Bayesian Inference: Miscellaneous

Charles A. Rohde
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Charles A. Rohde: Johns Hopkins University, Bloomberg School of Health

Chapter Chapter 16 in Introductory Statistical Inference with the Likelihood Function, 2014, pp 187-195 from Springer

Abstract: 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 p ( θ | y 1 , y 2 ) = f ( y 1 , y 2 ; θ ) g ( θ ) ∫ Θ f ( y 1 , y 2 ; θ ) g ( θ ) d θ = f ( y 2 ; θ ) f ( y 1 ; θ ) g ( θ ) ∫ Θ f ( y 2 ; θ ) f ( y 1 ; θ ) g ( θ ) d θ = f ( y 2 ; θ ) f ( y 1 ; θ ) g ( θ ) f ( y 1 ∫ Θ f ( y 2 ; θ ) f ( y 1 ; θ ) g ( θ ) f ( y 1 ) d θ = f ( y 2 ; θ ) p ( θ | y 1 ) ∫ Θ f ( y 2 ; θ ) p ( θ | y 1 ) d θ ) $$\displaystyle\begin{array}{rcl} p(\theta \vert y_{1},y_{2})& =& \frac{f(y_{1},y_{2};\theta )g(\theta )} {\int _{\Theta }f(y_{1},y_{2};\theta )g(\theta )d\theta } {}\\ & =& \frac{f(y_{2};\theta )f(y_{1};\theta )g(\theta )} {\int _{\Theta }f(y_{2};\theta )f(y_{1};\theta )g(\theta )d\theta } {}\\ & =& \frac{f(y_{2};\theta )\frac{f(y_{1};\theta )g(\theta )} {f(y_{1}} } {\int _{\Theta }f(y_{2};\theta )\frac{f(y_{1};\theta )g(\theta )} {f(y_{1})} d\theta } {}\\ & =& \frac{f(y_{2};\theta )p(\theta \vert y_{1})} {\int _{\Theta }f(y_{2};\theta )p(\theta \vert y_{1})d\theta )} {}\\ \end{array}$$ i.e., “yesterday’s posterior becomes today’s prior.”

Keywords: Bayesian Inference; Ancillary Statistic; Randomized Stopping Rule; Point Null Hypothesis; Positive Prior Probability (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-10461-4_16

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