What is the added value of a multivariate Bernoulli distribution over a multinomial distribution?

Question

I came across the multivariate Bernoulli distribution of Dai, Ding & Wahba (2013) that has the following form (in the bivariate case): $P(X_1,X_2)=p_{11}^{x_1 x_2} p_{10}^{x_1 (1-x_2)} p_{01}^{(1-x_1) x_2} p_{00}^{(1-x_1) (1-x_2)}$

This distribution is a multinomial distribution for $n=1$ with the powers expressed as the values on $x_1$ and $x_2$. Given that we need both $x_1$ and $x_2$ to compute the probability mass, we can express these power coefficients immediately as $x_{11}$, $x_{10}$, $x_{01}$ and $x_{00}$.

Then why would we use such a multivariate Bernoulli distribution rather than the multinomial distribution? Is there any theoretical or practical value?

Answer 1

Indeed, from a purely probabilistic point of view there is nothing new---a multivariate bernoulli distribution is a multinomial distribution. The utility comes from a modeling perspective. Lets concentrate on the bivariate case, with response $X_i=(X_{1i}, X_{2i})$ for $i=1,2,\dotsc,n$. $i$ could as an example index patients with some eye problems (or not), and the problem could affect one, both or none eyes.

If this were coded in a multinomial way with just $Y_i$ equal to 1,2,3 or 4 this way (dropping now index $i$): $Y=1 \iff X=(1,1), Y=2\iff X=(1,0), Y=3 \iff X=(0,1), Y=4\iff X=(0,0)$ then we lose the direct connection with each eye! making modeling more complicated. If you read the paper you linked too, you will see this is taken advantage of in formulating (and answering) questions about conditional distribitions, independence and so on among the component bernoulli variables, which cannot easily be formulated in the multinomial setting. We could equally have named this a structured multinomal distribution, a special case of multinomial where we can formulate and answer questions that do not give meaning in the general multinomial setting.

Look through this stoored google scholar search to see evidence for this.