Unrestricted model M₁
Section of Bayesian hypotheses and model comparison, part of Bayesian statistics
Unrestricted model M₁
| Type | Section |
| Taught in | Bayesian hypotheses and model comparison |
| Part | Bayesian statistics |
Under
{\boldsymbol{p}}\mid M_1\sim\operatorname{Dirichlet}(1,1,1,1,1,1),
the prior density is
\frac{\Gamma(6)}{\Gamma(1)^6}=120.
The fact that a uniform density equals 120 rather than 1 reflects the volume of the five-dimensional simplex.
Let C=501!/\prod_{k=1}^6x_k! be the multinomial coefficient. Then \begin{aligned} p({\boldsymbol{x}}\mid M_1) &=\int C\prod_{k=1}^6p_k^{x_k}\,120\,\mathrm{d}{\boldsymbol{p}}\\ &=120C\,B(x_1+1,\ldots,x_6+1)\\ &=120\frac{501!}{\prod_kx_k!} \frac{\prod_k\Gamma(x_k+1)}{\Gamma(501+6)}\\ &=120\frac{501!}{\prod_kx_k!} \frac{\prod_kx_k!}{\Gamma(507)}\\ &=\frac{501!\Gamma(6)}{\Gamma(507)}\\ &=\binom{506}{5}^{-1}\\ &=3.690091\times10^{-12}. \end{aligned} The cell factorials cancel, so this prior predictive probability is the same for every six-cell count vector with total 501.
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See also
Posterior hypothesis probability · Bayes action for two hypotheses · Posterior-odds factorisation · Marginal likelihood and Bayes factor · A hypothesis region is not a separate model · Probability statements and decisions · Poisson imperfect-fruit example · Bayes factors and posterior odds