---
title: "Equivalence to Conditional Gamma-Poisson Model"
author: "Patrick Brown"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{Equivalence to Conditional Gamma-Poisson Model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# Equivalence to Conditional Gamma-Poisson Model

The Dirichlet-multinomial distribution with parameter $\alpha$ is equivalent to a conditional Gamma-Poisson model. This equivalence provides an alternative interpretation and can be useful for understanding the properties of the distribution.

Consider a Gamma-Poisson hierarchy where:

1. $U_k \sim \text{Gamma}(\alpha, 1)$
2. $\lambda_k =  \mu_k U_k$
3. $y_k \mid \lambda_k \sim \text{Poisson}(\lambda_k)$ for $k = 1, \ldots, K$

Given $\lambda_k$, the distribution of $y_k$ is Poisson:

\begin{equation}
y_k \mid \lambda_k \sim \text{Poisson}(\lambda_k)
\end{equation}


The distribution of $\mathbf{y}$ conditioned on the sum given the $\lambda_k$ is multinomial:

\begin{equation}\label{eq:multinom}
P(\mathbf{y} \mid \boldsymbol{\lambda}, \sum_{k=1}^K Y_k = N)
=
\frac{N!}{\prod_{k=1}^K y_k!}
\prod_{k=1}^K
\left(\frac{\lambda_k}{\sum_{j=1}^K \lambda_j}\right)^{y_k}
\end{equation}

Integrating out the $U_k$ gives the Dirichlet-multinomial distribution:

\begin{equation}\label{eq:dm}
P(\mathbf{y} \mid \alpha, \boldsymbol{\mu}, N)
=
\frac{N!}{\prod_{k=1}^K y_k!}
\frac{\Gamma(\alpha)}{\Gamma(N+\alpha)}
\prod_{k=1}^K
\frac{\Gamma(y_k+\alpha \mu_k)}{\Gamma(\alpha \mu_k)}.
\prod_{k=1}^K
\frac{\Gamma(y_k+\alpha\mu_k)}{\Gamma(\alpha\mu_k)}.
\end{equation}

Note that $\text{E}(U_k) = 1$ and 
$$
\text{sd}(U_k) = 1/\sqrt(\alpha)
$$

## Simplificaiton

Here is the simplification of the Dirichlet-multinomial PMF using the gamma recurrence identity.

The key identity is the gamma recurrence over an integer number of steps:

\begin{equation}\label{eq:gamma_ratio}
\frac{\Gamma(x+n)}{\Gamma(x)}
=
x(x+1)\cdots(x+n-1)
\end{equation}


Equation \ref{eq:dm} becomes

\begin{equation}\label{eq:dm_ratio}
P(\mathbf{y} \mid \alpha, \boldsymbol{\mu}, N)
=
\frac{N!}{\prod_{k=1}^K y_k!}
\frac{1}{\prod_{j=0}^{N-1} (\alpha + j)}
\prod_{k=1}^K
\prod_{j=0}^{y_k-1} (\alpha \mu_k + j)
\end{equation}


substitute 
$$
\alpha = \frac{1}{\tau^2}
$$
to give
\begin{equation}\label{eq:dm_ratio_tau}
P(\mathbf{y} \mid \tau, \boldsymbol{\mu}, N) =
\frac{N!}{\prod_{k=1}^K y_k!}
\frac{\tau^{2N}}{\prod_{j=0}^{N-1} (1 + j \tau^2)}
\prod_{k=1}^K
\frac{\prod_{j=0}^{y_k-1} (\mu_k + j \tau^2)}{\tau^{2y_k}}
\end{equation}


The $\tau^2$ cancel.  On the log scale:

\begin{equation}\label{eq:dm_ratio_log}
\log P(\mathbf{y} \mid \tau, \boldsymbol{\mu}, N)=
\log(N!) - \sum_{k=1}^K \log(y_k!) - \sum_{j=0}^{N-1} \log(1 + j \tau^2) + \sum_{k=1}^K \sum_{j=0}^{y_k-1} \log(\mu_k + j \tau^2)
\end{equation}

Note that when $\tau = 0$ then 

\begin{equation}\label{eq:dm_ratio_log_zero}
\log P(\mathbf{y} \mid \tau=0, \boldsymbol{\mu}, N)=
\log(N!) - \sum_{k=1}^K \log(y_k!) + \sum_{k=1}^K y_k \log(\mu_k)
\end{equation}

which is the multinomial density



```{r testDirMn}
K <- 6
N <- 100 * K
mu <- 2
tau <- 0.2
mu_vec <- rep(mu, K)
alpha <- mu_vec / sum(mu_vec)

U <- rgamma(N, shape = tau^(-2), rate = tau^(-2))
c(mean(U), sd(U))
lambda <- mu * U
y <- rpois(N, lambda)

cc_matrix <- Matrix::sparseMatrix(
  i = 1:N, j = rep(seq(1, N / K), each = K), x = 1
)

mu_vec <- rep(mu, K)
alpha_norm <- mu_vec / sum(mu_vec)
alpha <- alpha_norm / tau^2

log_dens_with_gammas <- log_dens_simplified <- rep(NA, ncol(cc_matrix))

for (D in 1:ncol(cc_matrix)) {
  # Extract the indices where the column has non-zero values
  indices_here <- which(cc_matrix[, D] == 1)
  y_here <- y[indices_here]

  # Simplified log density using the sum formula
  # Handle zeros correctly by filtering them out for the sum
  y_double_sum <- mapply(seq,
    len = y_here,
    from = alpha_norm,
    MoreArgs = list(by = tau^2)
  )

  log_dens_simplified[D] <-
    lfactorial(sum(y_here)) -
    sum(lfactorial(y_here)) -
    sum(log(seq(from = 1, by = tau^2, len = sum(y_here)))) +
    sum(log(unlist(y_double_sum)))

  # Log density using Gamma functions
  # lgamma is defined for all non-negative integers, so no special handling needed for zeros
  log_dens_with_gammas[D] <-
    lgamma(sum(alpha)) + lgamma(sum(y_here) + 1) -
    lgamma(sum(y_here) + sum(alpha)) +
    sum(lgamma(y_here + alpha)) -
    sum(lgamma(alpha)) - sum(lgamma(y_here + 1))
}
```

```{r plot}
plot(log_dens_simplified, log_dens_with_gammas,
  xlab = "simplified", ylab = "gammas"
)
abline(0, 1)
```