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Problem:
X ~ N x|, 
iid
e(n)  x  {x1 , x2 ,..., xn }
1 n
1 n


2
1) Show that n; x   xi ; s   ( xi x ) 2  is a set of sufficient statistics
n i 1
n i 1


2) Being { ,  } location and scale parameters, take
(improper) prior and show that
x
T  n  1
 ~ St (t | n  1)
 s 
inferences on

 ( ,  )  1
as
 s2  2
Z  n  2  ~ χ ( z | n  1)
 
inferences on

E[T ]  0 (n  2)
E[Z ]  n  1 (n  1)
V [T ]  (n  1)(n  3) 1 (n  3)
V [Z ]  2(n  1) (n  1)
…
…
Problem: Comparison of means and variances




Let X1 ~ N x|1 , 1 and X 2 ~ N x|2 ,  2 be independent r.q.
Consider the samplings x1  {x11, x12 ,..., x1n } (iid ) x2  {x21, x22 ,..., x2 n

and the sufficient statistics n ; x  1
k
k
nk

1
nk
1
x
;
s


ki
k
nk
i 1
2
2
} (iid )

( xki xk ) 

i 1
k 1, 2
nk
2
1) Show that for the priors (improper)
Comparison of variances:  (1 ,  1 , 2 ,  2 )  1
 1 2
Comparison of means:
H1: same (but unknown) variances:
2
2
n2 (n1  1)   1 / s1 
 2 2  ~ Sn( z | n2  1, n1  1)
Z
n1 (n2  1)   2 / s2 
1   2  
 (1  2 )  ( x1  x2 ) 
 ~ St (t | n1  n2  2)
T  
1/ 2 
 s (1 / n1  1 / n2 ) 
 (1 , 2 ,  )  1
n s  n2 s2
s 1 1
n1  n2  2
2
H2: unknown and different variances:  (1 ,  1 , 2 ,  2 )  1
 1 2
W  1  2 ~
 s

p( w | x1 , x2 ) ~
1

2
 ( x1  w  u )

2  ( n1 1/ 2 )
s
2
2
 ( x2  u )
2

2 ( n2 1/ 2 )
(Behrens-Fisher problem)
du
PROBLEM: Correlation Coefficient of the Bivariate Normal Model
 1

t
p( x | μ, Σ )  det[ Σ ]1/ 2 exp   x  μ  Σ 1  x  μ 
 2

 
μ   1 
 2 
x 
x   1 
 x2 
  12
Σ  
  1 2
 1 2 

2
 2 
iid
e(n)  x  {( x11, x21 ), ( x12 , x22 ),..., ( x1n , x2 n )}
1) Show that a probability matching prior with
given by

the parameter of interest is
 (  , 1 , 2 ,  1 ,  2 )   11 2 1 (1   2 ) 1
2) Show that the posterior for the correlation coefficient is:


 (  | x )  (1   2 ) ( n 3) / 2 (1  r ) ( n 3 / 2) F 1 / 2,1 / 2, n  1 / 2,
sample correlation r 
 (x
1i
1  r 

2 
 x1 )( x1i  x1 )
i

2
2
(
x

x
)
(
x

x
)


1i
1
2i
2


i
 i

1/ 2
is a sufficient statistic for

PROBLEM: Poisson Distribution
p(n |  )  e
Consider X  e(k )  xk  n1 , n2 ,..., nk 
iid
Show that
f k  (θ,)  k 
1/ 2
(n  1)
  (θ )  1
and
and, in consequence
f k (θ ,)
1
 ( )  lim
 1/ 2
k  f ( ,)

k
0
N n
p(n |  , N )    (1   ) N n
n
PROBLEM: Binomial Distribution
Consider X  e(k )  xk  n1 , n2 ,..., nk 
iid
Show that
n

θ  (0,1)    (θ )  θ a 1 (1  θ )b1
and
f k (θ ,)
1
 ( )  lim
 1/ 2
1/ 2
k  f ( ,)

(
1


)
k
0
Hint: Analize the behaviour of
f k (θ ,)
expanding
log ( z,)
around
E[z ]
and considering the asymptotic behaviour of the Polygamma Function
 n ) ( z ) ~ an z  n  an 1 z  ( n 1)  ... , the moments of the Distribution,…
PROBLEM: Negative Binomial
( a  x )
 a (1   ) x
( x  1)(a)
a0
 X  {0,1,2,}
0  1
X ~ p( x |  , a) 
a: number of failures until experiment is stopped (fixed)
X: number of successes observed
θ: probability of failure
E[ X ]  a(1   ) 1
F ( )  a 2 (1   ) 1
 R ( )   J ( )   1 (1   ) 1/ 2
p( | x, a) 
PROBLEM: Weibull Distribution
1) Find the transformations
 a 1 (1   ) x 1/ 2
Be (a, x  1 / 2)
p( x |  ,  )   (x) 1 exp{(x) }1[0,) (x)
Z  Z(X )
and
φ  φ( ,  )
 ,   R
such that the new parameters are location and scale parameters and
transform them back to get the corresponding (improper) prior
2) Obtain the Fisher’s matrix and the Jeffrey’s prior
3) Find the reference prior
4) Show that it is a Probability Matching Prior
 ( ,  )   2  1
Problem: Linear Regression
(with uncertainty in x and y)
Linear Model:
{( xi , yi ); i  1,..., n}
y  a  bx
Model:
( X i , Yi ) ~ N ( xi , yi | xi0 , yi0 ,  xi ,  yi )
Data:
Assume precisions
( xi ,  yi ) are known and show that:

 xi yi

p(a, b | x, y ) ~  (a, b) 
2
2 2
i 1


b
 xi
yi

n
Take  (a, b)   (a) (b)  c

2
 1 n ( yi  a  bxi ) 

 exp 




2
2 2

2


b



i 1
yi
xi



and obtain p(a, b | x, y )
 y 
     x2
 x 
2
2
y
Problem:
1) Generate a sample :
X ~ Un( x 0,1)
x1 , x2 ,...xn 
n  106
3 2
X ~ p( x)  x I[ 1,1] ( x)
2
2) Get for each case the sampling distribution of
1
1
X ~ Ca( x 0,1) 
 1 x2
1 m
Ym   X k
m k 1
m  {2, 5, 10, 20, 50}
3) Discuss the sampling distribution of Ym in connection with the Law of Large
Numbers and the Central Limit Theorem
n
4) If X ~ Un( x 0,1) and Wn   U k  [0,1] How is Z n   log Wn distributed?
k 1
5) If
X k ~ Ga( x 0, k )
How is Z n 
6) If
X i ~ Ga( x |  , i )
How is
Xn
distributed?
Xn  Xm
Y   X i distributed?
i
(assumed to be independent random quantities)
PROBLEMS:
Gamma Distribution:
1) Show that if
X ~ Ga( x | a, b)
2) Show that if b  n  N
then
Y  aX ~ Ga( y | 1, b)
n
Y   Z i ~ Ga( y | 1, n)
i 1
Z i ~ Ga( z | 1,1)  Ex( z | 1)
( generate exponentials)
Beta Distribution:
 X 1 ~ Ga( x1 | a, b1 ) 
X1
1) Show that if 
 then Y  X  X ~ Be ( y | b1 , b2 )
1
2
 X 2 ~ Ga( x2 | a, b2 )
Problem 3D:
Sampling of Hidrogen
atom wave functions
n,l ,m r , ,   
(3,1,0)
( x, y )
Pr, ,  n, l , m 

(3,2,0)
 Rn ,l r Yl ,m  ,  
2
Evaluate the energy
using Virial Theorem
T 1 V
2
E n1 V n
2
e2 1
V (r )  
40 r

(3,2,±1)
 Rn,l r  Yl ,m  ,   r 2 sin 
2
( x, z )
( y, z )