

![0_image_0.png](0_image_0.png)

```
to improved 
             ontrol on average. However, we stress that 
                                                           urrently there is no theoreti
                                                                                        al
                                                                                          

```

underpinning for this observation and thus it may not be true for all belief states and any problem.

## 4.2.3 Extensions Of The Fast Informed Bound Method

```
The main idea of the fast informed bound method is to sele
                                              t the best linear fun
                                                               tion for every observation and every 
                            urrent state separately. This diers from the exa
                                                                             t update
                                                                                      

```

where we seek a linear fun tion that gives the best result for every observation and the ombination of all states. However, we observe that there is a great deal of middle ground between these two extremes. Indeed, one an design an update rule that hooses optimal

(maximal) linear fun tions for disjoint sets of states separately. To illustrate this idea, assume a partitioning S = fS1; S2;    ; Smg of the state spa
e S. The new update for S is:

$$\widehat{V}_{i+1}(b)=\max_{a\in A}\left\{\sum_{s\in S}\rho(s,a)b(s)\quad+\quad\gamma\sum_{o\in\Theta}\left[\max_{a^{\prime}\in\Pi}\sum_{s^{\prime}\in S}\sum_{s^{\prime}\in S}P(s^{\prime},o|s,a)b(s)\alpha_{i}(s^{\prime})+\cdots\right.\right.$$ $$\left.\left.\max_{\alpha_{i}\in\Pi^{\prime}}\sum_{s\in S_{i}}\sum_{s^{\prime}\in S}P(s^{\prime},o|s,a)b(s)\alpha_{i}(s^{\prime})+\cdots+\right.\right.$$ $$\left.\left.\max_{\alpha_{i}\in\Pi^{\prime}}\sum_{s\in S_{n}}\sum_{s^{\prime}\in S}P(s^{\prime},o|s,a)b(s)\alpha_{i}(s^{\prime})\right]\right\}$$

It is easy to see that the update upper-bounds the exa t update. Exploration of this

```
approa
     h and various partitioning heuristi
                                s remains an interesting open resear
                                                             h issue.
                                                                  

```
