What is the number of parameters in a transition matrix for an HMM with 3 states?Question 8Answera.9b.3c.12d.6

To determine the number of parameters in a transition matrix for a Hidden Markov Model (HMM) with 3 states, we can follow these steps:

1. Break Down the Problem

The transition matrix for an HMM describes the probabilities of moving from one state to another. For $N$ states, we need to define the parameters for each possible transition.

2. Relevant Concepts

The transition matrix $A$ for $N$ states is defined as: $A = \begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1N} \\ a_{21} & a_{22} & \ldots & a_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ a_{N1} & a_{N2} & \ldots & a_{NN} \end{pmatrix}$ Where $a_{ij}$ is the probability of transitioning from state $i$ to state $j$.

3. Analysis and Detail

For 3 states, the transition matrix will have $3 \times 3 = 9$ elements. However, each row must sum to 1 (the total probability must equal 1). Thus, for $N$ states, the number of independent parameters (degrees of freedom) is given by: $\text{Number of Parameters} = N^2 - N$ For 3 states ($N = 3$): $\text{Number of Parameters} = 3^2 - 3 = 9 - 3 = 6$

4. Verify and Summarize

We calculated the independent parameters to be 6, which corresponds to the 6 entries in the transition matrix after accounting for the constraints that the sum of each row equals 1.

Final Answer

The number of parameters in the transition matrix for an HMM with 3 states is 6. Thus, the correct answer is d. 6.