov Chain Monte Carlo </H1> <img alt="*" src="/dot_green.gif" align="left" width="550" height="10"> <br> ...

#### toc

Minnesota, V 116
Excerpt: ... Nonparametric locally efcient estimation of the treatment specic survival distribution with right censored data and covariates in observational studies Alan E. Hubbard, Mark J. van der Laan, and James M. Robins Estimation of disease rates in small areas: A new mixed model for spatial dependence Brian G. Leroux, Xingye Lei, and Norman Breslow Markov chain Monte Carlo methods for clustering in case event and count data in spatial epidemiology Andrew B. Lawson and Allan B. Clark ...

#### short

Maryville MO, LODDOA 070306
Excerpt: ... Bayesian Analysis of Multivariate Stochastic Volatility and Dynamic Models Antonello Loddo Dongchu Sun, Dissertation Supervisor ABSTRACT We consider a multivariate regression model with time varying volatilities in the error term. The time varying volatility for each component of the error is of unknown nature, may be deterministic or stochastic. We propose Bayesian stochastic search as a feasible variable selection technique for the regression and volatility equations. We develop Markov Chain Monte Carlo (MCMC) algorithms that generate a posteriori restrictions on the elements of both the regression coefficients and the covariance matrix of the error term. Efficient parametrization of the time varying covariance matrices is studied using different modified Cholesky decompositions. We propose a hierarchal approach for selection of the volatility equation's variance components. We extend the results of the first in order to apply the stochastic search algorithm to dynamic model settings. We develop a MCMC alg ...