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Assume that we have access to a generative model $G$, that takes as input a state-action pair $(s,a)$ and outputs a sample consisting of a successor state-observation pair $(s',o)$, that is, $(s',o)\sim G(s,a)$.

It can be assumed that the states, actions, and observations all take on a finite number of values (can be positive integers).

Is it possible to use the generative model to estimate the probability $P(o|s,s',a)$?


My thoughts:

Fix $s_0,a_0,s_0'$.

  1. Generate $(s',o)\sim G(s_0,a_0)$
  2. If $s'=s_0'$, add a particle to bin $b(o)$
  3. Return to step 1

Continue the above three steps for $M$ runs so that we have $M$ particles in different bins (each bin corresponding to an observation we have seen so far). Now how do we convert the set of particles to probabilities?

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Think of this as estimating a probability of some outcome, based on observations of many trials. Suppose we flip a coin 1000 times, and it comes up heads 327 times. What is your best estimate for the probability that the next flip will come up heads? $0.327$, right? You get that by dividing the number of heads by the number of trials.

What if you rolled a six-sized die 1000 times, and it came up a "one" 327 times? What is your best estimate for the probability that it comes up "one"? The same thing -- $0.327$, right?

It's the same here. Each time that you generate $(s',o)$ and find that $s'=s'_0$, think of that as a trial -- the outcome of the trial is the observed value of $o$. (Ignore all iterations where $s' \ne s'_0$; they're not even a trial.) So, if you have a particular bin, say the bin corresponding to $o=1$, then you count the number of trials where you observed that $o$ was $1$, and divide by the total number of trials. That's your estimate for the desired probability.

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