Beta-Binomial Distribution

Parametrization

The Beta-Binomial distribution for a random vector \(\pmb{y} = (y_1, y_2, \dots, y_n)\) of count observations arises from a hierarchical model. The probability of success \(p_i\) follows a Beta distribution:

\[\pi(p_i) = \frac{1}{B(\alpha, \beta)} p_i^{\alpha-1} (1-p_i)^{\beta-1}, \quad \alpha > 0, \; \beta > 0, \quad i = 1, 2, \dots, n\]

Given \(p_i\), the response \(y_i\) is Binomial:

\[\pi(y_i \mid p_i) = \binom{N_i}{y_i} p_i^{y_i} (1-p_i)^{N_i-y_i}, \quad y_i = 0, 1, \ldots, N_i\]

Marginalizing over \(p_i\), the distribution of \(y_i\) is Beta-Binomial:

\[\pi(y_i) = \binom{N_i}{y_i} \frac{B(y_i + \alpha, N_i - y_i + \beta)}{B(\alpha, \beta)}, \quad i = 1, 2, \dots, n\]

where \(B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\) is the Beta function.

Mean and Variance

The mean and variance of each observation are:

\[\mu_i = N_i \mu_p, \qquad \sigma_i^2 = N_i \mu_p (1 - \mu_p) \left(1 + (N_i - 1) \rho\right), \quad i = 1, 2, \dots, n\]

where:

• \(\mu_p = \frac{\alpha}{\alpha + \beta}\) is the mean probability from the Beta distribution.

• \(\rho = \frac{1}{\alpha + \beta + 1}\) is the overdispersion parameter (\(0 < \rho < 1\)).

The term \((1 + (N_i - 1)\rho)\) inflates the variance beyond the standard Binomial, capturing overdispersion.

Link Function

The mean probability \(\mu_p\) is linked to the linear predictor \(\pmb{\eta} = (\eta_1, \eta_2, \dots, \eta_n)\) using the logit link (default):

\[\mu_p = \frac{\exp(\eta_i)}{1 + \exp(\eta_i)}, \quad i = 1, 2, \dots, n\]

Possible link functions: default, logit, loga, cauchit, probit, cloglog, ccloglog, loglog, robit, sn.

Hyperparameters

The hyperparameter is the overdispersion parameter \(\rho\), which is represented as:

\[\rho = \frac{\exp(\theta)}{1 + \exp(\theta)}\]

where the prior is defined on \(\theta\), ensuring \(\rho \in (0, 1)\).

Key: rho

Default prior specification:

• Prior: gaussian

• Parameters: (mean = 0, precision = 0.4)

• Initial value: 0

Validation Rules

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

Ntrials (Required)

Key: Ntrials

The Ntrials argument is required for betabinomial:

• Must be provided for each observation

• Values must be positive integers

• Length must match the number of observations

# betabinomial requires Ntrials
model = {"response": "y", "fixed": ["1", "x"]}
result = pyinla(model=model, family="betabinomial", data=df, Ntrials=df["n"].to_numpy())

Hyperparameters

Key: control['family']['hyper']

Hyperparameter configuration is allowed for betabinomial. The rho hyperparameter controls the overdispersion.

Each entry in control['family']['hyper'] may contain these keys:

• id - Hyperparameter identifier (rho). Can be omitted for the first (and only) hyperparameter.

• prior - Prior distribution name

• param - Prior parameters (list)

• initial - Initial value on logit scale

• fixed - Whether to fix the hyperparameter (True/False)

# Custom hyperparameter configuration (default values shown)
model = {"response": "y", "fixed": ["1", "x"]}
control = {
    'family': {
        'hyper': [{
            'id': 'rho',
            'prior': 'gaussian',
            'param': [0.0, 0.4],
            'initial': 0,
            'fixed': False
        }]
    }
}
result = pyinla(model=model, family="betabinomial", data=df,
                Ntrials=df["n"].to_numpy(), control=control)

# Fixed overdispersion (not estimated)
control = {
    'family': {
        'hyper': [{
            'id': 'rho',
            'initial': 0.0,  # logit(0.5) = 0, so rho = 0.5
            'fixed': True
        }]
    }
}
result = pyinla(model=model, family="betabinomial", data=df,
                Ntrials=df["n"].to_numpy(), control=control)

Variant

Key: control['family']['variant']

The variant parameter is allowed for betabinomial:

• 0 (default): Standard parameterization

• 1: Alternative parameterization

Response Values

Response variable \(\pmb{y}\) must be non-negative integers with \(0 \leq y_i \leq N_i\) for each observation. pyINLA will raise PyINLAError if any response value is negative, non-integer, or exceeds its corresponding Ntrials value.

Exposure Not Allowed

The E (exposure) argument is not allowed for betabinomial. Use nbinomial or poisson if you need exposure.

Scale Not Allowed

The scale argument is not allowed for betabinomial.

Allowed Link Functions

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

These link functions are supported:

• logit (default)

• loga

• cauchit

• probit

• cloglog

• ccloglog

• loglog

• robit

• sn

# With probit link
model = {"response": "y", "fixed": ["1", "x"]}
result = pyinla(model=model, family="betabinomial", data=df,
                Ntrials=df["n"].to_numpy(), control={"family": {"link": "probit"}})

Specification

• family="betabinomial"

• Required arguments: \(\pmb{y}\) (response) and Ntrials (\(N_i\)).

• Optional: control['family']['variant'] (0 or 1), control['family']['hyper'], control['family']['link'].

• Beta-Binomial Regression