Is this expected utility correct?

Question

I'm learning Bayesian Networks and Decision Making but I'm not very good with notation.

I have written down this expected utility (more info here):

$$UE=\sum_s\max_b\sum_{s,r}\max_q\sum_tP(t)P(s|t)P(r|t:b)U(t,b,q)$$

For this graph:

Yo can find the image on this paper, page 62.

My doubt is about the second sum: $$\sum_{s,r}$$

I don't if it is correct to put $s,r$ or if I can shorten it with the first sum because it is an s summation also, $\sum_s$, getting the following formula:

$$UE=\sum_s\max_b\sum_r\max_q\sum_tP(t)P(s|t)P(r|t:b)U(t,b,q)$$

Is that summation ($\sum_{s,r}$) correct?

Answer 1

The variable elimination algorithm is based on marginalizing one by one all potentials depending on each variable. So the correct expression is:

$UE=\sum_s\max_b\sum_r\max_q\sum_tP(t)P(s|t)P(r|t:b)U(t,b,q)$

So the expression $\sum_{s,r}$ is out of scope here, as it's trying to marginalize the potentials on both at the same time, while you're trying to sum it up again at the end of the process (the first $\sum_{s}$ in both expressions).

Answer 2

There must be a typo. The sub-expression $$\max_b\sum_{s,r}\max_q\sum_tP(t)P(s|t)P(r|t:b)U(t,b,q)$$ doesn't depend on $s$. So for each $s$, this sub-expression is the same. So you sum the same thing for each s.

Literally, it means: $$UE=card(s)\cdot\max_b\sum_{s,r}\max_q\sum_tP(t)P(s|t)P(r|t:b)U(t,b,q)$$ where $card(s)$ is the cardinality of the set from which $s$ is taken from.