People often decide their outdoor activities according to the weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by W = (W₁, W2, W3), is unknown. His activities option, denoted by V = (V₁, V2, V3), is decided by the weather conditions. The initial state of the weather is = [0.3, 0.4,0.3]. Given the Hidden Markov model 0 = (A, B,T), calculate the probability that you observe a specific activity sequence O = [v2, v2, V1, V3] of your friend over the past four days, where A₁ is the transition probability from w; to wŋ, Bi̟j is the probability of observing the activity v¡ under the state wą.
People often decide their outdoor activities according to the weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by W = (W₁, W2, W3), is unknown. His activities option, denoted by V = (V₁, V2, V3), is decided by the weather conditions. The initial state of the weather is = [0.3, 0.4,0.3]. Given the Hidden Markov model 0 = (A, B,T), calculate the probability that you observe a specific activity sequence O = [v2, v2, V1, V3] of your friend over the past four days, where A₁ is the transition probability from w; to wŋ, Bi̟j is the probability of observing the activity v¡ under the state wą.
A First Course in Probability (10th Edition)
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Author:Sheldon Ross
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![## Understanding Hidden Markov Models in Decision Making
People often decide their outdoor activities according to weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by \( W = (\omega_1, \omega_2, \omega_3) \), are unknown. His activity choices, denoted by \( V = (v_1, v_2, v_3) \), depend on these weather conditions.
### Weather and Activities
The initial state of the weather is represented by the probability distribution \( \pi = [0.3, 0.4, 0.3] \). Given the Hidden Markov Model (HMM) \( \theta = (A, B, \pi) \), you need to calculate the probability of observing a specific activity sequence \( O = [v_2, v_2, v_1, v_3] \) over the past four days. In this context:
- \( A_{i,j} \) is the transition probability from \( \omega_i \) to \( \omega_j \).
- \( B_{i,j} \) is the probability of observing activity \( v_j \) under the weather state \( \omega_i \).
### Transition and Observation Matrices
**Transition Matrix (\( A \))**:
\[
A = \begin{bmatrix}
0.3 & 0.2 & 0.5 \\
0.1 & 0.4 & 0.5 \\
0.2 & 0.5 & 0.3
\end{bmatrix}
\]
**Observation Matrix (\( B \))**:
\[
B = \begin{bmatrix}
0.4 & 0.5 & 0.1 \\
0.2 & 0.4 & 0.4 \\
0.3 & 0.1 & 0.6
\end{bmatrix}
\]
These matrices allow us to model and predict the probability of the observed sequence of activities based on the hidden weather conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5d860cd5-2859-4da0-be1a-b0f5f49f917c%2F279fbecf-629c-4ca7-97d3-df66d83ddde4%2Fr7xm5pn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Understanding Hidden Markov Models in Decision Making
People often decide their outdoor activities according to weather conditions. Suppose you have a friend in London, where the weather conditions, denoted by \( W = (\omega_1, \omega_2, \omega_3) \), are unknown. His activity choices, denoted by \( V = (v_1, v_2, v_3) \), depend on these weather conditions.
### Weather and Activities
The initial state of the weather is represented by the probability distribution \( \pi = [0.3, 0.4, 0.3] \). Given the Hidden Markov Model (HMM) \( \theta = (A, B, \pi) \), you need to calculate the probability of observing a specific activity sequence \( O = [v_2, v_2, v_1, v_3] \) over the past four days. In this context:
- \( A_{i,j} \) is the transition probability from \( \omega_i \) to \( \omega_j \).
- \( B_{i,j} \) is the probability of observing activity \( v_j \) under the weather state \( \omega_i \).
### Transition and Observation Matrices
**Transition Matrix (\( A \))**:
\[
A = \begin{bmatrix}
0.3 & 0.2 & 0.5 \\
0.1 & 0.4 & 0.5 \\
0.2 & 0.5 & 0.3
\end{bmatrix}
\]
**Observation Matrix (\( B \))**:
\[
B = \begin{bmatrix}
0.4 & 0.5 & 0.1 \\
0.2 & 0.4 & 0.4 \\
0.3 & 0.1 & 0.6
\end{bmatrix}
\]
These matrices allow us to model and predict the probability of the observed sequence of activities based on the hidden weather conditions.
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