Economics Chapter 16 Homework This Simplifies Considerably The Combinatorial And Gamma

Chapter 16

Bayesian Estimation and Inference

◼ Exercises

1 (a) The likelihood function is

(b) The posterior is

The product of factorials will fall out. This leaves

1

0

exp( ) (1 / )

( | , , )

exp( ) (1 / )

ii

y

ny

n

p y y

nd

  

   





−

=−



Application

a. p(Fi|Ki,) =

−



 − 



(1 )

i i i

iF K F

i

K

F

so the log-likelihood function is

11

1

()

(1 ) (1 )

( ) ( ) .

i i i

niF K F ab

i

Kab

Fab



−−−

=



 +

 −  − 



 







This simplifies considerably. The combinatorials and gamma functions fall out, leaving

11 () 11

1

(1 ) (1 ) (1 ) (1 )

i i i i i i i i

nF K F ab F K F ab

i

−−−   − −−

=



 −   −   −   − 



   



The denominator is a beta integral, so the posterior density is

The denominator simplifies slightly;

( ) ( 1) [ ( )] ( 1)

[( ) ( 1)] [( ( )) ( 1)]

( | ) (1 )

[( ) ( 1) ( 1)]

i i i i i

i i i i i F a K F b

ii

F a K F b

pK a b

 + −  − + −

  + −   − + −

 =  − 





  + − + −

y