## Probability Definitions Below you will find: - `P_init`: The initial probabilities of the weather state, whether sunny (0), rainy (1) or foggy (2) - `P_transition`: The transition probabilities or the probability that one weather state one day transitions to another weather state - `P_emission`: The 'emission' probabilities or the probabilities that an observation is made, conditioned on the underlying state.

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### DO NOT CHANGE ###

import numpy as np

# prior probability on weather states # P(sunny) = 0.5 P(rainy) = 0.25 P(foggy) = 0.25

P_init = np.array([0.5, 0.25, 0.25])

# states (0=sunny, 1=rainy, 2=foggy)

S = [0, 1, 2]

# transition probabilities # tomorrrow # today sunny rainy foggy # sunny 0.8 0.05 0.15 # rainy 0.2 0.6 0.2 # foggy 0.2 0.3 0.5

P_transition = np.array([ [0.8, 0.05, 0.15], [0.2, 0.6, 0.2], [0.2, 0.3, 0.5] ])

# conditional probabilities of evidence (observations) given weather # sunny rainy foggy # P(weather | umbrella=no) 0.9 0.2 0.7 # P(weather | umbrella=yes) 0.1 0.8 0.3

P_emission = np.array([ [0.9, 0.2, 0.7], [0.1, 0.8, 0.3] ])

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### [Question - State Transitions] If today is sunny what is the most likely forecast for the next two days if you have no umbrella observations to work with?

### [Question - Emission probabilities ] If on the first day you see no umbrella, what is the probability that it is rainy, foggy or sunny (Hint: remember the importance of overall probability of each state)? What is the most likely weather (hidden state)?