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I'm working on a game theory model of incomplete information, where players observe certain attributes via noisy signals. I am looking to solve for two different probability functions, though I think the math should be very similar:

  1. Say there is some random variable $X \sim U(a - \epsilon, a + \epsilon)$. You observe a draw from this distribution, call it $S_1$, but you do not know $a$. Given $S_1$, what is the probability that another draw, $S_2$, from the same distribution, will be greater than or equal to an arbitrary number $b$? That is, what is:

$$P(S_2 \geq b \mid S_1)$$

  1. (Note these problems are separate, so, for example, $b$ here does not mean the same thing as $b$ in part 1). Say that $b$ and $c$ are two independent draws from $U(0,1)$, the standard uniform distribution. Now, $B \sim U(b - \epsilon, b + \epsilon)$ and $C \sim U(c - \epsilon, c + \epsilon)$. You observe one draw from $B$ and one draw from $C$, but you do not know $b$ or $c$. What is the probability that a new draw from $B$ will be greater than a new draw from $C$? That is, if $S_1^b$ is your observed draw from $B$ and $S_1^c$ is your observed draw from $C$, what is:

$$P(S_2^b \geq S_2^c \mid S_1^b, S_1^c)$$

Any help with this would be much appreciated! I've read a bit about convolutions and posterior predictive distributions, but I don't have the grasp on it that I need to solve for these functions.

Thanks!

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In order to even be able to define the probabilities you want to compute, you need to have a Bayesian setting in which you put a prior on the parameters (the endpoints) of your uniform distributions. See http://www.amstat.org/publications/jse/v6n3/rossman.html

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  • $\begingroup$ In (1), $a$ can be any number on the interval $[0,1]$, and is chosen by a person. How can you form a prior distribution if there is no randomness? In (2), the prior is given: $b$ and $c$ are drawn from $U(0,1)$, so that would be the prior. $\endgroup$
    – sundance
    Feb 3, 2016 at 16:34
  • $\begingroup$ if you do not know a, then either you make a guess about its value (which means that you put a dirac delta prior on it) or you properly model the uncertainty by putting a proper prior, with a non-zero dispersion. Otherwise, what is the meaning of unknown? $\endgroup$
    – user4422
    Feb 3, 2016 at 21:25
  • $\begingroup$ Okay, I suppose you could just guess that it is the mean of the distribution (.5). After obtaining these priors, what is the next step? How do you update the distribution with the realized signals and then compute the function at the end of the post? $\endgroup$
    – sundance
    Feb 4, 2016 at 0:50
  • $\begingroup$ if your prior is conjugate (as suggested in the link above) then you have an analytical solution for the posterior density and you can use it to compute the probability you need. $\endgroup$
    – user4422
    Feb 4, 2016 at 14:47
  • $\begingroup$ This claim that a "Bayesian setting" is needed is incorrect. Question (2) is fully answerable; question (1) has an answer that depends on $a.$ $\endgroup$
    – whuber
    Jan 16, 2021 at 16:47

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