Maximum likelihood estimation of a Dirichlet distribution multivariate parameters

Is it necessary to find the 'maximum likelihood estimates' of prior dirichlet parameters after finding their initial values through the 'method of moments ' to find posterior probabilities through bayesian analysis (using dirichlet -multinomial conjugate pairs)? If yes, I need to know an easy way to find the maximum likelihood estimates of dirichlet parameters. I did the bayesian analysis with the initial guess made through the 'method of moments ' without finding MLE . Is it correct to do so? Eg. Let A,B,C be the three dependent variables with a prior probability of occurrence (Pij's). So, j = 1,2,3 for the three variables. I have three observations of three different time periods for each variable A,B,C. So, i = 1,2,3 for the three observations.(The data given below is only an example; not original). j=1 implies the variable A. j=2 implies the variable B. j=3 implies the variable C. Let 0.1, 0.4 , and 0.2 be the three observations (for i=1,2,3) of variable A. Then 0.6, 0.5, and 0.7 be the three observations of variable B. Then 0.3, 0.1, and 0.1 be the three observations of variable C. I fitted this data with a dirichlet distribution. With this prior Pij's I made the initial guess of dirichlet parameters (alphas) by the 'method of moments' from a reference. In the reference they use this intial guess of alphas to find the 'maximum likelihood estimates' of alphas. But I skipped the MLE method and did the bayesian analysis with a multinomial data as my likelihood function (frequency counts of each variable) and used the initial guess of alphas to estimate the posterior probability of occurrence of the three dependent variables. Is MLE necessary for prior alphas?

• Yes, I meant a trivariate distribution and the individual variables are dependent as clarified in the above answer. I made the initial guess (prior values for the alphas) of the prior dirichlet parameters only through the 'Method of moments' without using 'MLE' and found the posterior parameters. With these posterior parameters I estimated the posterior mode and mean. My doubt is actually that whether the prior parameters found only through ' method of moments' without using 'MLE ' is sufficient/valid in my bayesian analysis. – Anagha Raveendran Dec 15 '17 at 15:38

You seem to be confusing many things in your question.

Is there any quick solution (either by any statistical software or manual workout) to find the maximum likelihood estimates of alpha of three independent variables of a dirichlet distribution

First of all, Dirichlet distribution is a multivariate distribution, I assume that you mean trivariate distribution in here. Obviously, the individual variables are not independent, it should be obvious at least from the fact that if $x_1,x_2,\dots,x_k$ are draws from Dirichlet distribution, then $\sum_{i=1}^k x_i = 1$, so they need to be dependent to meet the constraint.

provided that the initial values of the three parameters are found by the method of moments.

What do you mean by "method of moments" in here? There are many ways of computing the parameters of Dirichlet distribution (see, e.g. Minka, 2000; Huang, 2005), that in most cases maximize the likelihood numerically, and there is no simple, closed-form solution.

Or, Is the initial values sufficient to obtain posterior estimates in multinomial-dirichlet bayesian analysis?

To obtain posterior in Bayesian analysis you do not need to find the maximum likelihood estimates of the parameters. Maximum likelihood and Bayesian approaches are two different approaches to estimating parameters. Maximum likelihood is about finding such combination of parameters that maximize the likelihood function. In Bayesian case, you estimate the parameters in terms of the likelihood function and the priors. In Dirichlet-multinomial model (this is not the same as Dirichlet distribution), this is straightforward since Dirichlet is a conjugate prior for the multinomial distribution, and we have a closed form solution. The posterior estimate for $k$-th $\alpha$ is $\alpha_k + y_k$ where $\alpha_k$ is your prior guess for $\alpha_k$ and $y_k$ is the observed number of successes for $k$-th category in the multinomial distribution.

Huang, J. (2005). Maximum likelihood estimation of Dirichlet distribution parameters. CMU Technique Report.

Minka, T. (2000). Estimating a Dirichlet distribution. Online draft.

• The prior guess for alphas is made through the 'method of moments' by referring to the paper on ''A note on parameter estimation in the multivariate beta distribution'' by A. Narayanan, (1992). I am not using the 'method of maximum likelihood' explained in this paper for the prior guess for alphas. Is this approach correct to find posterior estimates through bayesian method while using dirichlet - multinomial conjugate pairs? – Anagha Raveendran Dec 16 '17 at 5:14
• @AnaghaRaveendran As I said in my answer, prior is not something that is estimated from the data. Moreover, I can't see how exactly you could estimate the prior parameters using method of moments: for this you would need to the the Dirichlet distributed data, i.e. the data on your parameters, while from your description it seems that you have multinomialy distributed data and assume Dirichlet distribution for the unobserved parameters. – Tim Dec 16 '17 at 11:02
• I am thanking you for your reply. I estimated the prior dirichlet parameters from the prior probability data (Pij's)that I have. j=3 for the three dependent variables and i=3 for three observations of each variable. For this I choose the ' method of moments' from the reference I mentioned early. To estimate those I have'nt taken any multinomial data. – Anagha Raveendran Dec 16 '17 at 16:07
• The 'MLE method ' in the aforesaid reference uses the initial parameter values obtained from 'method of moments' to find a maximum likelihood parameter values. But I did'nt use the latter method. So I need to know whether my former method alone is sufficient for the prior parameter estimation to do the bayesian analysis. – Anagha Raveendran Dec 16 '17 at 16:23
• @AnaghaRaveendran then I am afraid that your description of the problem is insufficient and the question is unclear. What data do you have exactly? What do you want to estimate? How does this relate to Dirichlet-multinomial model (you either have data on multinomial counts, or on "probabilities")? – Tim Dec 16 '17 at 17:05