Difference in pdf formula between Dirichlet and Multinomial distributions The pdf for Dirichlet distribution seems to be
$$
Dir(\alpha_1,\alpha_2,\ldots\alpha_k) \text{ is defined as} 
$$
$$
pdf(θ_1,θ_2,\ldots,θ_k )= \frac{Γ(\alpha_0)}{Γ(∝_1 )Γ(∝_2 )\cdots Γ(∝_k )} θ_1^{∝_1-1} θ_2^{∝_2-1} \cdots θ_k^{∝_k-1}
$$
$$
\text{ over the region where } θ_i \gt 0 \text{ and } θ_1 + θ_2 + ... + θ_k = 1 
$$
$$
\text{ and } ∝_0 = ∝_1+ ∝_2+\cdots+∝_k
$$
The formula for a multinomial seems to be
$$
pmf(φ_1,φ_2,\ldots, φ_k;n_1,n_2,… n_k)=  \frac{Γ(n_0+1)}{Γ(n_1+1)Γ(n_2+1)\cdots Γ(n_k+1)} φ_1^{n_1} φ_2^{n_2}\cdots φ_k^{n_k}
$$
$$
\text{ over the region where } φ_i \ge 0 \text{ and } φ_1 + φ_2 + ... + φ_k = 1 
$$
$$
\text{ and } n_0 = n_1+ n_2+\cdots+n_k
$$
I am unable to see the difference in the formulas? Did I miss some other conditions? 
(I am aware that Dirichlet is a conjugate-prior to multinomial)
 A: The pdf for the multinomial distribution is a discrete probability mass function, supported on the set of tuples $(n_1,\ldots,n_k)$ of non-negative integers satisfying the constraint $n_1+\cdots+n_k=n$. The notation you use obscures that.  Which distribution it is depends on parameters $(\alpha_1,\ldots,\alpha_k)$.  The input to the function is a tuple $(n_1,\ldots,n_k)$ and the output is a probablity.
The pdf for the Dirichlet distribution is a continuous probability density function, supported on the set of tuples $(\theta_1,\ldots,\theta_k)$ satisfying the constraint $\theta_1+\cdots+\theta_k=1$.  Obviously these are real arguments, not integer arguements, and they vary continuously, not discretely. The output is not a probabilty, and in particular it can be far more than $1$.
A lot of confusion can be caused by messy notation, and yours is pretty bad.  You're actually using the same letter, $p$ to refer to two quite different functions, and then you're using $p$ with subscripts to refer to something altogether different, and you don't at all make it clear that the input to the probability mass function for the multinomial distribution is a tuple of integers $n_1,\ldots,n_k$ rather than a tuple of real numbers, nor do you make clear the relationship between the arguments $p_1,\ldots,p_k$ and the parameters $\theta_1,\ldots,\theta_k$.
