Stat 516, 2014 Homework 2, Solution Sketch
1. (a) Independence of X and Y implies covariance XY = 0. Conditional independence of X and Y
given implies a zero in the inverse of the covariance matrix, which yields XY ZZ XZ Y Z = 0.
We deduce that XZ Y Z = 0
Stat 516, 2014 Homework 1
Due date: Tuesday, October 7.
1. Gamma-Poisson mixture
Recall that if Z Poisson(), then E(Z) = var(Z) = , where > 0 is the intensity parameter.
This property of the Poisson distribution is restrictive since often in practice exce
Introduction
Likelihood Evaluation
Hidden State Inference
Parameter Estimation
Model Selection
Earthquake Example
Forecasting
516: Stochastic Modeling of Scientic Data
Hidden Markov Models
Mathias Drton
Department of Statistics
University of Washington
In
Stat 516, 2014 Homework 3, Solution Sketch
1. This problem can be solved by rst-step analysis similar to the problem of absorption probabilities
treated in class, thinking about the 4-state Markov chain for the location of both Cat and Rat. (Read
Brmaud S
Stat 516, 2014 Homework 6, Solution Sketch
1. (a) Let (x) be the density of N (0, 1), and let M =
(x)
1
2
= exp x2 2|x|
g (x)
2
2
2
2
e /2 .
2
=
It holds that
1
1
2
exp (|x| )2 + 2 .
2
2
2
Thus, the normal density is bounded as
(x) M g (x) x R,
and t
Stat 516, 2014 Homework 5, Solution Sketch
1. (a) The log-likelihood and the score function are
= y log E + const.
y
S() =
E.
l()
The Fisher-information is
I() = E
y
E
= .
2
MLE is = y/E, var() = I()1 = /E.
(b) Posterior is
p(|y) p(y|)p() y eE a1 eb = y
Stat 516, 2014 Homework 6
Due date: Tuesday, November 25.
Note: Do this homework individually. Do not include any R code in your main handout just include as
an appendix, in compact form.
1. Suppose you want to generate a draw from N (0, 1) using a reject
Stat 516, 2014 Homework 5
Due date: Thursday, November 13.
Note: Do this homework in pairs; two students turning in a single joint solution. No two Statistics or
Biostatistics Ph.D. students may work together. No two QERM Ph.D. students may work together,
Stat 516, 2014 Homework 4, Solution Sketch
1. Fix any two states i, k S. For later reference, note that since
(pij pkj ) = 1 1 = 0,
jS
we have that
(pij pkj ) =
j:pij >pkj
(pij pkj ) =
(pkj pij ).
(1)
(pij ) = 1 d.
j:pij <pkj
(2)
j:pij <pkj
These non-neg
Introduction
Motivating Examples
Probability Review
516: Stochastic Modeling of Scientic Data
Introduction and Overview
Mathias Drton
Department of Statistics
University of Washington
Introduction
Motivating Examples
Probability Review
Logistics
Instruct
Unrestricted Transition Probabilities
Parametric Transition Probabilities
516: Stochastic Modeling of Scientic Data
Inference for Discrete-Time Markov Chains
Mathias Drton
Department of Statistics
University of Washington
Unrestricted Transition Probabili
Overview
Importance Sampling
Direct Sampling
Markov chain Monte Carlo
Simulated Annealing
516: Stochastic Modeling of Scientic Data
Monte Carlo Methods
Mathias Drton
Department of Statistics
University of Washington
Overview
Importance Sampling
Direct Sam
Introduction
Markov Property
First-Step Analysis
Classication
Recurrence
516: Stochastic Modeling of Scientic Data
Discrete-Time Markov Chain Theory
Mathias Drton
Department of Statistics
University of Washington
Introduction
Markov Property
First-Step An
EM Algorithm
Bayesian Data Augmentation
516: Stochastic Modeling of Scientic Data
Methods for Missing Data
Mathias Drton
Department of Statistics
University of Washington
EM Algorithm
Bayesian Data Augmentation
Section Outline
The EM Algorithm
Bayesian Da
Limiting behavior
Reversibility
Fundamental matrices
516: Stochastic Modeling of Scientic Data
Discrete-Time Markov Chain Theory
Mathias Drton
Department of Statistics
University of Washington
Limiting behavior
Reversibility
Section Outline
Limiting Behav
Stat 516, 2014 Homework 2
Due date: Tuesday, October 14.
Note: Do this homework in pairs; two students turning in a single joint solution. No two Statistics or
Biostatistics Ph.D. students may work together. No two QERM Ph.D. students may work together, e
Stat 516, 2014 Homework 3
Due date: Tuesday, October 21.
Note: Do this homework individually.
1. (Brmaud 2.3.1) Rat and Cat move between two rooms, using dierent paths. Their motions are
e
independent, governed by their respective transition matrices
1
2
Stochastic Processes
Finite-Dimensional Distributions
Properties of Sample Paths
516: Stochastic Modeling of Scientic Data
A Few Words About Stochastic Processes
Mathias Drton
Department of Statistics
University of Washington
1 / 12
Stochastic Processes
F