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where Zi2 k2 and Z1 |
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p-value = P 0 [ (X) > (x)] P r2 q > (x) |
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Multinomial LRT
mle: pbn = Xn1 ; : : : ; Xnk
T (X) = |
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The approximate size LRT rejects H0 when (X) 2k 1;
Pearson Chi-square Test
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where |
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D2
T ! k 1
p-value = P 2k 1 > T (x)
D2
Faster ! Xk 1 than LRT, hence preferable for small n
Independence testing
I rows, J columns, X multinomial sample of size n = I J
mles unconstrained: pij = |
Xij |
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mles under H0 |
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D2
LRT and Pearson ! k , where = (I 1)(J 1)
iid
N (0; 1)
14 Bayesian Inference
Bayes' Theorem
f( |
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f(x j )f( ) |
= |
f(x j )f( ) |
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( )f( ) |
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f(xn) |
R f(x j )f( ) d |
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De nitions
Xn = (X1; : : : ; Xn)
xn = (x1; : : : ; xn)
Prior density f( )
Likelihood f(xn j ): joint density of the data
n
Y
In particular, Xn iid =) f(xn j ) = f(xi j ) = Ln( )
i=1
Posterior density f( j xn)
Normalizing constant cn = f(xn) = R f(x j )f( ) d
Kernel: part of a density that depends on
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f( j x |
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R |
Ln( )f( ) |
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Posterior mean n = |
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n( )f( ) d |
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14.1Credible Intervals
Posterior interval
Z b
P [ 2 (a; b) j xn] = f(
a
Equal-tail credible interval
Z a Z 1
f( j xn) d = f( j xn) d = =2
1 b
Highest posterior density (HPD) region Rn
1.P [ 2 Rn] = 1
2.Rn = f : f( j xn) > kg for some k
Rn is unimodal =) Rn is an interval
14.2Function of parameters
Let = '( ) and A = f : '( ) g.
Posterior CDF for
Z
H(r j xn) = P ['( ) j xn] = f( j xn) d
A
Posterior density
h( j xn) = H0( j xn)
Bayesian delta method
j Xn N '(b); seb '0(b)
14