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Bayesian Probability Calculation

This integration uses Bayesian probability to determine the likelihood of an area being occupied based on the current states of configured sensors.

Core Concept

Instead of a simple binary "motion detected = occupied" logic, this integration calculates a probability score (0% to 100%) representing the confidence that the area is occupied.

It relies on Bayes' Theorem, which mathematically describes how to update a belief (the probability of occupancy) given new evidence (sensor states).

Calculation Steps

  1. Identify Active Sensors: The system first checks which of the configured sensors are currently in an "active" state (e.g., motion sensor on, media player playing, door sensor open, etc.). The definition of "active" is based on the integration's configuration and learned patterns (see Prior Learning). Sensors that are unavailable or not in an active state are ignored for the current calculation cycle.
  2. Retrieve Probabilities for Each Active Sensor: For each active sensor, the system retrieves its associated probabilities, which are learned over time (or use defaults):
    • Likelihood P(Active | Occupied): The probability the sensor is active given the area is occupied (prob_given_true).
    • Likelihood P(Active | Not Occupied): The probability the sensor is active given the area is not occupied (prob_given_false).
    • Prior P(Occupied): The baseline probability of the area being occupied, typically derived from the primary sensor's history or using a default. Note that the specific prior used in the individual sensor calculation might be entity-specific, type-specific, or the overall prior depending on the implementation details, but it represents the belief before considering this specific sensor's evidence.
  3. Calculate Unweighted Posterior Probability (per Sensor): Using the probabilities above and Bayes' theorem, the system calculates the posterior probability of occupancy suggested by that single active sensor, before considering its weight. This answers: "Based only on this sensor being active and our prior belief, what is the new probability of occupancy?"
  4. Apply Weight: The unweighted posterior probability from step 3 is then multiplied by the configured weight for its sensor type (e.g., weight_motion, weight_media). This scales the influence of each sensor type, resulting in a weighted_probability for that sensor's evidence.
  5. Combine Weighted Probabilities (Complementary Method): The weighted_probability values from all active sensors are combined to get a single overall probability. This integration uses a complementary probability approach, assuming independence between the weighted sensor evidence:
    • For each active sensor, it calculates the probability that the area is not occupied, given that sensor's weighted evidence: P(Not Occupied | Sensor Active) = 1 - weighted_probability.
    • It multiplies these individual "not occupied" probabilities together: Combined P(Not Occupied) = Product(1 - weighted_probability) for all active sensors.
    • The final result is the complement of this combined "not occupied" probability: Final P(Occupied) = 1 - Combined P(Not Occupied). This yields the combined probability of the area being occupied, considering all active sensor evidence together.
  6. Apply Bounds: The final calculated probability is clamped between minimum (e.g., 1%) and maximum (e.g., 99%) values defined in the constants (MIN_PROBABILITY, MAX_PROBABILITY) to prevent extreme values.

Output

The result of this calculation (step 6) is the value shown by the Occupancy Probability sensor.

The Occupancy Status binary sensor compares this probability to the configured Occupancy Threshold to determine its on or off state.

The Maths

Bayesian Update (Single Sensor)

The integration uses Bayes' theorem within the bayesian_update function to calculate the unweighted posterior probability of Occupancy (O) given that a single sensor is Active (A):

\[P(O|A) = \frac{P(A|O) \times P(O)}{P(A)}\]

The denominator, \(P(A)\) (the overall probability of the sensor being active), is expanded using the law of total probability:

\[P(A) = P(A|O) P(O) + P(A|\neg O) P(\neg O)\]

Where: - \(P(O|A)\) is the posterior probability of occupancy given the sensor is active (the result of bayesian_update). - \(P(A|O)\) is the likelihood the sensor is active given occupancy (prob_given_true). - \(P(O)\) is the prior probability of occupancy (prior). - \(P(A|\neg O)\) is the likelihood the sensor is active given the area is not occupied (prob_given_false). - \(P(\neg O)\) is the prior probability of the area not being occupied (\(1 - P(O)\)).

Substituting the expanded \(P(A)\) into Bayes' theorem gives the formula used in the code: [ P(O|A) = \frac{P(A|O) P(O)}{P(A|O) P(O) + P(A|\neg O) (1 - P(O))} ]

Weighting

The result \(P(O|A)\) is then weighted: [ P_{weighted} = P(O|A) \times W_{type}] Where \(W_{type}\) is the configured weight for the sensor's type.

Combining Probabilities (Complementary Method)

To combine the weighted probabilities (\(P_{weighted, i}\)) from multiple active sensors (\(i = 1 \text{ to } n\)), the complementary method is used:

  1. Calculate the probability of not being occupied suggested by each sensor: \(1 - P_{weighted, i}\)
  2. Multiply these probabilities together: \(\prod_{i=1}^{n} (1 - P_{weighted, i})\)
  3. Subtract the result from 1 to get the final combined probability: [ P_{final} = 1 - \prod_{i=1}^{n} (1 - P_{weighted, i}) ]

Decay Calculation

When Probability Decay is active, the probability \(P_{decayed}\) at a given time is calculated based on the probability when decay started (\(P_{start}\)) and the time elapsed since decay began (\(t_{elapsed}\)).

The core decay factor (\(f_{decay}\)) is calculated using an exponential function:

\[ f_{decay} = e^{ -\lambda \times \frac{t_{elapsed}}{T_{window}} } \]

Where: - \(e\) is the base of the natural logarithm (Euler's number). - \(\lambda\) (DECAY_LAMBDA) is a decay constant that influences the steepness of the decay curve. - \(t_{elapsed}\) is the time in seconds since the decay process started. - \(T_{window}\) (decay_window) is the configured time window in seconds over which the decay primarily occurs.

The decayed probability is then calculated by applying this factor to the starting probability, ensuring it stays within the defined bounds: [ P_{decayed} = \max( P_{min}, \min( P_{start} \times f_{decay}, P_{max} ) ) ]

Where: - \(P_{start}\) is the probability value recorded just before decay began. - \(P_{min}\) (MIN_PROBABILITY) is the minimum allowed probability (e.g., 0.01). - \(P_{max}\) (MAX_PROBABILITY) is the maximum allowed probability (e.g., 0.99).

This ensures the probability decays exponentially but is clamped within reasonable limits.