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Combination of results from similar experiments

Results may be combined in a natural way making an interactive use of Bayesian inference. As a first case we assume several experiments having the same efficiency and exposure time. As an exercise, let us analyse the two extreme cases, starting from a uniform prior:
$ \boxed{\sum_ix_i=0}$
if none of the $ n$ similar experiments has observed events we have
$\displaystyle f(\lambda\,\vert\,n$ expts$\displaystyle , 0$ evts$\displaystyle )$ $\displaystyle =$ $\displaystyle n e^{-n\,\lambda}$  
$\displaystyle F(\lambda\,\vert\,n$ expts$\displaystyle , 0$ evts$\displaystyle )$ $\displaystyle =$ $\displaystyle 1-e^{-n\,\lambda}$  
$\displaystyle \lambda_u$ $\displaystyle =$ $\displaystyle -\frac{\ln(1-P_u)}{n}$   with probability $\displaystyle P_u\,.$  

$ \boxed{\sum_ix_i\mbox{ \lq\lq large''}}$
If the number of observed events is large (and the prior flat), the result will be normally distributed:

$\displaystyle f(\lambda) \sim {\cal N}(\mu_\lambda, \sigma_\lambda)\,.$

Then, in this case it is more practical to use maximum likelihood methods than to make integrals (see Section [*]). From the maximum of $ f(\lambda)$, in correspondence of $ \lambda=\lambda_m$, we easily get:

$\displaystyle \mu_\lambda =$   E$\displaystyle (\lambda)
\approx \lambda_{m}= \frac{\sum_{i=1}^n x_i}{n}\,,$

and from the second derivative of $ \ln f(\lambda)$ around the maximum:
$\displaystyle \left.\frac{\partial ^2 \ln f(\lambda)}{\partial \lambda^2}\right
\vert _{\lambda_{m}}$ $\displaystyle =$ $\displaystyle \frac{-n^2}{\sum_{i=1}^n x_i}$  
$\displaystyle \sigma_\lambda^2$ $\displaystyle \approx$ $\displaystyle -\left(\left.\frac{\partial ^2 \ln f(\lambda)}{\partial
\lambda}\right \vert _{\lambda_{m}}\right)^{-1}
= \frac{1}{n}\frac{\sum_{i=1}^n x_i}{n}$  
$\displaystyle \sigma_\lambda$ $\displaystyle \approx$ $\displaystyle \frac{\sqrt{\mu_\lambda}}{\sqrt{n}}\, .$  


next up previous contents
Next: Combination of results: general Up: Poisson model: dependence on Previous: Dependence on priors   Contents
Giulio D'Agostini 2003-05-15