Exercício 30. Mostre que a priori de Jeffreys p(\boldsymbol{\theta}) \propto|\mathcal{I}(\boldsymbol{\theta})|^{1 / 2} é invariante a transformações 1 a 1 , ou seja, se \boldsymbol{\varphi}=\boldsymbol{\varphi}(\boldsymbol{\theta}) for uma transformação 1 a 1 de \boldsymbol{\theta}, então a priori de Jeffreys para \varphi será p(\varphi) \propto|\mathcal{I}(\varphi)|^{1 / 2}.

Passo 1

O enunciado pede para mostrar que a priori de Jeffreys
p(\boldsymbol{\theta})\propto\bigl|\mathcal{I}(\boldsymbol{\theta})\bigr|^{1/2}
é invariante por transformação bijetora \boldsymbol{\varphi}=\boldsymbol{\varphi}(\boldsymbol{\theta}).
O conteúdo envolve (1) transformação de variáveis em densidades e (2) regra de transformação da matriz de Fisher.

Passo 2

Quando fazemos a mudança de parâmetro \boldsymbol{\theta}=\boldsymbol{\theta}(\boldsymbol{\varphi}), a densidade a priori em \boldsymbol{\varphi} obedece
p(\boldsymbol{\varphi}) = p\bigl(\boldsymbol{\theta}(\boldsymbol{\varphi})\bigr)\,\Bigl|\det\Bigl(\tfrac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\varphi}}\Bigr)\Bigr|
porque toda densidade transforma multiplicando o valor original pela razão de volume \bigl|\det(\partial\boldsymbol{\theta}/\partial\boldsymbol{\varphi})\bigr|.

Passo 3

Pela definição da informação de Fisher,
\mathcal{I}_{\boldsymbol{\varphi}}(\boldsymbol{\varphi}) =\E\Bigl[\bigl(\nabla_{\!\boldsymbol{\varphi}}\log f(X;\boldsymbol{\varphi})\bigr)\bigl(\nabla_{\!\boldsymbol{\varphi}}\log f(X;\boldsymbol{\varphi})\bigr)^{T}\Bigr].
Como \nabla_{\!\boldsymbol{\varphi}}\log f=\bigl(\tfrac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\varphi}}\bigr)^{T}\nabla_{\!\boldsymbol{\theta}}\log f, obtemos
\mathcal{I}_{\boldsymbol{\varphi}}(\boldsymbol{\varphi}) =\Bigl(\tfrac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\varphi}}\Bigr)^{T} \mathcal{I}_{\boldsymbol{\theta}}\bigl(\boldsymbol{\theta}(\boldsymbol{\varphi})\bigr) \Bigl(\tfrac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\varphi}}\Bigr).

Passo 4

Usando a identidade \det(A^{T}BA)=\det(A)^{2}\det(B) para matrizes quadradas A e B, segue
\bigl|\mathcal{I}_{\boldsymbol{\varphi}}(\boldsymbol{\varphi})\bigr| =\Bigl|\det\bigl(\tfrac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\varphi}}\bigr)\Bigr|^{2} \bigl|\mathcal{I}_{\boldsymbol{\theta}}(\boldsymbol{\theta}(\boldsymbol{\varphi}))\bigr|.
Logo
\bigl|\mathcal{I}_{\boldsymbol{\varphi}}(\boldsymbol{\varphi})\bigr|^{1/2} =\Bigl|\det\bigl(\tfrac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\varphi}}\bigr)\Bigr|\, \bigl|\mathcal{I}_{\boldsymbol{\theta}}(\boldsymbol{\theta}(\boldsymbol{\varphi}))\bigr|^{1/2}.

Passo 5

Substituindo em p(\boldsymbol{\varphi})=p(\boldsymbol{\theta})\,|\det(\partial\boldsymbol{\theta}/\partial\boldsymbol{\varphi})| com
p(\boldsymbol{\theta})\propto|\mathcal{I}_{\boldsymbol{\theta}}|^{1/2}, vem
p(\boldsymbol{\varphi}) \propto\bigl|\mathcal{I}_{\boldsymbol{\theta}}\bigr|^{1/2} \Bigl|\det\bigl(\tfrac{\partial\boldsymbol{\theta}}{\partial\boldsymbol{\varphi}}\bigr)\Bigr| =\bigl|\mathcal{I}_{\boldsymbol{\varphi}}(\boldsymbol{\varphi})\bigr|^{1/2}.

Resposta

Mostramos que sob qualquer transformação bijetora \boldsymbol{\varphi}=\boldsymbol{\varphi}(\boldsymbol{\theta}), a priori de Jeffreys em \boldsymbol{\varphi} é
p(\boldsymbol{\varphi}) \propto \bigl|\mathcal{I}(\boldsymbol{\varphi})\bigr|^{1/2},
isto é, a forma fica invariante.

Exercícios relacionados