Editing Gamma distribution (section)

====Bayesian minimum mean squared error====
With known {{mvar|α}} and unknown {{mvar|θ}}, the posterior density function for theta (using the standard scale-invariant [[prior probability|prior]] for {{mvar|θ}}) is

<math display=block>P(\theta \mid \alpha, x_1, \dots, x_N) \propto \frac 1 \theta \prod_{i=1}^N f(x_i; \alpha, \theta)</math>

Denoting

<math display=block> y \equiv \sum_{i=1}^Nx_i , \qquad P(\theta \mid \alpha, x_1, \dots, x_N) = C(x_i) \theta^{-N \alpha-1} e^{-y/\theta}</math>

where the {{mvar|C}} (integration) constant does not depend on {{mvar|θ}}. The form of the posterior density reveals that {{math|1 / ''θ''}} is gamma-distributed with shape parameter {{math|''Nα'' + 2}} and rate parameter {{mvar|y}}. Integration with respect to {{mvar|θ}} can be carried out using a change of variables to find the integration constant

<math display=block>\int_0^\infty \theta^{-N\alpha - 1 + m} e^{-y/\theta}\, d\theta = \int_0^\infty x^{N\alpha - 1 - m} e^{-xy} \, dx = y^{-(N\alpha - m)} \Gamma(N\alpha - m) \!</math>

The moments can be computed by taking the ratio ({{mvar|m}} by {{math|1=''m'' = 0}})

<math display=block>\operatorname{E} [x^m] = \frac {\Gamma (N\alpha - m)} {\Gamma(N\alpha)} y^m</math>

which shows that the mean ± standard deviation estimate of the posterior distribution for {{mvar|θ}} is

<math display=block> \frac y {N\alpha - 1} \pm \sqrt{\frac {y^2} {(N\alpha - 1)^2 (N\alpha - 2)}}. </math>