# Correlation between Beta distributions

I have a Computer Science background and not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. Anyways, the problems is that we have two Bernoulli variables $X_1,X_2$ that generate sequences of values for $n$ consecutive Bernoulli trials. Now we have $m$ independent observations of those trials in discrete independent time-slots. For example:

let: $n=4, m=3$

The result of the aforementioned trials is:

for $X_1: ((1,0,0,1),(0,0,1,1),(0,0,0,1))$
for $X_2: ((0,0,1,0),(0,1,1,0),(0,1,0,0))$

where each of "inner-sequences" is one of $m$ independent sequence of Bernoulli trials. Now we aggregate each of those "inner-sequences" into a Beta distribution function to represent the posterior probability of success/failure of each variable in different observation. For instance the above sequence transforms to the following

$X_1:(B_{11},B_{12},B_{13})$ $X_2:(B_{21},B_{22},B_{23})$

where each $B_{ij}$ is a Beta distribution function associated with the corresponding sequence of trials in the previous part of the example.

Now we a have two sequences Beta Distribution where we want to use in order to find the correlation between $X_1$ and $X_2$ preferably producing a final beta curve that shows the degree of correlation between $X_1$ and $X_2$ incorporating the factor of uncertainty, or a Gaussian curve. A very simple approach is to find the correlation based on the mean of the curves and using the Pearson's correlation method. However, this is not precise enough. Is there any method to find the correlation between two Beta distribution functions. An easier question is where can I find useful information about the detection of correlation based on two distribution functions of any kind (easiest should be Gaussian functions). Thank you so much in advance.

Amir

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