While reading about reinforcement learning, I have come across the following expression for expected rewards in terms of a summation, the denominator of which I am not able to account for.

The formula given is:

enter image description here

According to what I understand, the formula should have been correct without the denominator (that I have highlighted). How is this formula correct?

Source of image: Andrew G and Sutton's book on RL.


Expectation of reward after taking action $a$ in state $s$ and ending up in state $s'$ would simply be \begin{equation} r(s, a, s') = \sum_{r \in R} r \cdot p(r|s, a, s') \end{equation} The problem with this is that they do not define probability distribution for rewards separately, they use joint distribution $p(s', r|s, a)$ which represents probability for ending up in state $s'$ with reward $r$ after taking action $a$ in state $s$. This probability can be separated in 2 parts using product rule \begin{equation} p(s', r|s, a) = p(s'|s, a)\cdot p(r|s', s, a) \end{equation} which represent probability for getting to state $s'$ from $(s, a)$ and then probability for getting reward $r$ after ending up in $s'$.
If we define reward expectation through joint distribution we would have \begin{align} r(s, a, s') &= \sum_{r \in R} r \cdot p(s', r|s, a)\\ &= \sum_{r \in R} r \cdot p(s'|s, a) \cdot p(r|s', s, a) \end{align} but this would not be correct since we have this extra $p(s'|s, a)$ so we divide everything by it to get expression with only $p(r|s', s, a)$. So, in the end we have

\begin{equation} r(s, a, s') = \sum_{r \in R} r \frac{p(r, s'|s, a)}{p(s'|s, a)} \end{equation}

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$\frac{p(s', r \mid s, a)}{p(s' \mid s, a)}$ represents the probability of observing reward $r$ in state $s'$, given that state $s'$ is the next state transitioned to. The equation assumes a probability distribution of rewards $r$ over state $s'$, meaning that a different reward might be observed whenever a state transitions from $s$ to $s'$. In most cases, if $r(s, a, s')$ is a deterministic reward then $p(s', r \mid s, a) = p(s' \mid s,a )$.

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  • $\begingroup$ But probability of observing reward r in state s', given that state s' is the next state transitioned to can be given by: Σ p(s',r|s,a) over set of all actions. I am not able to understand how these two expressions are equivalent. $\endgroup$ – SAGALPREET SINGH Apr 14 at 8:02
  • $\begingroup$ Actions in the case of the equation above is fixed rite ? $A_{t-1}$ = a. The equation does not talk about sum over all set of actions, but that a single action can lead to multiple rewards observed for state $s'$ $\endgroup$ – calveeen Apr 14 at 8:05
  • $\begingroup$ Yes, in the question, the action is fixed but in the first line of your answer you wrote that the fraction represents probability of observing reward r in state s', given that state s' is next state transitioned to. I was referring to that. $\endgroup$ – SAGALPREET SINGH Apr 14 at 8:08
  • $\begingroup$ In fact that is what my question was. Since, action is fixed the summation in Σ p(s',r | s,a) reduces simply to p(s',r | s,a), then why divide it by something? $\endgroup$ – SAGALPREET SINGH Apr 14 at 8:10
  • $\begingroup$ I see what you mean. Maybe you could consider the example whereby $p(s'|s,a) = 0.5$ and there are 2 rewards to be observed, $r_1$ and $r_2$, where $p(s',r_1|s,a) = 0.3$ and $p(s',r_2|s,a) = 0.2$. If u calculate expectations without dividing by the denominator, you would get $E(R_t) = 0.3R_1 + 0.2R_2$, which is not correct because in actual fact, you would observe $r_1$ about 0.6 of the time and $r_2$ 0.4 of the time when you transition to state $s'$ from state $s$, action $a$. $\endgroup$ – calveeen Apr 14 at 8:38

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