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Statistics & estimation

Hidden Markov model

the classic 2-state Hidden Markov Model ("a patient is Healthy or has a Fever; a doctor only observes symptoms") used to hand-verify HMM forward/Viterbi implementations against a well-known reference answer.

catalog/qu_hmm_healthy_fever.qu

What it exercises

• hmm(transition, emission, initial) -- validates both matrices are
• row-stochastic
• model.forward(obs) -- total probability of an
• observation sequence
• model.viterbi(obs) -- most likely hidden-state path

How it works

transition = [0.7, 0.3; 0.4, 0.6]
emission   = [0.5, 0.4, 0.1; 0.1, 0.3, 0.6]
initial    = [0.6, 0.4]

model = hmm(transition, emission, initial)

obs = [0, 1, 2]

likelihood = model.forward(obs)
path = model.viterbi(obs)

print("P(observing normal, cold, dizzy) = {likelihood:.5f}")   # hand-computed: 0.03628
print("Most likely state sequence: {path}")                    # hand-computed: [0, 0, 1]
print("  (0 = Healthy, 1 = Fever)")

expected_likelihood = 0.03628
expected_path = [0, 0, 1]
if abs(likelihood - expected_likelihood) < 1e-6 then
    print("forward() matches the hand-computed likelihood.")
else
    print("MISMATCH: forward() disagrees with the hand-computed likelihood!")
end if
if sum(abs(path - expected_path)) < 1e-9 then
    print("viterbi() matches the hand-computed most-likely path.")
else
    print("MISMATCH: viterbi() disagrees with the hand-computed path!")
end if

Results

P(observing normal, cold, dizzy) = 0.03628
Most likely state sequence: [0, 0, 1]
  (0 = Healthy, 1 = Fever)
forward() matches the hand-computed likelihood.
viterbi() matches the hand-computed most-likely path.

The whole program

Runnable as it stands: qu run catalog/qu_hmm_healthy_fever.qu

# qu_hmm_healthy_fever.qu -- the classic 2-state Hidden Markov Model

# ("a patient is Healthy or has a Fever; a doctor only observes symptoms")

# used to hand-verify HMM forward/Viterbi implementations against a

# well-known reference answer.

#

# Constructs exercised (all real, verified-working builtins):

#   hmm(transition, emission, initial)  -- validates both matrices are

#                                           row-stochastic

#   model.forward(obs)                  -- total probability of an

#                                           observation sequence

#   model.viterbi(obs)                  -- most likely hidden-state path


backend auto

#%% States: Healthy=0, Fever=1. Symptoms: normal=0, cold=1, dizzy=2.

transition = [0.7, 0.3; 0.4, 0.6]
emission   = [0.5, 0.4, 0.1; 0.1, 0.3, 0.6]
initial    = [0.6, 0.4]

model = hmm(transition, emission, initial)

#%% Three days of symptoms: normal, then cold, then dizzy.

obs = [0, 1, 2]

likelihood = model.forward(obs)
path = model.viterbi(obs)

print("P(observing normal, cold, dizzy) = {likelihood:.5f}")   # hand-computed: 0.03628

print("Most likely state sequence: {path}")                    # hand-computed: [0, 0, 1]

print("  (0 = Healthy, 1 = Fever)")

#%% Sanity check against the hand-derived reference values.

expected_likelihood = 0.03628
expected_path = [0, 0, 1]
if abs(likelihood - expected_likelihood) < 1e-6 then
    print("forward() matches the hand-computed likelihood.")
else
    print("MISMATCH: forward() disagrees with the hand-computed likelihood!")
end if
if sum(abs(path - expected_path)) < 1e-9 then
    print("viterbi() matches the hand-computed most-likely path.")
else
    print("MISMATCH: viterbi() disagrees with the hand-computed path!")
end if