5.5 Distribuição a priori

All deductions are made a priori. (Wittgenstein 1921, 5.133)

5.5.1 Priori de Jeffreys

A law of chance is invariant for all transformations of the parameters when the law is differentiable with regard to all parameters. (Jeffreys 1946, 453)

(Jeffreys 1946) sugere uma forma invariante para a probabilidade a priori em problemas de estimação. É definida por

\[\begin{equation} \pi_J \propto \sqrt{\det I(\theta)} \tag{5.3} \end{equation}\]

\(I(\theta)\) é a Informação de Fisher (Ly et al. 2017), dada por

\[\begin{equation} I(\theta) = -E \left[ \frac{\partial^2 \log f(x|\theta)}{\partial \theta^2} \right] \tag{5.4} \end{equation}\]

A Informação de Fisher é a curvatura da log-verossimilhança, e \(I(\theta) \ge 0\).

Exemplo 5.7 Seja \(X|\theta \sim B(n,\theta)\) com fmp \[p(x|\theta) = {n \choose x}\theta^{x}(1-\theta)^{n-x}, \;\; 0 \le \theta \le 1\] conforme Eq. (3.49). Assim \[\begin{align*} \log p(x|\theta) &= x \log \theta + (n-x) \log (1-\theta) \\ \frac{\partial \log p(x|\theta)}{\partial \theta} &= \frac{x}{\theta} - \frac{n-x}{1-\theta} \\ \frac{\partial^2 \log p(x|\theta)}{\partial \theta^2} &= -\frac{x}{\theta^2} - \frac{n-x}{(1-\theta)^2} \\ \end{align*}\]

Como \(E(X|\theta) = n\theta\), \[\begin{align*} I(\theta) &= -E \left[ \frac{\partial^2 \log p(x|\theta)}{\partial \theta^2} \right] \\ &= -E \left[ -\frac{x}{\theta^2} - \frac{n-x}{(1-\theta)^2} \right] \\ &= \frac{n\theta}{\theta^2} + \frac{n-n\theta}{(1-\theta)^2} \\ &= \frac{n}{\theta} + \frac{n(1-\theta)}{(1-\theta)^2} \\ &= \frac{n}{\theta} + \frac{n}{(1-\theta)} \\ I(\theta) &= \frac{n}{\theta(1-\theta)} \end{align*}\]

Desta forma \[\pi_J = \sqrt{\frac{n}{\theta(1-\theta)}} \propto \theta^{-\frac{1}{2}}(1-\theta)^{-\frac{1}{2}} \propto \theta^{\frac{1}{2}-1}(1-\theta)^{\frac{1}{2}-1},\] que é o núcleo de uma \(Beta\left(\frac{1}{2},\frac{1}{2}\right)\).

5.5.2 Priori de referência

A procedure is proposed to derive reference posterior distributions which approximately describe the inferential content of the data without incorporating any other information. (Bernardo 1979)

Definição 5.4 Seja \(x\) o resultado de um experimento \(\varepsilon=\{X,\Theta,p(x|\theta)\}\) e seja \(C\) a classe de prioris admissíveis. A posteriori de referência de \(\theta\) após se observar \(x\) é definida por \(\pi(\theta|x)=\lim \pi_k(\theta|x)\), onde \(\pi_k(\theta|x) \propto p(x|\theta)\pi_k(\theta)\) é a densidade a posteriori correspondendo àquela priori \(\pi_k(\theta)\) que maximiza \(I^{\theta}\{\varepsilon(k),p(\theta)\}\) em \(C\). Uma priori de referência para \(\theta\) é uma função positiva \(\pi(\theta)\) que satisfaz \(\pi(\theta|x) \propto p(x|\theta)\pi(\theta)\).

5.5.3 Priori subjetiva

The diversity of man’s beliefs is as wide as the uncounted millions that have been or are now cluttered upon earth. (Lea 1909, 3)

Regra de Cromwell

Almost all thinking people agree that you should not have probability 1 (or 0) for any event, other than one demonstrable by logic, like \(2 \times 2 = 4\). The rule that denies probabilities of 1 or 0 is called Cromwell’s rule, named after Oliver Cromwell who said to the Church of Scotland, “think it possible you may be mistaken”. (Lindley 2006, 91)


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