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Erschienen in:
2014 | OriginalPaper | Buchkapitel
16. Bayesian Inference: Miscellaneous
verfasst von : Charles A. Rohde
Erschienen in: Introductory Statistical Inference with the Likelihood Function
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Abstract
Suppose you have obtained a posterior distribution for θ based on data y
1. At a later date you are given data y
2 whose distribution depends on the same parameter and is independent of the previous data. Then we have that i.e., “yesterday’s posterior becomes today’s prior.”
$$\displaystyle\begin{array}{rcl} p(\theta \vert y_{1},y_{2})& =& \frac{f(y_{1},y_{2};\theta )g(\theta )} {\int _{\Theta }f(y_{1},y_{2};\theta )g(\theta )d\theta } {}\\ & =& \frac{f(y_{2};\theta )f(y_{1};\theta )g(\theta )} {\int _{\Theta }f(y_{2};\theta )f(y_{1};\theta )g(\theta )d\theta } {}\\ & =& \frac{f(y_{2};\theta )\frac{f(y_{1};\theta )g(\theta )} {f(y_{1}} } {\int _{\Theta }f(y_{2};\theta )\frac{f(y_{1};\theta )g(\theta )} {f(y_{1})} d\theta } {}\\ & =& \frac{f(y_{2};\theta )p(\theta \vert y_{1})} {\int _{\Theta }f(y_{2};\theta )p(\theta \vert y_{1})d\theta )} {}\\ \end{array}$$