Exchangeability
Definition taught in Hierarchical Bayes, part of Bayesian statistics
Exchangeability
| Type | Definition |
| Taught in | Hierarchical Bayes |
| Part | Bayesian statistics |
| Worked examples | 2 |
| Questions using it | 1 |
The parameters \theta_1,\ldots,\theta_J are exchangeable if their joint density is invariant to every permutation \pi:
p(\theta_1,\ldots,\theta_J) =p(\theta_{\pi(1)},\ldots,\theta_{\pi(J)}).
Conditional iid sampling from a common Beta distribution implies exchangeability:
p({\boldsymbol{\theta}}\mid\alpha,\beta) =\prod_{j=1}^Jp(\theta_j\mid\alpha,\beta).
The scientific claim is that study labels do not encode systematic differences in strain, laboratory, protocol, calendar time, or tumour definition. The sample sizes n_j may differ; exchangeability requires the latent rates not to be systematically related to those sizes.
Examples
Rat-tumour hierarchy
Worked example from Hierarchical Bayes
Tumour rates in female F344 rats supply a second hierarchical example. In a current control study, y=4 of n=14 rats developed one or more tumours over two years. A one-study model is
Y\mid\theta\sim\operatorname{Bin}(14,\theta), \qquad \theta\sim\operatorname{Beta}(\alpha,\beta),
giving
\theta\mid Y=4 \sim\operatorname{Beta}(\alpha+4,\beta+10).
The question is how to choose (\alpha,\beta) when 70 earlier control studies are available.
Let study j contribute y_j tumour-bearing rats from n_j animals. The hierarchical model is Y_j\mid\theta_j,n_j\sim\operatorname{Bin}(n_j,\theta_j), \qquad \theta_j\mid\alpha,\beta \stackrel{\mathrm{iid}}{\sim}\operatorname{Beta}(\alpha,\beta), \quad j=1,\ldots,J. Each study has its own rate, but all study rates are drawn from a common population distribution.
C9. Pooling
Question page: C9. Pooling
How does partial pooling in a hierarchical model differ from complete pooling and no pooling?
Definitions/notation. Complete pooling uses one common group parameter, no pooling estimates each group separately, and partial pooling connects distinct group parameters through a shared population distribution. This hierarchy allows information to be shared, typically shrinking less precisely estimated group effects more strongly toward the population centre.
Used in
- Taught in Hierarchical Bayes, where the derivation and examples are.
- Also used in Categorical responses · Marginal models, and two responses at once
- Questions that use it (1) — C39 (answer)
- All 9 questions on Hierarchical Bayes
See also
Joint probability density · Beta distribution · Hierarchical Bayesian model · Hyperparameter · Normal hierarchical shrinkage · Empirical Bayes · Study-rate full conditional · Marginal hyperposterior