The following diagram summarizes conjugate prior relationships for a number of common sampling distributions.

Arrows point from a sampling distribution to its conjugate prior distribution. The symbol near the arrow indicates which parameter the prior is unknown.

*These relationships depends critically on choice of parameterization*, some of which are uncommon. This page uses the parameterizations that make the relationships simplest to state, not necessarily the most common parameterizations. See footnotes below.

**Click on a distribution** to see its parameterization. **Click on an arrow** to see posterior parameters.

See this page for more diagrams on this site including diagrams for probability and statistics, analysis, topology, and category theory.

## Parameterizations

Let C(n, k) denote the

binomial coefficient(n, k).

The **geometric** distribution has only one parameter, p, and has PMF f(x) = p (1-p)^{x}.

The **binomial** distribution with parameters n and p has PMF f(x) = C(n, x) p^{x}(1-p)^{n-x}.

The **negative binomial** distribution with parameters r and p has PMF f(x) = C(r + x – 1, x) p^{r}(1-p)^{x}.

The **Bernoulli** distribution has probability of success p.

The **beta** distribution has PDF f(p) = Γ(α + β) p^{α-1}(1-p)^{β-1} / (Γ(α) Γ(β)).

The **exponential** distribution parameterized in terms of the rate λ has PDF f(x) = λ exp(-λ x).

The **gamma** distribution parameterized in terms of the rate has PDF f(x) = β^{α} x^{α-1}exp(-β x) / Γ(α).

The **Poisson** distribution has one parameter λ and PMF f(x) = exp(-λ) λ^{x}/ x!.

The **normal** distribution parameterized in terms of precision τ (τ = 1/σ^{2})

has PDF f(x) = (τ/2π)^{1/2} exp( -τ(x – μ)^{2}/2 ).

The **lognormal** distribution parameterized in terms of precision τ has PDF f(x) = (τ/2π)^{1/2} exp( -τ(log(x) – μ)^{2}/2 ) / x.

## Posterior parameters

For each sampling distribution, assume we have data x_{1}, x_{2}, …, x_{n}.

If the sampling distribution for x is **binomial**(m, p) with m known, and the prior distribution is **beta**(α, β), the posterior distribution for p is **beta**(α + Σx_{i}, β + mn – Σx_{i}). The **Bernoulli** is the special case of the binomial with m = 1.

If the sampling distribution for x is **negative binomial**(r, p) with r known, and the prior distribution is **beta**(α, β), the posterior distribution for p is **beta**(α + nr, β + Σx_{i}). The **geometric** is the special case of the negative binomial with r = 1.

If the sampling distribution for x is **gamma**(α, β) with α known, and the prior distribution on β is gamma(α_{0}, β_{0}),

the posterior distribution for β is **gamma**(α_{0} + n, β_{0} + Σx_{i}). The **exponential** is a special case of the gamma with α = 1.

If the sampling distribution for x is **Poisson**(λ), and the prior distribution on λ is **gamma**(α_{0}, β_{0}), the posterior

on λ is **gamma**(α_{0} + Σx_{i}, β_{0} + n).

If the sampling distribution for x is **normal**(μ, τ) with τ known, and the prior distribution on μ is

**normal**(μ_{0}, τ_{0}), the posterior distribution on μ is **normal**((μ_{0} τ_{0} + τ Σx_{i})/(τ_{0} + nτ), τ_{0} + nτ).

If the sampling distribution for x is **normal**(μ, τ) with μ known, and the prior distribution on τ is

**gamma**(α, β), the posterior distribution on τ is **gamma**(α + n/2, (n-1)S^{2}) where S^{2} is the sample variance.

If the sampling distribution for x is **lognormal**(μ, τ) with τ known, and the prior distribution on μ is

**normal**(μ_{0}, τ_{0}), the posterior distribution on μ is **normal**((μ_{0} τ_{0} + τ Πx_{i})/(τ_{0} + nτ), τ_{0} + nτ).

If the sampling distribution for x is **lognormal**(μ, τ) with μ known, and the prior distribution on τ is

**gamma**(α, β), the posterior distribution on τ is **gamma**(α + n/2, (n-1)S^{2}) where S^{2} is the sample variance.

## References

A compendium of conjugate priors by Daniel Fink.

See also Wikipedia’s article on conjugate priors.