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I'm simplifying my research question and want to know whether the question can be properly modeled or not.

Suppose we have two coins $X_1,X_2$ and assume that the outcomes are possibly correlated. Denote $p_i=\mathbb P[X_i=1]$ and $p_{12}=\mathbb P[X_1=1\wedge X_2=1]$.

We have to estimate $p_i$, $p_{ij}$, $p_{X_1|X_2}$ and $p_{X_2|X_1}$ from data, a table of heads/tails of each trial. What is more is that, in some trials, only one coin is tossed. For example, the data we have is

Trial 1: (head,head), Trial 2: (head,no tossing), Trial 3: (tail, tail), Trial 4: (tail,head) and so on.

Would there be a prior distiribution(such as the Beta conjugate in independent variable casees) that we can handle Bayesian inference for this bivariate Bernoulli trials?

Any comments will be very much appreciated.

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Since you've got a known number of outcomes (9, if I'm counting correctly, which arguably is not very many), you could model this data in a general way as a categorical distribution. In that case, you would have a vector of probabilities, each corresponding to the discrete outcomes {(head, head), (head, tail), (head, none), (tail, head), (tail, tail), (tail, none), (none, head), (none, tail), (none, none)}. In that case, you can use a Dirichlet as your conjugate prior distribution.

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  • $\begingroup$ Thanks for your comment. Does Dirichlet work for correlated trials as well? In my model set up, two coins are possibly correlated. $\endgroup$ – Andeanlll Aug 5 at 4:29
  • $\begingroup$ I don't think I understand - you're saying two different things: 1) The coins are correlated, ie coin flips within a trial are not independent 2) The trials are correlated, ie this trial can affect the next one Which one is it? If it's the first one, there's no issue, each of the 9 outcomes under the categorical distribution is mutually exclusive. If it's the second one, then there may be an issue. $\endgroup$ – Louis Cialdella Aug 5 at 14:05
  • $\begingroup$ Oh, I get your point. So, basically, multivariate multinoulli can be modeled using multinomial dist. right? and the categorical dist. can be a good option because it's a conjugate prior for multinomial. Do I understand your point correctly? $\endgroup$ – Andeanlll Aug 7 at 3:22
  • $\begingroup$ Very close, yes - you are correct that I'm imagining your 9 outcomes as being multinomially distributed. The only correction I would make to your comment is that the Dirichlet is the prior here (the categorical is the multiple outcome bernoulli equivalent). $\endgroup$ – Louis Cialdella Aug 7 at 15:48
  • $\begingroup$ That's correct. Thank you for your correction and also for your detailed answer! $\endgroup$ – Andeanlll Aug 8 at 3:19

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