Confusion about hidden Markov models definition and graphical notation?

Question

In Wikipedia the following diagram has been sketched and the following text has been written:

From the diagram, it is clear that the conditional probability distribution of the hidden variable x(t) at time t, given the values of the hidden variable x at all times, depends only on the value of the hidden variable x(t − 1): the values at time t − 2 and before have no influence.

Now the problem is I don't understand why this is true for $X_1$ as there is a loop there and as a result I cannot see why this follows: $$p(X_1|X_2,X_3)=p(X_1|X_0=\emptyset)=p(X_1)$$ But it actually should be: $$p(X_1|X_2,X_3)=p(X_1|X_2)$$

Answer 1

I think the problem is a misinterpretation of the notation. $X_1$ stands for $X(t)=1$, not for $X(t=1)$, and $a_{ij}=P(X(t)=j| X(t-1)=i)$. $X_1, X_2$ and $X_3$ are the three possible values of the variable $X$ at a time $t$, not three variables a $t=1, t=2$ and $t=3$.

So at time $t=1$ you have an initial value. Let's take, for the sake of the example, $X(1)=1$. the probability of $X(2)=2$ is $a_{12}$ and the probability of $X(2)=1$ is $1-a_{12}$.

The value of $X(t)$ is only dependant of the value of $X(t-1)$ because there is no arrow pointing from a $Y$ node to an $X$ node.

Answer 2

Your confusion is coming from different graphical notations about HMM. Specifically, there are two types of notations.

• Type 1 is using nodes to represent possible values of random variables and use arrows to represent transitions
• Type 2 is using nodes to represent random variables, and use errors to represent conditional dependencies.

Here is the same HMM using two different notation

Type 1 notation (Note, for the model shown below, random variable $X$ has $3$ possible values, and random variable $Y$ has $4$ possible values. But it does not show how many observations.)

Type 2 notation (Note, the model shown below has $N$ random variables / observations, but not show how many possible values for $X$ and $Y$)

PS,I think the figure in Wikipedia is confusing, if you want to use vertex to represent possible values, you should not capitalize $X$, but make it consistent with $y$.

Answer 3

Transition probability is a potential for switching between any 2 states, not a statement about the relationship of events that have happened. These transition probabilities are independent of each other, as shown in the diagram by the fact that $a_{12}$ has no explicit relationship to $a_{21}$ or $a_{23}$.

The diagram is showing that there are probabilities of transition between hidden states $X_1$ and $X_2$. Moreover, any transition probability is referent to 2 states: the initial state and the state that will be transitioned to.

A loop in this case means that transitions between states 1 and 2 can happen in both directions (to and from $X_1$, to and from $X_2$) with some probabilities $a_{12}$, $a_{21}$. So, a process can start at $X_1$ and has the probability of moving to $X_2$ with $P(X_1 \rightarrow X_2) = a_{12}$, which, if it were to do so, would then have the probability of moving back to $X_1$ with $P(X_2 \rightarrow X_1) = a_{21}$.

This is what is meant when it is said the state at time $t$, $x(t)$, is dependent on the immediately prior state, which was at $t-1$, $x(t-1)$: a given state will be moved out of only with the probabilities of its immediately adjacent states.