What is the name of the following (Markov-like) chain?

Question

We know that the random sequence $X_1,X_2,\ldots$ forms a Markov chain if $$f(X_n\mid X_1,\ldots,X_{n-1})=f(X_n\mid X_{n-1}).$$ Now suppose that we have two random sequences $X_1,X_2,\ldots$ and $Y_1,Y_2,\ldots$ that satisfy $$f(Y_n\mid X_1,\ldots,X_n)=f(Y_n\mid X_n).$$ As we see it is somehow similar to the Markov chain, but I could not find their specific name in the Markov-chain literature. Any help?

Answer 1

Maybe you have in mind a Hidden Markov Model, but the equation for that process is $$f(Y_n|X_1,Y_1,...,X_n,Y_n)=f(Y_n|X_n),$$ where $\{Y_n\}$ is the observed process and $\{X_n\}$ is an unobserved Markov chain.