SMC Procedure
SMC Evaluating
The marginal likelihood is defined by
Monte Carlo methods can directly estimate this integral, but the approximation is usually poor in practice. Alternatively, the marginal likelihood (
) is decomposed as
For 
, it is straightforward to have
These terms can be estimated by the SMC algorithms with the weighted particles at time step 
as
SIR Algorithm
In the SIR algorithm, this approximation can be further simplified by sampling 
(
) from 
for 
:
Moreover, 
can be written as the following integral with the importance sampling scheme:
In the SIR algorithm, 
are sampled from 
. Therefore, the basic Monte Carlo approximation of is
BF Algorithm
In the BF algorithm, this approximation can be simplified by sampling () from 
for :
Moreover, the 
in the BF algorithm is computed in the same way as the one in the SIR algorithm.
APF Algorithm
In the APF algorithm, because 
, the approximation can be written as
The auxiliary index variables 
are sampled according to 
, and are sampled from . The weight for each auxiliary index is 
. The preceding approximation is thus simplified as
It is further simplified to 
if you use APF(ADP). Moreover, the in the APF algorithm is computed in the same way as the one in the SIR algorithm.