Recap: Normal model
Data: X = (X1 , X2 , . . . , Xn ). Data model: X1 , X2 , . . . , Xn N (, ) p(x|, ) = 2
n 2
iid
1 exp - 2
n
(xi - )2
i=1
We have considered two cases 1. Unknown mean , known precision
Prior: p() = N (0 , 0 )
x Posterior: p(|x) = N ( nn
'
$
A short diversion into the theory of Markov chains, with a view to Markov chain Monte Carlo methods
by Kasper K. Berthelsen and Jesper Mller
&
June 2004
2004-01
%
D EPARTMENT
A ALBORG U NIVERSITY 2 Fredrik Bajers Vej 7 G DK - 9220 Aalborg st Denmark P
Exercises for the 3rd Morning 1 Posterior prediction - normal case
Assume that x1 , . . . , xn are independent observations from a normal distribution with unknown mean and known precision . A priori we assume that N (0 , 0 ). What is the posterior predic
Motivating example
Heights of some Copenhageners in 1995
Histogram of height
500 0 140 100 200 300 400
150
160
170
180
190
200
210
Assume: Heights are normal, X N (, ). For now: Assume precision known.
1/31
Bayesian Idea
Parameter of interest: (e.g. mean
1
1.1
Examples of Bayesian inference
Binomial likelihood
Assume that we have perform n independent experiments where each experiment has probability for success. Let x cfw_0, 1, . . . , n denote the random number of successes. The number of successes foll
A brief introduction to (simulation based) Bayesian inference
The basic idea of Bayesian inference is to setup a full probability model for both observed and unobserved quantities. Inference is then based on the so-called posterior density - that is the c
Intro
Bayesian Statistics, Simulations and Software Each day: 9:00 to 16:00 Lunch: 12:15 to 12:45 To pass: Active participation in 10 of 12 half-days. Today: AM: Probability brush-up PM: Introduction to R software
1/6
Example: Test for rare disease
Events
INSTITUT FOR MATEMATISKE FAG
AALBORG UNIVERSITET
FREDRIK BAJERS VEJ 7 G URL: www.math.auc.dk E-mail: [email protected] 9220 AALBORG ST Tlf.: 96 35 88 63 Fax: 98 15 81 29
Exercises: Basics of probability theory
Exercise 1
A fair coin is tossed n times (where