Exercício 26. Seja \left(X_{1}, X_{2}\right) \sim \operatorname{Trinomial}\left(n, \pi_{1}, \pi_{2}\right), em que
p\left(x_{1}, x_{2} \mid \pi_{1}, \pi_{2}\right)=\frac{n!}{x_{1}!x_{2}!\left(n-x_{1}-x_{2}\right)!} \pi_{1}^{x_{1}} \pi_{2}^{x_{2}}\left(1-\pi_{1}-\pi_{2}\right)^{n-x_{1}-x_{2}} \mathbb{1}_{\{0,1, \ldots, n\}}\left(x_{1}\right) \mathbb{1}_{\left\{0,1, \ldots, n-x_{1}\right\}}\left(x_{2}\right)𝟙𝟙.
Mostre que a priori não informativa de Jeffreys para ( \pi_{1}, \pi_{2} ) será
p\left(\pi_{1}, \pi_{2}\right) \propto\left[\pi_{1} \pi_{2}\left(1-\pi_{1}-\pi_{2}\right)\right]^{-1 / 2} .

Passo 1

O modelo é (X_{1},X_{2})\sim Trinomial (n,\pi_{1},\pi_{2}) com \pi_{3}=1-\pi_{1}-\pi_{2} e x_{3}=n-x_{1}-x_{2}. Queremos a priori de Jeffreys
p(\pi_{1},\pi_{2})\propto\sqrt{\det I(\pi_{1},\pi_{2})},
onde I(\pi) é a matriz de Fisher. Para isso vamos:
  1. escrever a log-verossimilhança \ell,
  2. calcular as segundas derivadas de \ell,
  3. obter I(\pi) pela expectativa do negativo dessas derivadas,
  4. computar \det I(\pi),
  5. extrair a priori.

Passo 2

A log-verossimilhança (ignorando constantes em \pi) é
\ell(\pi_{1},\pi_{2}) = x_{1}\log\pi_{1} + x_{2}\log\pi_{2} + x_{3}\log\pi_{3}.

Passo 3

As segundas derivadas são, para i,j\in\{1,2\}:
\frac{\partial^{2}\ell}{\partial\pi_{i}^{2}} = -\frac{x_{i}}{\pi_{i}^{2}} - \frac{x_{3}}{\pi_{3}^{2}}, \quad \frac{\partial^{2}\ell}{\partial\pi_{1}\partial\pi_{2}} = -\frac{x_{3}}{\pi_{3}^{2}}.

Passo 4

Pela definição,
I_{ij}(\pi) \;=\; -\mathbb{E}\Bigl[\frac{\partial^{2}\ell}{\partial\pi_{i}\partial\pi_{j}}\Bigr].
Como \mathbb{E}[x_{i}]=n\pi_{i} e \mathbb{E}[x_{3}]=n\pi_{3}, obtemos
I_{ii} = n\Bigl(\frac{1}{\pi_{i}}+\frac{1}{\pi_{3}}\Bigr), \quad I_{12}=I_{21} = \frac{n}{\pi_{3}}.
Em forma matricial,
I(\pi)=n \begin{pmatrix} 1/\pi_{1}+1/\pi_{3} & 1/\pi_{3}\\\\ 1/\pi_{3} & 1/\pi_{2}+1/\pi_{3} \end{pmatrix}.

Passo 5

Seja C=I(\pi)/n. Então
\det C = \Bigl(\frac{1}{\pi_{1}}+\frac{1}{\pi_{3}}\Bigr)\Bigl(\frac{1}{\pi_{2}}+\frac{1}{\pi_{3}}\Bigr) - \Bigl(\frac{1}{\pi_{3}}\Bigr)^{2} = \frac{1}{\pi_{1}\pi_{2}}+\frac{1}{\pi_{1}\pi_{3}}+\frac{1}{\pi_{2}\pi_{3}} = \frac{1}{\pi_{1}\pi_{2}\pi_{3}}.
Logo
\det I(\pi) = n^{2}\,\det C = \frac{n^{2}}{\pi_{1}\,\pi_{2}\,\pi_{3}}.

Passo 6

A priori de Jeffreys é
p(\pi_{1},\pi_{2})\propto \sqrt{\det I(\pi)} \;\propto\; \frac{n}{(\pi_{1}\,\pi_{2}\,\pi_{3})^{1/2}} \;\propto\; (\pi_{1}\,\pi_{2}\,\pi_{3})^{-1/2},
onde \pi_{3}=1-\pi_{1}-\pi_{2}.

Resposta

p(\pi_{1},\pi_{2}) \propto \bigl[\pi_{1}\,\pi_{2}\,(1-\pi_{1}-\pi_{2})\bigr]^{-1/2}.

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