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Poisson Distribution

The Poisson distribution is used for modeling count data where events occur independently within a fixed period of time. The parameter lambda, representing the expected number of events, controls the rate of occurrence. This distribution is commonly applied in fields such as telecommunications and traffic engineering.

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Parameterization

The Poisson distribution for a random vector \(\pmb{y} = (y_1, y_2, \dots, y_n)\) is defined by the probability mass function:

\[f(y_i \mid \lambda_i) = \frac{\lambda_i^{y_i} e^{-\lambda_i}}{y_i!}, \quad y_i = 0, 1, 2, \dots, \quad i = 1, 2, \dots, n\]

where \(\pmb{\lambda} = (\lambda_1, \lambda_2, \dots, \lambda_n)\) is a vector of rate parameters, with \(\lambda_i > 0\) representing the mean number of events for each observation \(y_i\).
Assume \(n\) = 1. To illustrate how the Poisson PMF changes for different values of the rate parameter \(\lambda\), Figure 1 displays two subplots. The left subplot shows \(\lambda\) in \(\{1,2,3\}\), while the right subplot covers larger rates (\(\lambda \in \{5,10,15\}\)).

Poisson PMF with different rate parameters \(\lambda\). Left plot: \(\lambda\in\{1,2,3\}\). Right plot: \(\lambda\in\{5,10,15\}\). The x-axis indicates the number of events \(k\), and the y-axis gives the PMF value.

For smaller values of \(\lambda\), the distribution is more concentrated around lower counts. As \(\lambda\) grows, the distribution shifts rightward and spreads out, reflecting the higher expected number of events.

The canonical link function for the Poisson distribution is the log link:

\[\eta_i = \log(\lambda_i), \quad i = 1, 2, \dots, n\]

The relationship between the linear predictor \(\pmb{\eta} = (\eta_1, \eta_2, \dots, \eta_n)\) and the mean \(\pmb{\lambda}\) is given by:

\[\lambda_i = \exp(\eta_i), \quad i = 1, 2, \dots, n\]

or in vector form:

\[\pmb{\lambda} = \exp(\pmb{\eta})\]

Hyperparameters

The Poisson likelihood has no hyperparameters. The rate parameter \(\lambda\) is fully determined by the linear predictor \(\eta\) through the log link function.

Validation Rules

pyINLA enforces several validation rules for Poisson models to ensure correct specification:

No Hyper Configuration

The Poisson likelihood has no hyperparameters. Do NOT provide control['family']['hyper'] configuration.

Exposure (E)

Key: E

When providing the E (exposure) argument:

  • All values must be strictly positive (> 0)

  • Length must match the number of observations

# With exposure
result = pyinla(model={'response': 'y', 'fixed': ['1', 'x']}, data=df, family="poisson", E=df["exposure"])

Offset Alternative

Key: offset

Instead of E, you can use offset = log(E). When using offset:

  • Values must not contain Inf; NaN is allowed and treated as 0 (no offset for that observation)

  • Length must match the number of observations

import numpy as np

# Using offset instead of E
result = pyinla(model={'response': 'y', 'fixed': ['1', 'x']}, data=df, family="poisson",
                offset=np.log(df["exposure"]))

Response Values

Response variable \(\pmb{y}\) must be non-negative integers (counts: 0, 1, 2, ...).

Allowed Link Functions

Key: control['family']['link']

These link functions are supported:

  • log (default)

  • logoffset

  • quantile

Specification

  • family="poisson"

  • Required arguments:

    • \(\pmb y\) (integer-valued counts)

  • Optional arguments:

    • E: exposure vector (positive values)

    • offset: log-exposure offset

Worked examples