MATH-440
Examples of Bayesian methods in my research
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Slide 1
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MATH-440
Examples of Bayesian methods in my research
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Gene expression microarray
Slide 3
Single-stranded DNA fragments deposited at discrete sites.
Rely on hybridiza
MATH-440
Bayesian Analysis of Probit Model
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Reading Assignment: Chapter 12
Probit link for binary outcome
vspace2mm Recall the latent variable formulation for binary
outcomes.
Slide 1
Suppose yi is binary and a continuous latent variable Zi exists
suc
MATH-440
Generalized Linear Models
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Generalized linear models (GLM)
Slide 1
This is a broad class of models that includes linear models as
well as models for non-normal response distributions.
In GLM, Yi is assumed to follow an exponential family
dis
MATH-440
Multivariate Normal & Wishart Models
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Reading Assignment:
Chapter 7
Multivariate Normal Distribution
Let Y = (Y1 , . . . , Yk )T be a k -dimensional vector following a
Slide 1
multivariate normal distribution with mean = (1 , . . . , k )T and
MATH-440
Hierarchical Linear Model
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Reading Assignment: Chapter 11
Hierarchical models arise naturally in the analysis of data
Slide 1
obtained by cluster sampling. For example,
when observations are taken on related individuals
when data are gather
MATH-440
Hierarchical Model
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Reading Assignment:
Chapter 8
The exercise of specifying a model over several levels is called
hierarchical modeling, with each new distribution forming a
new level in the hierarchy.
Slide 1
In a hierarchical model, the obs
MATH-440
Model Selection
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Reading Assignment: Chapter 9
In regression analysis, there is often a large number of potential
covariates, some of which have little, if any, predictive value.
Slide 1
We generally wish to include in the model only variabl
MATH-440
Hypothesis Testing & Bayes Factor
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Reading Assignment:
Bayes factor
Section 2.2
is used to test hypotheses and compare models
in the Bayesian framework.
Slide 1
Suppose we have two candidate models,
respective parameter vectors
The
1
and
M1
a
MATH-440
Linear Regression
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Reading Assignment:
Chapter 9
The question of interest is to investigate how a response variable y
varies as a function of explanatory variables, x1 , . . . , xk .
yi = x i + i ,
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i N (0, 2 ),
E [yi |, X ] = 1 xi1 + .
MATH-440
MCMC Convergence Diagnostics
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Reading Assignment: Chapters 6 & 10
An MCMC algorithm has converged at iteration T when its
output can be safely thought to arise from the true stationary
distribution of the Markov chain for all t > T .
Slide 1
MATH-440
Metropolis Algorithm
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Reading Assignment:
Chapter 10
Metropolis algorithm
Slide 1
Let p(|y ) be a target distribution computed up to a
normalizing constant.
Let Jt ( |(t1) ) be a symmetric proposal distribution, that is,
Jt (a |b ) = Jt (b |
MATH-440
Gibbs Sampler
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Chapter 6
Reading Assignment:
Markov chain Monte Carlo
The idea of Markov chain simulation is to simulate a random
walk in the space of
,
which converges to a stationary
distribution (also called equilibrium distribution or
Slid
Y =F
1
(U )
U
F
(0, 1)
F
Y
Y
U
(0, 1)
Y = F 1 ( U )
Y
i = 1, . . . , n FY (y ) = i:yi y pi
U
(0, 1)
U p1
Y = y1
FY (yi1 ) < U FY (yi )
P ( Y = yi ) = pi
Y
Y = yi
i = 2, . . . , n
( , )
X
FX (x) = 1 e(x/ )
X 1 , . . . , XN
E [X i ] =
X1 + . . . + XN
N
( )
f ( |y )
N ( 0 , 2 )
2
2
( )
( ) d =
y1 , . . . , yn N (, 1)
< <
( ) 1
f ( |y )
1
exp
(yi )2 1
2i
1
= exp n2 2
yi
2
i
n
1
exp ( y )2 ,
y=
yi
2
ni
1
|y N y , n
() [I ()]
1 /2
2 log(f (y |)
I ( ) = E
2
I ( )
Y
f (y | )
p( )
= h( )
d 1
y |
(n, )
L() y (1 )ny
(a, b)
( ) =
(a + b) a1
(1 )b1 ,
(a)(b)
0 1.
|y
(a + y, b + n y )
P
f ( y | )
( ) P f ( | y ) P .
p(y |) = h(y )c() expcfw_t(y ),
t(y )
p() = (n0 , t0 )c()n0 expcfw_n0 t0 .
t0
t(Y )
n0
Y
(1, )
y
p(y |) = (1 )
= log
1
1
MATH-440
Review of Probability Concepts
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Reading Assignment: Chapters 1 & 2
Slide 1
Experiment: phenomenon where outcomes are uncertain
e.g., single throws of a six-sided die.
Sample space: set of all outcomes of the experiment
S = cfw_1, 2, 3, 4,
MATH-440
Introduction to Bayesian Paradigm
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Statistical Inference
Statistical inference is the process of learning about the
general characteristics of a population from a subset of
members of that population (sample).
Slide 1
Numerical values of pop