Tracking a State We Cannot Observe Directly

Botie sees an alarm from a pump, but cannot directly see its internal condition. The pump can become faulty between inspections, and the alarm can occasionally be wrong.

There are two separate questions: what condition is plausible now, given the previous condition, and how plausible is this alarm under each current condition? Filtering answers them in that order.

Total weight = 0.076 + 0.216 = 0.292
Fault probability = 0.216 / 0.292 ≈ 0.740
Working probability = 0.076 / 0.292 ≈ 0.260

First allow the pump's condition to change according to the transition model. Then compare an informative alarm with a reading equally likely in both states.

The belief vector gives probabilities over hidden states. The transition matrix describes how a state changes before the next observation. The emission likelihood gives the probability of the observation under each possible current state.

Use one fixed state order throughout. Transition rows must sum to one, but likelihoods for one observed alarm across different states do not need to sum to one.

A likelihood compares the same observation under different explanations. It is not yet a posterior.

With old states as rows and new states as columns, a row belief vector multiplies the transition matrix. This sums all ways to arrive at each new state.

Old belief = [0.8, 0.2]
Transition rows = [[0.9, 0.1], [0.2, 0.8]]
Predicted belief = [0.76, 0.24]

Transposing the matrix without changing the vector convention silently changes the calculation.

Multiply each predicted state probability by that state’s alarm likelihood. Sum these weights to obtain the observation probability, then divide each weight by the sum.

The corrected belief becomes the starting point for the next step. Repeating prediction and correction lets evidence accumulate while still allowing the state to change.

Weights = [0.76 × 0.1, 0.24 × 0.9] = [0.076, 0.216]
Fault posterior = 0.216 / 0.292 ≈ 0.740

A zero total weight means the model assigns no probability to what was observed; normalising cannot fix that.

import numpy as np
belief = np.array([1., 0.])
transition = np.array([[0.75, 0.25], [0.2, 0.8]])
reading_likelihood = np.array([0.2, 0.8])
predicted = belief @ transition
supported = predicted * reading_likelihood
print(predicted.tolist())
print(np.round(supported / supported.sum(), 3).tolist())

[0.75, 0.25]
[0.429, 0.571]

Filtering uses observations available up to the current time. Smoothing uses later observations to revise earlier state estimates. Viterbi decoding seeks a most likely whole state sequence, a different question again.

Our states and likelihoods are supplied. Learning them from data requires additional estimation and checks that the hidden states correspond to something useful.

The workshop performs a filtering step and never uses a future observation.

The pump's state vector stores probabilities for working and faulty. A transition matrix first moves that belief forward in time. Only then does the current alarm provide evidence about the current state.

Multiplying by alarm likelihoods changes the relative support for each state. Normalise after that multiplication so the entries again sum to one. The normalising sum is also the probability assigned to the observed alarm under the predictive belief.

The first report is an alarm; the second is quiet. Both matter, but the second report should not erase the transition model or reset us to the initial belief.

Rows of the transition matrix describe the old state and columns the next state. With a row-vector belief, multiply on the right. Changing orientation silently computes a different operation.

import numpy as np
belief = np.array([.9,.1])
transition = np.array([[.9,.1],[.2,.8]])
alarm = np.array([.1,.8])
for likelihood in [alarm, 1-alarm]:
    predicted = belief@transition
    weight = predicted*likelihood
    belief = weight/weight.sum()
    print(np.round(belief, 4).tolist())

[0.379, 0.621]
[0.7966, 0.2034]

If every state assigns zero likelihood to a report, the normalising sum is zero. Dividing anyway does not produce a valid posterior. The observation contradicts the supplied model or an input convention is wrong.

Detect that case explicitly. With many small probabilities, numerical underflow can create a similar symptom, so stable scaling or log-space calculations become useful. Distinguish a numerical representation failure from a genuinely impossible observation.

Filtering asks what we can infer now using reports available now. Smoothing revisits earlier hidden states after later reports have arrived. A later repair inspection can improve our account of yesterday's pump condition, but it was unavailable to yesterday's decision.

Keep that distinction when evaluating an online detector. Scoring a historical decision using a smoothed belief can give it information from the future and exaggerate how well it could have acted at the time.