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Say we have a "3 faces coin" that will be flipped infinitely many times.

Face one is Heads (H) that comes up with probability $(1-m) p$,

Face two is Tails (T) that comes up with probability $(1-m)(1-p)$, and

Face three is Empty (E) that comes up with probability $m$.

Both $m$ and $p$ are between 0 and 1.

Say the coin flipper never misreads H and T. However, whenever they face E, they will interpret it, with probability $g \geq 0.5$, as evidence in favor of the face that came up more often until that point. (i.e. it will randomly transform the observed E into H or T, depending on the observed relative frequency of T and H).

Particular example: if, at any point in time, the agent has observed 5 realizations in favor of H and 3 in favor of T, then if they face E, they will interpret as H with probability $g$ or as T with probability $1-g$. Once the interpretation is stored, it will also influence the way the future E realizations will be interpreted.

QUESTION IS: what is the probability that the number of H realizations/interpretations is greater than (less than) the number of number of T realizations/interpretations after infinitely many flips? (i.e., I am looking for an analytical/closed-form solution for such probability and that does depend on the parameters $m,p,g$.

Obs. 1: For simplicity, if the number of H's and T's are the same, one could use a tie break that favors H. I conjecture this is not wlog, but it makes things easier.

Obs. 2: This problem seems to be similar to the Gambler's ruin, Polya's urn or some other branching process. But I haven't been able to solve it properly (not a mathematician here). thx!

Marcos RF
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  • What have you tried? – Mars Mar 12 '21 at 17:02
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    @Mars: I have tried 2 things. 1) Forward Induction, but there is no clear pattern arising. Plus, there seems to have an asymmetry between odds and even periods. 2) I computed the Probability analytically and I got something like: $P\left(\sum_{t}1{x_{t} = H} - \sum_{t}1{x_{t} = T} + \sum_{t}\left[1{x_{t} = E} (\dots) \right] < 0\right)$ – Marcos RF Mar 12 '21 at 17:12
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    @Mars: The expression in $(\dots)$ is a bit cumbersome to type, but the idea is that it is counting the number of instances when the agent conformed the interpretation with the state that appeared more minus the number of instances when the agents did not conform. The expression $1{\cdot}$ is a standard indicator function that is being used to count the events. – Marcos RF Mar 12 '21 at 17:16
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    Now I'm confused about the question you're asking. What does seriously "number of evidences" mean? What do you mean by that? What is the outcome that you want the probability of? Is it that the relative frequency of heads is greater than the relative frequency of tails? What do periods of anything have to do with it? Is the idea that an observed heads should raise your assessment of the probability of heads, but an Empty should (should, or does?) raise your assessment of the probability of heads with probability g? – Mars Mar 12 '21 at 23:33
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    It might help us to visualise things better if you gave us an idea of where this problem came from. (Also, if simplicity is your goal, I would think it more natural to assume that if H=T then the coin flipper chooses H or T with probability $\frac12$. That keeps things symmetrical, at least.) – TonyK Mar 12 '21 at 23:45
  • @Mars: Not sure why this is confusing, since the question seems to be properly posted. Evidence is the realization of the random variable I described. And yes, I want to know what is the probability that the number of H is greater than the number of T as $t \to \infty$ (periods are equivalent to trials: 1 period, 1 trial), as a function of the parameters described ($m, g, p$). Feel free to put it aside if that is still confusing. – Marcos RF Mar 18 '21 at 19:16
  • Hi @TonyK. Thx for your comment. At least when I try to iterate this forward, imposing this rule makes things more complicated as it creates more branches in this probabilistic tree. I agree that keeping things symmetric almost invariably helps. But since there is a recursion in this problem (my future interpretation depends on the previous interpretations), the asymmetry reduces significantly the dimensionality of the tree. The intuition of this is that people may observe informative signals, but every once in a while they mess up bcs they conform the interpretation with their priors. – Marcos RF Mar 18 '21 at 19:20
  • Hi Marcos. The word "evidence" never is plural ("evidences") in mainsteam English, so that was confusing. Thanks for explaining--now I understand. The use of "confound" in the question is confusing too; I think you meant "confront" or "encounter". (I'm not criticizing your use of English--everyone here lacks knowledge of parts of something--just trying to help clarify.) I'm still a little confused by "evidence toward the state". Does this mean an estimate of the probability of heads on the next toss, or an estimate of the value of $p$ (or tails and $(1-p)$ if tails came up more often)? – Mars Mar 19 '21 at 15:02
  • Would it make sense to think of the problem this way? There's a coin that only comes up H with probability $p$, or T with probability $(1-p)$. But there is a separate process s.t. with probability $m$, after any toss the observer looks at frequencies of H in past tosses and estimates the probability of H. In that case, this is statistical inference, and you can use frequentist or Bayesian statistics, but I think the best estimate on most models will be the relative frequency of heads so far. This estimate changes given that observations occur at random times, so it's a random process. – Mars Mar 19 '21 at 15:10
  • In the example, why is the estimate $g$ given five H and 3 T? The question also says that an estimate at one time will influence an estimate at a later time. How would it do that? Is the idea that the estimate at time 2 is some adjustment of the estimate at time 1, based on relative frequencies in times between them? Or over all past time? What sort of adjustment are you thinking of? – Mars Mar 19 '21 at 15:14
  • Could you clarify the specific question you have about this model? The "the number of H realizations ... after infinitely many flips" could be infinite. Maybe you want the expected value of the limit of the ratio of the observation counts? Or the limit of the probability of observing H? – Karl Mar 22 '21 at 17:46
  • @Karl: The question is in bold. I would like to know the analytical form of the (limiting) probability that the number of H's is greater than the number of T's. Not sure how to be more clear. I am confident one can find a mathematical expression that combines the parameters $m, p, g$ that pins down this probability. – Marcos RF Mar 22 '21 at 18:17
  • Ah, it looked like you were asking for the probability that the infinite sequence itself has some property, not the limiting probability that a condition is satisfied at the $n$th step. – Karl Mar 22 '21 at 18:24

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