Natural parameter
Topic of Introduction to generalised linear models, part of Generalised linear models · Also in Exponential families and conjugacy
Natural parameter
| Type | Topic |
| Taught in | Introduction to generalised linear models · Exponential families and conjugacy |
| Part | Generalised linear models · Bayesian statistics |
| Questions using it | 3 |
The parameter that multiplies the data, or the sufficient statistic, in the exponent of an exponential-family density. For the Poisson it is \log\lambda, for the Bernoulli the log-odds \log\{p/(1-p)\}. A canonical link sets it equal to the linear predictor.
In Introduction to generalised linear models
Two reductions have names. If b(\theta)=\theta, the density is in canonical form, and \theta is already the natural parameter. If a(y)=y, the family is a linear exponential family, and b(\theta) is its natural parameter. In the linear canonical case both happen at once, and Equation (6) in Introduction to generalised linear models simplifies to f(y;\theta)=\exp\left\{\frac{y\theta+c(\theta)}{e(\phi)}+d(y;\phi)\right\}. You can always rename the parameter, or replace Y by a(Y), to reach this form. Definitions that start from Equation (7) in Introduction to generalised linear models have already done that renaming. We will use Equation (6) in Introduction to generalised linear models when the natural parameter should appear by itself, as \log\lambda does for the Poisson, and Equation (7) in Introduction to generalised linear models once that parameter has been named \theta.
In Exponential families and conjugacy
Completeness must not be claimed merely because a model has been written in exponential-family notation. For a full-rank exponential family whose natural parameter space contains an open set of the appropriate dimension, the natural sufficient statistic is complete under standard integrability conditions. Curved or restricted subfamilies need not inherit this property.
Used in
- Taught in Introduction to generalised linear models · Exponential families and conjugacy
- Also used in Conjugate analysis for normal models · Exponential families and conjugacy · Smooth means
- Questions that use it (3) — Q57 · C4 · C17 (answer)
- All 9 questions on Introduction to generalised linear models
See also
Exponential family · Natural exponential family · Canonical link · Sufficient statistic · Logit link · Linear predictor · Log-partition moment identities · Mean and variance of an exponential family