Identifiability of discrete HMM with categorical observations My setup is simple. I have two categorical distributions with probabilities $p$ and $\tilde{p}$ that generate an observation depending on whether the hidden state is 1 or 0, respectively. In other words (abusing some notation), $P(x_{t+1}|x_t)$ is a square matrix of size  2, and $P(y_t|x_t) = x_t*p+(1-x_t)\tilde{p}$.
I'd like to know if this model is identifiable. My intuition tells me no. In particular, if the state dynamics have no memory, ie $x_t\sim\text{Bern}(\delta),0<\delta<1,$ then this is just a mixture of categorical distributions, and I don't see how this can possibly be identifiable.
On the other hand, I've been doing a literature search, and it appears for many HMMs, there are positive identifiability results. See Petrie 1969 for example. That case is finite state and observation space, which is what I want. The problem is the math is way over my head to the point that I don't even know if my conditions apply to the assumptions of the paper.

You have to know that you have four parameters $\theta=(p,\overline{p}, a, b)$ where $P=\begin{pmatrix}
a & 1-a\\
b & 1-b
\end{pmatrix}$ is the transition matrix of the Markov chain $x_t$.
When you suppose that $x_t$ have no memory, this mean that the probability of going from 1 to 0 is the same as the one of going from 0 to 0. So the transition matrix is no more a $2\times2$ matrix but should be like $P=\begin{pmatrix}
a & 1-a\\
a & 1-a
\end{pmatrix}$ or just $P=\begin{pmatrix}
P(x_t=0)=a & P(x_t=1)=1-a
\end{pmatrix}$ . With this kind of model, if you used the $4$ previous parameters, you will over-parametrize the model and it will lead to an non identifiability.
