1 Probability Refresher
1.1 Notations
\(\Omega\): A sample space, a set of possible outcomes of a random experiment.
\(X\): A random variable, a function from the sample space to the real numbers: \(X: \Omega \to \R\).
Stochastic Process
A stochastic process is a family of random variables, \(\{X(t): t\in T\},\) where \(t\) usually denotes time. That is, at every time \(t\) in the set \(T\), a random number \(X(t)\) is observed.
- Discrete-time process: \(T=\{0,1,2,3\}\), the discrete process is \(\{X(0), X(1), X(2), \dots\}\)
- Continuous-time process: \(T=[0, \infty]\) or \(T=[0, K]\) for some \(K\).
The state space, \(S\), is the set of real values that \(X(t)\) can take.
You can think of “conditioning” as “changing the sample space.”
From unconditional to conditional
\[ \P (B) = \P(B\mid \Omega) \]
\(\Omega\) denotes the sample space, \(\P (B) = \P(B\mid \Omega)\) just means that we are looking for the probability of the event \(B\), out of all possible outcomes in the set \(\Omega.\)
Partition Theorem
\[ \P(A) = \sum_{i=1}^m \P(A\cap B_i) = \sum_{i=1}^m \P(A\mid B_i) \P(B_i) \]
where \(B_i, i=1,\dots,m,\) are a partition of \(\Omega.\) The intuition behind the Partition Theorem is that the whole is the sum of its parts.
A partition of \(\Omega\) is a collection of mutually exclusive events whose union is \(\Omega.\)
That is, sets \(B_1, B_2, \dots, B_m\) form a partition of \(\Omega\) if
\[ \begin{split} B_i \cap B_j &= \emptyset \;\text{ for all $i, j$ with $i\ne j,$} \\ \text{and } \bigcup_{i=1}^m B_i &= B_1 \cup B_2 \cup \dots \cup B_m = \Omega. \end{split} \]
Bayes’ Theorem
Bayes’ Theorem allows us to invert a conditional statement, i.e., the express \(\P(B\mid A)\) in terms of \(\P(A\mid B).\)
For any events \(A\) and \(B\):
\[ \P(B\mid A) = \frac{\P(A\cap B)}{\P(A)} = \frac{\P(A\mid B)\P(B)}{\P(A)} \]
Generalized Bayes’ Theorem
For any partition member \(B_j\),
\[ \P(B_j\mid A) = \frac{\P(A\mid B_j)\P(B_j)}{\P(A)} = \frac{\P(A\mid B_j)\P(B_j)}{\sum_{i=1}^m\P(A\mid B_i)\P(B_i)} \]