5. Conjugate Families

One way to avoid needing to calculate the normalizing constant is to make use of conjugate pairs. These are likelihood-prior pairs where the posterior will be the same family as the prior.

Conjugate table

From Vidakovic [2017] p. 341. This is not an exhaustive list but it has some of the most common conjugate pairs.

Likelihood	Prior	Posterior
\(X_i \mid \theta \sim \mathcal{N} (\theta, \sigma^2)\)	\(\theta \sim\mathcal{N}(\mu, \tau^2)\)	\(\theta\mid X\sim \mathcal{N}\left(\frac{\tau^2}{\tau^2 +\sigma^2/n}\bar{X} + \frac{\sigma^2/n}{\tau^2 + \sigma^2/n}\mu, \frac{\tau^2\sigma^2/n}{\tau^2+ \sigma^2/n}\right)\)
\(X_i\mid \theta \sim Bin(m,\theta)\)	\(\theta \sim Be(\alpha,\beta)\)	\(\theta\mid X \sim Be\left(\alpha + \sum_{i=1}^{n}X_i,\beta + mn - \sum_{i=1}^{n}X_i\right)\)
\(X_i\mid \theta \sim Poi(\theta)\)	\(\theta \sim Ga(\alpha,\beta)\)	\(\theta\mid X \sim Ga\left(\alpha+\sum_{i=1}^{n}X_i,\beta+n\right)\)
\(X_i\mid \theta \sim \mathcal{NB}(m,\theta)\)	\(\theta \sim Be(\alpha,\beta)\)	\(\theta\mid X \sim Be\left(\alpha+mn,\beta+\sum_{i=1}^{n}X_i\right)\)
\(X_i\mid \theta \sim Ga\left(1/2,1/2\theta\right)\)	\(\theta \sim IG(\alpha, \beta)\)	\(\theta \mid X \sim IG\left(\alpha + n/2, \beta + \frac{1}{2}\sum_{i=1}^{n}X_i\right)\)
\(X_i\mid \theta \sim U(0,\theta)\)	\(\theta \sim Pa(\theta_0,\alpha)\)	\(\theta\mid X \sim Pa\left(\max\{\theta_0,X_1,...,X_n\},\alpha+n\right)\)
\(X_i\mid \theta \sim \mathcal{N}(\mu,\theta)\)	\(\theta \sim IG(\alpha, \beta)\)	\(\theta\mid X \sim IG\left(\alpha+n/2,\beta+\frac{1}{2}\sum_{i=1}^{n}(X_i - \mu)^2\right)\)
\(X_i\mid \theta \sim Ga(\nu,\theta)\)	\(\theta \sim Ga(\alpha,\beta)\)	\(\theta \mid X \sim Ga\left(\alpha +n\nu, \beta + \sum_{i=1}^{n}X_i\right)\)
\(X_i\mid \theta \sim Pa(c,\theta)\)	\(\theta \sim Ga(\alpha,\beta)\)	\(\theta \mid X \sim Ga\left(\alpha + n, \beta + \sum_{i=1}^{n}\log\left(X_i/c\right)\right)\)