ExponentialClassFamily: support for quasi-exponential families

[LeonidElkin]

ExponentialClassFamily: support for quasi-exponential families

Task

Design how ExponentialClassFamily should handle “quasi-exponential” families, where one or more parameters explicitly define the support, so that the support depends on the parameters.

These are families that are not strictly exponential-class under the usual definition (support independent of θ), but are still often treated using similar machinery.

Requirements

1. Support-dependent parameters

Extend the design so that it is possible to distinguish:

• structural (natural) parameters that enter the exponential part via η(θ) and T(x),
• support parameters that only affect the support (domain of x) but not the exponential form.

Examples:

• uniform on [0, θ],
• shifted exponential with support [x0, ∞).

2. API design

Propose and implement a simple mechanism in ExponentialClassFamily to:

• store which parameters define the support,
• ensure that:
  • T(x) and η(θ) refer only to the structural (exponential) part,
  • support conditions are checked during density evaluation.

Possible approaches:

• a dedicated field that describes support parameters vs natural parameters,
• an internal split of theta into (theta_exponential, theta_support).

3. Limitations and flags

Document and encode possible limitations:

• for quasi-exponential families, generic conjugate prior and posterior predictive formulas may not hold as-is;
• some methods (e.g. posterior_predictive) may:
  • either be disabled,
  • or explicitly documented as “not supported” unless additional assumptions are met.

Add a simple flag or property such as:

• is_strict_exponential_class vs is_quasi_exponential_class,

so that higher-level code can make decisions based on this.