PARTICLE FILTERING - Adaptive Signal Processing: Next Generation Solutions - page 280

The example shows that if we sample from f (x( n ) j y(1 : n )) a large number
of particles M , we will be able to estimate E ( h (X( n )) with arbitrary accuracy. In
practice, however, the problem is that we often cannot draw samples directly from
the a posteriori PDF f (x( n ) j y(1 : n )). An attractive alternative is to use the concept
of importance sampling [58]. The idea behind it is based on the use of another function
for drawing particles. This function is called importance sampling function or
proposal distribution, and we denote it by p (x( n )).
When the particles are drawn from p (x( n )), the estimate of E ( h (X( n ))) in (5.11) can
be obtained either by

M X

M

1

w ( m ) ( n ) h (x

( m ) ( n ))

E ( h (X( n ))) (5 : 15)

m¼ 1

or by

E ( h (X( n ))) X

M

w ( m ) ( n ) h (x

( m ) ( n ))

(5 : 16)

m¼ 1

where

f (x

( m ) ( n ) j y(1 : n ))
p (x

w ( m ) ( n ) ¼ (5 : 17)

( m ) ( n ))

and

w ( m ) ( n )
P i¼ 1 w ( i ) ( n ) w ( m ) ( n ) ¼ (5 : 18)

where

w ( m ) ( n ) ¼ cw ( m ) ( n )

with c being some unknown constant. The symbols w ( m ) ( n ) and w ( m ) ( n ) are known as
true and normalized importance weights of the particles x

( m ) ( n ), respectively. They are
introduced to correct for the bias that arises due to sampling from a different function
than the one that is being approximated, f (x( n ) j y(1 : n )). The estimate in (5.15) is
unbiased whereas the one from (5.16) is with a small bias but often with a smaller
mean-squared error than the one in (5.15) [55]. An advantage in using (5.16)
over (5.15) is that we only need to know the ratio f (x( n ) j y(1 : n )) =p (x( n )) up to a
multiplicative constant and not the exact ratio in order to compute the estimate of
the expectation of h (X( n )).
How is (5.18) obtained? Suppose that the true weight cannot be found and instead
we can only compute it up to a proportionality constant, that is

( m ) ( n ) j y(1 : n ))
p (x

w ( m ) ( n ) ¼ c f (x

( m ) ( n ))

¼ cw ( m ) ( n )

(5 : 19)