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Probability Vector 📂Mathematical Statistics

Probability Vector

Definition

A vector that satisfies the following condition p=[p1pn]T\mathbf{p} = \begin{bmatrix}p_{1} & \cdots & p_{n} \end{bmatrix}^{\mathsf{T}} is called a probability vector.

0pi1(1in)andi=1npi=1 0 \le p_{i} \le 1 \quad (1 \le i \le n)\quad \text{and} \quad \sum_{i=1}^{n} p_{i} = 1

Explanation

A probability vector is a vector that represents the probabilities of each state when there are nn states. Conceptually, it is analogous to a probability mass function. If the probability mass function of a discrete random variable XX is pXp_{X}, then the probability vector is as follows.

p=[p1pn]=[pX(1)pX(n)]=[p(1)p(n)] \mathbf{p} = \begin{bmatrix}p_{1} \\ \vdots \\ p_{n} \end{bmatrix} = \begin{bmatrix}p_{X}(1) \\ \vdots \\ p_{X}(n) \end{bmatrix} = \begin{bmatrix}p(1) \\ \vdots \\ p(n) \end{bmatrix}

If the probability that the jj-th state changes to the ii-th state is qij=q(ij)q_{ij} = q(i | j), then the probability vector q(j)=[q1jqnj]T\mathbf{q}^{(j)} = \begin{bmatrix}q_{1j} & \cdots & q_{nj} \end{bmatrix}^{\mathsf{T}} represents the probability that the jj-th state changes to other states. The matrix of these column vectors becomes a transition matrix.

Q=[q(1)q(2)q(n)]=[q11q12q1nq21q22q2nqn1qn2qnn] Q = \begin{bmatrix} \vert & \vert & & \vert \\ \mathbf{q}^{(1)} & \mathbf{q}^{(2)} & \cdots & \mathbf{q}^{(n)} \\ \vert & \vert & & \vert \end{bmatrix} = \begin{bmatrix} q_{11} & q_{12} & & q_{1n} \\ q_{21} & q_{22} & \cdots & q_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ q_{n1} & q_{n2} & \cdots & q_{nn} \end{bmatrix}

If p\mathbf{p} is the probability vector for the current state, then p=Qp\mathbf{p}^{\prime} = Q \mathbf{p} is the probability vector for the next state, given that the probabilities for the current state are provided as p\mathbf{p}.

p=Qp=[jqijpjjq2jpjjqnjpj]=[jq(1j)p(j)jq(2j)p(j)jq(nj)p(j)]=[p(1)p(2)p(n)] \mathbf{p}^{\prime} = Q \mathbf{p} = \begin{bmatrix} \sum\limits_{j} q_{ij} p_{j} \\ \sum\limits_{j} q_{2j} p_{j} \\ \vdots \\[1em] \sum\limits_{j} q_{nj} p_{j} \end{bmatrix} = \begin{bmatrix} \sum\limits_{j} q(1 | j) p(j) \\ \sum\limits_{j} q(2 | j) p(j) \\ \vdots \\[1em] \sum\limits_{j} q(n | j) p(j) \end{bmatrix} = \begin{bmatrix}p^{\prime}(1) \\[1em] p^{\prime}(2) \\[1em] \vdots \\[1em] p^{\prime}(n) \end{bmatrix}

Notation

If we let the probability row vector for the current state be π\pi and the transition matrix be P=[p(1)p(n)]P = \begin{bmatrix} - \mathbf{p}^{(1)} - \\ \vdots \\ - \mathbf{p}^{(n)} - \end{bmatrix}, the following notation is frequently used.

πP=π \pi P = \pi^{\prime}

In this case, Pij=P(ji)P_{ij} = P(j | i), where the element in the ii-th row and the jj-th column represents the probability of changing from the ii-th state to the jj-th state. The reason for using a row vector notation is not entirely clear. A notation utilizing the letter t for transition, such as TT, is also commonly used for the transition matrix.