
## 2.5 Sis With Rao-Bla Kwellisation (Rb-Sis)

Sin e SIS is a form of BIS, Rao-Bla kwellisation an also be used to improve its performan e (Liu & Chen, 1998; Dou et et al., 2000b). Let us onsider again the problem of estimating the expe tation f =
R

f (~xt)p(~xt jo~t), where ea
h variable xt is the joint produ
t of two variables (zt ; rt). We shall restri
t ourselves to the 
ase where x~t is Markov and ot is an observation of xt , i.e., when (~xt ; o~t) 
an be represented by a DBN. In addition, we only onsider f that depends only on the 
urrent variable xt , i.e., 
f is an expe
tation over the ltering distribution p(xt jo~t). For example, if A is a \future" event, i.e., an event that depends on fxt 0 jt 0  tg, we an estimate p(Ajo~t) by letting f (xt) = p(Ajxt) so that f =
R

xt p(Ajxt)p(xt jo~t) = p(Ajo~t).

Applying Rao-Bla kwellisation to this setting, we an let h(~rt) =
R

zt f (zt ; rt)p(zt jr~t ; o~t),
so that f = h =
R

r~t h(~rt)p(~rt jo~t). Thus, if we use SIS to estimate h, we also obtain an estimator for f :

$${\vec{f}}\approx{\hat{f}}_{R B S I S}={\hat{h}}_{S I S}=\sum_{i=1}^{N}h({\hat{r}}_{t}^{[i]}){\hat{w}}({\hat{r}}_{t}^{[i]})$$

$$\left({\boldsymbol{4}}\right)$$

The benet of doing this is the in rease in the a ura y of the estimator, as we now only need to sample the variables r~t . The down side is that for ea
h sample r~t , we need to ompute h(~rt ) using some exa
t inferen
e method. Furthermore, the SIS pro
edure to estimate h might require some additional omplexity sin e the sequen e r~t is generally non-
Markov, and ot no longer depends only on rt . Overall, in 
omparison with the normal SIS
estimator ^
fSIS (Eq. 1), for the same number of samples N, ^
fRBSIS is more a

urate but is also more omputationally demanding to ompute.

To see more learly what is involved in implementing the RB-SIS method, let us look at the Rao-Bla kwel lised belief state, i.e., the belief state of the dynami pro ess when the Rao-Bla kwellising variables an be observed: Rt = p(zt ; rt ; ot jr~t1; o~t1 ) and its posterior Rt+ = p(zt jr~t ; o~t). All the entities needed in the RB-SIS pro
edure 
an be 
omputed from these two distributions. Indeed, the fun tional h an be rewritten in terms of Rt+ as:

$$h\big(\vec{r}_{t}\,\big)=\int_{z_{t}}f(z_{t},r_{t})p\big(z_{t}|\vec{r}_{t},\partial_{t}\,\big)=\int_{z_{t}}f(z_{t},r_{t})\mathcal{R}_{t+}\big(z_{t}\,\big)$$

In addition, while performing SIS to estimate h
, from Eq. (3), the weight wt and the sampling distribution qt 
an be 
omputed from Rt :

$$w_{t}q_{t}=p(r_{t},o_{t}|\tilde{r}_{t-1},\delta_{t-1})={\cal R}_{t}(r_{t},o_{t})=\int_{z_{t}}{\cal R}_{t}(z_{t},r_{t},o_{t})\tag{6}$$

$$\left({\bar{5}}\right)$$

Thus, omputing the RB belief state Rt and its posterior Rt+ is an essential step in the RB-SIS method. Sin e we have to maintain an RB belief state for ea h sample of the RB variables r~t, it is 
ru
ial that this 
an be done eÆ
iently using an exa
t inferen
e method. If xt is 
omposed of many variables, as in the 
ase of a DBN, our 
hoi
e of the Rao-Bla kwellising variables should be so that the Rao-Bla kwellised belief state an be maintained in a tra table way. Hen e, Rao-Bla kwellisation is espe ially useful when the set of variables in a DBN 
an be split into two parts su h that onditioning on the rst part makes the stru ture of the se ond part tra table and amenable to exa t inferen e.

