I was reading this article over here on Hidden Markov Models (https://en.wikipedia.org/wiki/Hidden_Markov_model).
First, the definition for a Discrete Hidden Markov Model is provided:
Let ${\displaystyle X_{n}}X_{n}$ and ${\displaystyle Y_{n}}Y_{n}$ be discrete-time stochastic processes and ${\displaystyle n\geq 1}n\geq 1$. The pair ${\displaystyle (X_{n},Y_{n})}{\displaystyle (X_{n},Y_{n})}$ is a hidden Markov model if
${\displaystyle X_{n}}X_{n}$ is a Markov process whose behavior is not directly observable ("hidden"); ${\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{1}=x_{1},\ldots ,X_{n}=x_{n}{\bigr )}=\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{n}=x_{n}{\bigr )},}{\displaystyle \operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{1}=x_{1},\ldots ,X_{n}=x_{n}{\bigr )}=\operatorname {\mathbf {P} } {\bigl (}Y_{n}\in A\ {\bigl |}\ X_{n}=x_{n}{\bigr )},} for every {\displaystyle n\geq 1,}{\displaystyle n\geq 1,} {\displaystyle x_{1},\ldots ,x_{n},}{\displaystyle x_{1},\ldots ,x_{n},}$ and every Borel set ${\displaystyle A}$.
Next, the definition of a Continues Hidden Markov Model is provided:
Let $ {\displaystyle X_{t}}X_{t}$ and ${\displaystyle Y_{t}}Y_{t}$ be continuous-time stochastic processes. The pair ${\displaystyle (X_{t},Y_{t})}(X_{t},Y_{t})$ is a hidden Markov model if
${\displaystyle X_{t}}X_{t}$ is a Markov process whose behavior is not directly observable ("hidden"); ${\displaystyle \operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid \{X_{t}\in B_{t}\}_{t\leq t_{0}})=\operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid X_{t_{0}}\in B_{t_{0}})}{\displaystyle \operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid \{X_{t}\in B_{t}\}_{t\leq t_{0}})=\operatorname {\mathbf {P} } (Y_{t_{0}}\in A\mid X_{t_{0}}\in B_{t_{0}})}$, for every ${\displaystyle t_{0},}{\displaystyle t_{0},}$ every Borel set ${\displaystyle A,}A$, and every family of Borel sets ${\displaystyle \{B_{t}\}_{t\leq t_{0}}.}{\displaystyle \{B_{t}\}_{t\leq t_{0}}.}$
In both of these cases, I had the following question about the meaning of "Borel Sets". Here is my guess:
If we take a Discrete Hidden Markov Model - on some level, this is similar to a Discrete Stochastic Process. A Discrete Stochastic Process characterizes the probability of being in a certain "state" - given that you previously were in some other "state(s)".
It seems that a Hidden Markov Model characterizes the relationship between two Stochastic Processes $X_n$ and $Y_n$.
More specifically, it seems that Hidden Markov Model characterizes the probability of the Stochastic Process Yn being contained in "some collection of states A" at "time n" only depends on which "state" the Stochastic Process Xn was at in "time n". Here, this "collection of states" ("A") is being referred to as a "Borel Set".
Thus, it appears that in this case, a "Borel Set" is simply a "collection of states"?
This brings me to my question:
Is my general understanding of Borel Sets somewhat correct in the context of Hidden Markov Models? Is a "Borel Set" simply a "collection of states"?
Assuming that I am correct - why does "A" need to be defined as a "Borel Set"? Why can't "A" just simply be referred to as a "collection of states"? What is so necessary in this situation that "A" absolutely must be a "Borel Set" and not simply some other "type of set" - for some specific problem where the collection of all possible states that the Stochastic Process Yn can assume are only "A = a1 OR a2 OR a3 OR a4", do we still need to refer to "A" as a "Borel Set"?
Thank you!
Yes, you can think of “Borel set” here as meaning, any set of states for which the definition makes sense. The sets $A, B_t$, have to have probability measures! That is, $P(y\in A)$ has to be defined, and similarly for $B_t$. This is not the case for an arbitrary set of states $A$, although you won’t be able to explicitly define such a pathological set. The Borel requirement says, don’t worry about those.
The Borel sets are a generous collection, containing the open intervals, and any set you could build from them in countably many steps using basic set operations.