Imagine you have k binary variables that you observe over time. There is some (Markov like) persistence in each variable. But they also have a sum constraint s.

Each x can be modelled as a Bernoulli and the sum of x at each time step is Binomial() or Poisson-Binomial().

So first we determine the sum of x for time t and then we sample each x according to its previous value.

How would you model this?

x \in \{0, 1\}

\sum_i x_{it} = s_t

s_t \sim s_{t-1}

x_{it} | s_t \sim x_{it-1}

It’s a bit like a multivariate HMM.