Bayesian Logistic Regression Models for Credit Scoring by Gregg Webster

 
Definition 3.3 
A probability distribution 
with density 
is called a stationary distribution 
for a Markov chain 
( 
)
with transition kernel 
if
( ) ∫ ( ) ( )
for all 
except on a set 
with 
( )
. 

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55 
Such a distribution is called an invariant distribution. We now move onto the so-called 
detailed balance condition. 
 
Lemma 3.3 
Let 
( 
)
be a Markov chain with transition kernel 
If, for a 
density function 
p
, we have the detailed balance condition 
( ) ( ) ( ) ( )
for all 
then 
is the density of a stationary distribution of the Markov chain. 
Proof:
We have 
∫ ( ) ( ) ∫ ( ) ( )
( ) ∫ ( ) ( )
 
Definition 3.4 
Let 
( 
)
be a Markov chain with continuous state space 
. Let 
be a probability distribution on 
. The Markov chain is called 
-irreducible if for all 
and all 
with 
( )
there is 
such that 
( 
| 
) ∫
( )
( 
)
. 
If 
then the Markov chain is called strongly 
-irreducible. This property of a Markov 
chain implies that any set with a positive probability 
( )
can be visited from any 
in finite time. Thus, if this property holds, all states communicate.
Now, let 
∑
( 
)
denote the number of visits of the Markov chain in the set 
.
 
Definition 3.5 
Let 
( 
)
be a Markov chain and let 
. We then call 
-
the set 
recurrent if for all 
we have 
( 
| 
)
. 

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56 
-
the Markov chain recurrent if it is 
-irreducible for some probability distribution 
and 
whenever 
( )
, then 
is recurrent.
A stronger definition of recurrence is now given.