Marginal likelihood
Z
f(xn j Hi) = f(xn j ; Hi)f( j Hi) d
Posterior odds (of Hi relative to Hj) |
|
|
|
|
|
|
|
|
|
|||||||||
|
P [Hi j xn] |
= |
|
f(xn j Hi) |
|
|
|
P [Hi] |
|
|||||||||
|
P [Hj j xn] |
f(xn j Hj) |
|
|
P [Hj] |
|||||||||||||
|
|
|
|
|
||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
Bayes Factor BF |
|
prior odds |
||||||||||||
Bayes factor |
|
| {z } |
|
ij |
| {z } |
|||||||||||||
|
|
log10 BF10 |
BF10 |
evidence |
|
|||||||||||||
|
|
0 0:5 |
|
1 1:5 |
Weak |
|||||||||||||
|
|
0:5 1 |
|
1:5 10 |
Moderate |
|||||||||||||
|
|
1 2 |
|
10 100 |
Strong |
|||||||||||||
> 2 |
|
> 100 |
|
|
Decisive |
|||||||||||||
|
|
|
p |
BF10 |
|
|
||
|
|
|
1 p |
n |
|
|||
p = |
1 + |
p |
BF10 |
where p = P [H1] and p = P [H1 j x |
] |
|||
|
1 p |
|
||||||
|
|
|
|
|
|
|
||
15 Exponential Family
Scalar parameter
fX(x j ) = h(x) exp f ( )T (x) A( )g = h(x)g( ) exp f ( )T (x)g
Vector parameter |
i( )Ti(x) A( )) |
fX(x j ) = h(x) exp ( s |
|
=1 |
|
Xi |
|
= h(x) exp f ( ) T (x) A( )g
= h(x)g( ) exp f ( ) T (x)g
Natural form
fX(x j ) = h(x) exp f T(x) A( )g
=h(x)g( ) exp f T(x)g
=h(x)g( ) exp T T(x)
16 Sampling Methods
16.1The Bootstrap
Let Tn = g(X1; : : : ; Xn) be a statistic.
1.Estimate VF [Tn] with VFbn [Tn].
2.Approximate VFbn [Tn] using simulation:
(a)Repeat the following B times to get Tn;1; : : : ; Tn;B, an iid sample from the sampling distribution implied by Fbn
i.Sample uniformly X1 ; : : : ; Xn Fbn.
ii.Compute Tn = g(X1 ; : : : ; Xn).
(b)Then
vboot = VFn |
= B |
=1 |
Tn;b B r=1 Tn;r |
! |
2 |
|||
|
||||||||
|
1 |
B |
1 |
B |
|
|
||
b b |
|
|
Xb |
|
|
X |
|
|
|
|
|
|
|
|
|||
16.1.1Bootstrap Con dence Intervals
Normal-based interval
Tn z =2sebboot
Pivotal interval
1.Location parameter = T (F )
2.Pivot Rn = bn
3.Let H(r) = P [Rn r] be the cdf of Rn
4.Let Rn;b = bn;b bn. Approximate H using bootstrap:
B
Hb(r) = B1 XI(Rn;b r)
b=1
5.= sample quantile of (bn;1; : : : ; bn;B)
6.r = sample quantile of (Rn;1; : : : ; Rn;B), i.e., r = bn
7. Approximate 1 con dence interval Cn = a;^ ^b where |
|
|
|
|
|
||||||||||||||||
a^ = |
n |
|
H 1 |
1 |
|
|
= |
n |
|
r |
|
|
= |
2 n |
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
||||||||||||
^b = |
b |
|
|
|
|
|
|
|
= |
b |
|
|
=2 = |
|
|
|
=2 |
||||
nb |
H 1 |
|
|
n |
1 |
r |
b2 n |
1 |
|
||||||||||||
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
b |
b |
|
|
|
|
b |
|
|
|
|
|
b |
|
|
|
||||
|
|
|
2 |
|
|
|
|
|
|
|
|
||||||||||
Percentile interval
Cn = ;
=2 1 =2
16