Log-partition moment identities
Proposition taught in Exponential families and conjugacy, part of Bayesian statistics
Log-partition moment identities
| Type | Proposition |
| Taught in | Exponential families and conjugacy |
| Part | Bayesian statistics |
Whenever c(\theta) can be used as a local parameter and differentiation under the normalising integral is valid,
\nabla_c\log a(\theta) =\mathbb{E}_\theta\{T(X)\}, \qquad \nabla_c^2\log a(\theta) =\operatorname{Cov}_\theta\{T(X)\}.
Here derivatives are taken with respect to c=c(\theta), so a(\theta) is regarded as a function of c through the chosen parametrisation. Consequently \log a(\theta) is convex as a function of c, and the one-observation Fisher information in the natural parameter is \nabla_c^2\log a(\theta).
Used in
- Taught in Exponential families and conjugacy, where the derivation and examples are.
- All 18 questions on Exponential families and conjugacy
See also
Natural exponential family · Statistic and estimator · Sufficient statistic · Fisher–Neyman factorisation · Minimal sufficiency · Complete statistic · Ancillary statistic · Basu’s theorem