# Using categorical data to build a Dirichlet distribution

I am building a graphical model. I have some categorical data $$\boldsymbol{\mu}$$ where they are generated by $$p(\boldsymbol{\mu}|\boldsymbol{s},\mathbf{A})=\prod_k\prod_j\mathbf{A}_{ij}^{\mu_is_j}$$. I'd like to use $$\boldsymbol{\mu}$$ to be mapped to a Dirichlet distribution $$P(\boldsymbol{\pi}|\boldsymbol{\alpha})=\mathrm{Dir}(\alpha)$$.

I thought I can use the categorical distribution as a base distribution with some $$\alpha$$ scalar value. But after investigating more, actually it is not possible because the concentration parameter vector $$\boldsymbol{\alpha}$$ must be values greater than zero.

Can anybody suggest how I can solve this problem?

• Do you have a reference to a place where your categorical distribution is explained. I find that definition confusing. Oct 10 '20 at 13:15
• @SextusEmpiricus you can see this paper, section 4. Basically the sampling that happens through dirichlet distribution mimics the behaviour of the base distribution which in my example is a multinomial distribution and $\alpha$ defines how close the samples from the dirichlet distribution should be to the multinomial distribuion. Oct 10 '20 at 16:56
• it sounds like you only need a Dirichlet-multinomial bayesian model, at least from the question title. What you wnat to express with your equations is unclear and it's unclear how it's related to non-parametric bayesian statistics (which is the field of that paper). Oct 17 '20 at 16:19
• @carlo I have a hierarchical HMM model and what I am trying to do is that let's assume $K$ is the number of higher level states and I have $N$ lower level states where $K\lll N$ and higher level states maps to precision over lower level states. Basically I want to build a mixture of Gamma distributions to get the weighted average of top level states and map it to the precisions and I want to use variational inference meanwhile keep the conjugacy property. Oct 17 '20 at 16:55