# Does this graphical model describe a hidden Markov model?

I'm facing the problem of visual tracking in computer vision. I have some observation (image blobs by background subtraction) produced by some moving object, and the task is to infer the state (position, velocity) of the object given the observation. I assume that each state depends only on the immediate predecessor (Markov assumption), ruled by a temporal model (the dynamics) that describes how a state $X_t$ evolve in $X_{t+1}$, and that the measurement model do not depends on time, so that $y_t$ do not depends on $X_{t-1}$ but only on $X_t$.

Everywhere in the web I have found that this problem can be described by this graphical model: For inferring the posterior distribution, after receiving new measurements, a Kalman filter or Particle filter is used.

But anywhere I explicitly read that this is an Hidden Markov Model.

Is this a HMM? Why? Why not?

## 2 Answers

In short: you don't know yet. The plate diagram for a State Space Model and a Hidden Markov Model are the same (and they are what you have presented). This is deliberate.

The distinction between the two models is in the nature of X. If X is multinomial with K states then this is an K-state HMM of order 1. If X is (possibly multivariate and) continuous, then it is a SSM. If X also happens to be Normal then it is a linear Normal SSM, perhaps the most common variety, for which the Kalman filter and RT smoother was designed.

For typological purposes it doesn't apparently matter what Y is or the form of the conditional distributions, although clearly these strongly affect how easy it is to filter, smooth and estimate parameters for.

If it looks like a duck, swims like a duck, and quacks like a duck, then it probably is a duck.

It looks like an HMM to me.

• @omidi your reasoning is flawless. – daknowles Feb 6 '14 at 3:43
• I'm sorry it seemed that I made a mistake! – omidi Feb 6 '14 at 9:17