MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Correlation between Beta distributions

Hi,

Before you read the following question, I want to mention that I have a Computer Science background and am not very knowledgeable in Probability and Statistics. So excuse me if my question,notation, or language is flawed. This question has been viewed by a considerable number of people in this forum and yet no one has indicated anything. Considering the caliber of the experts in this forum, it's really improbable that no one can answer this. Therefore, I suspect that there is something wrong with my presentation of the problem that makes it not understandable. If this is the case I desperately ask for your kind comments and tips so I can revise my question. Maybe I am way off, if so, please don't hesitate to bash me if you like. That will also be more helpful. At least I will understand where I am located.

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

-
 Hi, I think the best way is to look at chapters of books on multivariate statistical analysis such as "An Introduction to Multivariate Statistical Analysis" by Anderson (for example see Theorem 4.2.2 for the density of the distribution of the correlation coefficient in a sample of N bivariate normal distribution) and "Aspects of Multivariate Statistical Theory" by Muirhead. – kakuritsu Mar 10 2011 at 1:08