The Beta distribution is the two-category Dirichlet
Proposition taught in Categorical data and the Dirichlet distribution, part of Bayesian statistics
The Beta distribution is the two-category Dirichlet
| Type | Proposition |
| Taught in | Categorical data and the Dirichlet distribution |
| Part | Bayesian statistics |
When K=2, write p_1=p and p_2=1-p. Restricting the Dirichlet density to the one-dimensional simplex gives
p(p\mid\alpha_1,\alpha_2) =\frac{\Gamma(\alpha_1+\alpha_2)} {\Gamma(\alpha_1)\Gamma(\alpha_2)} p^{\alpha_1-1}(1-p)^{\alpha_2-1},
which is exactly \operatorname{Beta}(\alpha_1,\alpha_2). The Dirichlet is therefore a genuine extension of the Beta family rather than merely a similar-looking density.
Used in
- Taught in Categorical data and the Dirichlet distribution, where the derivation and examples are.
- All 7 questions on Categorical data and the Dirichlet distribution
See also
Dirichlet distribution · Categorical and multinomial models · Multinomial central limit theorem · Dirichlet moments · Interior Dirichlet mode · Dirichlet–Multinomial conjugacy