Markov Switching and Hidden Markov Models

Are the two interchangeable terms? I have been reading about markov-switching models and am struggling to see the difference with HMM models.

Markov Switching Models are the same thing as Regime Switching Models. A Hidden Markov Switching Model or a Hidden Regime Switching Model (both of which are commonly called a Hidden Markov Model) is different.

A Hidden Markov Model (HMM) is a doubly stochastic process. There is an underlying stochastic process that is not observable (hidden), the results of which can be observed (these results being the second stochastic process). The underlying stochastic process that is hidden is what makes this model different.

Consider this coin toss example starting on page 5: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1165342

If the man behind the curtain flips 1 coin and tells you the results, you have 2 states (state 1: heads, state 2: tails). Thus, by knowing the results of each coin flip, you can determine the state. Therefore, you cannot construct a HMM.

If the man behind the curtain flips 2 coins and tells you the results, you still have 2 states (state 1: coin 1, state 2: coin 2) but they are not uniquely tied to heads or tails. This is a hidden process because you can't tell which state (coin 1 or coin 2) led to the observation (heads or tails). Therefore, you can construct a HMM.

• Would you mind elaborately some more or giving more examples? Because this seems like a misconception that happens often, e.g. the statsmodels documentation says: Markov switching models (MSAR), also known as Hidden Markov Models (HMM) – guy Apr 19 '17 at 3:29

A hidden Markov model is a bivariate stochastic process $$\{Y_t, X_t\}_{t=1,2,...}$$ where $$\{X_t\}$$ is an unobserved Markov chain and, conditional on $$\{X_t\}$$, $$\{Y_t\}$$ is an observed sequence of independent random variables such that the conditional distribution of $$Y_t$$ only depends on $$X_t$$.

When $$Y_t$$ depends both on $$X_t$$ and the lagged observations $$Y_{t−1}$$, it is called Markov Switching Model.

An, Y., Hu, Y., Hopkins, J., & Shum, M. (2013). Identifiability and inference of hidden Markov models. Technical report