Stats 225 Homework 1 Solutions
1. Let D be the event that George has the disease, T1 be the event that
Test 1 is positive, and T2 be the event that Test 2 is positive. We have
c
P (D) = 0.001, P (T1 |Dc ) = 0.1, P (T1 |D) = 0.06, P (T2 |Dc ) = 0.03, and
c
STATS 225 Midterm Solutions
1. This problem follows exactly the same logic as the original Monty Hall
problem. We let S be the event that the contestant chooses door 1, M
be the event that Monty chooses doors 3 and 4, and Ci be the event
that the car is a
Statistics 218 Final Project
Marc Coram
For my project I consider a simple decryption problem and implement a
solution via the Metropolis algorithm. The problem was motivated when an
enocoded message was brought to my attention that had apparently been wr
Stats 225Bayesian Statistics
A Few Examples
October 16, 2007
Main Concepts So Far
Background Info on Probability Theory (lecture 2)
Bayes Theorem (3)
The Basic Setup: Posterior = Prior * Likelihood (3)
Conjugate Priors (3)
Common Models (Binomial, Poisson
STATS 225: Bayesian Analysis
Session 17: Introducing some advanced topics
Babak Shahbaba
Autumn 2007
Introduction
In this session, we introduce two advanced topics: Gaussian
process for regression and Dirichlet process for mixture
modeling.
Note that de
STATS 225: Bayesian Analysis
Session 16: Decision theory
Babak Shahbaba
Autumn 2007
Decision theory
In the Bayesian paradigm, hypothesis testing and model
evaluation are special cases of decision problems. In fact many
topics such as point estimation and
STATS 225: Bayesian Analysis
Session 15: Generalized linear models (continued)
Babak Shahbaba
Autumn 2007
Multinomial logistic model
This is a generalization of logistic regression when the
outcome could have multiple values (i.e., could belong to one
of
STATS 225: Bayesian Analysis
Session 14: Generalized linear models (continued)
Babak Shahbaba
Autumn 2007
Example 1: Snoring and heart disease
The objective of this study (Norton and Dunn, 1985, British
Medical Journal; Agresti, 2002) is to investigate w
STATS 225: Bayesian Analysis
Session 13: Generalized linear models (GLM)
Babak Shahbaba
Autumn 2007
Variable transformation
For the ordinary linear model, we assume that the outcome
variable is normally distributed and its mean is a linear
function of pr
STATS 225: Bayesian Analysis
Session 12: Linear regression models (continued),
model checking and prediction
Babak Shahbaba
Autumn 2007
Reminder: Bayesian linear regression models
In the last session, we showed that the linear regression model
is equival
STATS 225: Bayesian Analysis
Session 11: Linear regression models
Babak Shahbaba
Autumn 2007
Reminder: Linear regression models
We discussed the ordinary liner regression model in its
simplest form dened as follows:
y |, 2 , x N (x , 2 In )
y is a colum
STATS 225: Bayesian Analysis
Session 10: Hierarchical Bayesian models
(continued), and linear regression models
Babak Shahbaba
Autumn 2007
Reminder: Hierarchical Bayesian models
We discussed hierarchical models where the parameters of the
model depend on
STATS 225: Bayesian Analysis
Session 9: Hierarchical Bayesian models
Babak Shahbaba
Autumn 2007
Reminder: Exchangeability
We discussed exchangeability before. Informally, we
mentioned that a set of observations y = (y1 , ., yn ) are
exchangeable if in co
STATS 225: Bayesian Analysis
Session 8: Convergence, the Gibbs sampler and
slice sampling
Babak Shahbaba
Autumn 2007
Reminder: the Metropolis algorithm
Last session, we discussed how we can use the Metropolis
algorithm for sampling from the posterior dis
STATS 225: Bayesian Analysis
Session 7: Using Metropolis for sampling from
the posterior distribution
Babak Shahbaba
Autumn 2007
Reminder
We mentioned that if a Markov chain is irreducible and
aperiodic and has a stationary distribution cfw_i , then for
Addendum to session 6
STATS 225: Bayesian Analysis
Example 1 - Since reversibility is a very importance concept in Markov chains and MCMC, I
would provide some more examples here. First, lets start with a very simple example. Consider a
Markov chain with
STATS 225: Bayesian Analysis
Session 5: Monte Carlo simulation and Markov
chains
Babak Shahbaba
Autumn 2007
Reminder
Remember that our main objective in Bayesian analysis is to
obtain posterior distributions, P (|y ), based on P () and
P (y |):
P ()P (y
STATS 225: Bayesian Analysis
Session 2: A quick review of rigorous probability
Babak Shahbaba
Autumn 2007
Reminder
We discussed probability as a measure of uncertainty.
Note that in practice, Bayesian and frequentist methods may
provide supercially simi
STATS 225: Bayesian Analysis
Week 1: Introduction
Babak Shahbaba
Autumn 2007
Outline
Course outline
Subjective probability
Decision making under uncertainty
Some toy examples
The cancer map
Conclusions
Subjective probability
Coins dont have probabi
Assignment 2, due data: November 27 (in class)
STATS 225: Bayesian Analysis
1. To detect disease D, we rely on a specic blood measurement y . This is not a very accurate
test of course. It is known that for people who dont have this disease, the distribut
Assignment 1, due data: October 23 (in class)
STATS 225: Bayesian Analysis
1. George goes for his annual check up, and unfortunately the result shows that he has a dangerous disease. The doctor tells him the probability of having this disease is 0.001. He