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Effetto del background nella misura dell'intensità di un processo di Poisson

$\displaystyle f(n_c\,\vert\,r,r_b) =\frac{e^{-(r+r_b)\,T}((r+r_b)\,T)^{n_c}}{n_c!}\,,$ (14.1)

and, making use of Bayes' theorem, we get

$\displaystyle f(r\,\vert\,n_c,r_b) \propto \frac{e^{-(r+r_b)\,T}((r+r_b)\,T)^{n_c}}{n_c!} f_\circ(r)\,.$ (14.2)

$\displaystyle f(r\,\vert\,n_c,r_b,f_\circ(r)=k) = \frac{e^{-r\,T}((r+r_b)\,T)^{n_c} } {n_c!\,\sum_{n=0}^{n_c}\frac{(r_b\,T)^n}{n!} }\,.$ (14.3)

Figura: The upper plot shows some reasonable priors reflecting the positive attitude of researchers: uniform distribution (continuous); triangular distribution (dashed); half-Gaussian distribution (dotted). The lower plot shows how the results of Fig. [*], obtained starting from an improper uniform distribution, (do not) change if, instead, the priors of the upper plot are used.
\begin{figure}\begin{center}
\begin{tabular}{\vert c\vert}\hline
\epsfig{file=fi...
...a.eps,width=0.65\linewidth,clip=}\\ \hline
\end{tabular}\end{center}\end{figure}

$\displaystyle f(r\,\vert\,n_c,r_b) \propto f(n_c\,\vert\,r,r_b)\cdot f_\circ(r)\,,$ (14.7)

and consider only two possible values of $ r$, let them be $ r_1$ and $ r_2$. From (14.7) it follows that

$\displaystyle \frac{f(r_1\,\vert\,n_c,r_b)}{f(r_2\,\vert\,n_c,r_b)} = \underbra...
...r_b)} }_{\mbox{\it Bayes factor}}\,\cdot\, \frac{f_\circ(r_1)}{f_\circ(r_2)}\,.$ (14.8)

The Bayes factor can be extended to a continuous set of hypotheses $ r$, considering a function which gives the Bayes factor of each value of $ r$ with respect to a reference value $ r_{REF}$. The reference value could be arbitrary, but for our problem the choice $ r_{REF}=0$, giving

$\displaystyle {\cal R}(r;n_c,r_b) = \frac{f(n_c\,\vert\,r,r_b)}{f(n_c\,\vert\,r=0,r_b)}\,,$ (14.9)

is very convenient for comparing and combining the experimental results [#!ci!#,#!zeus!#,#!higgs!#]. The function $ {\cal R}$ has nice intuitive interpretations which can be highlighted by reordering the terms of (14.8) in the form

$\displaystyle \frac{f(r\,\vert\,n_c,r_b)}{f_\circ(r)}\left/ \frac{f(r=0\,\vert\...
...ht. = \frac{f(n_c\,\vert\,r,r_b)}{f(n_c\,\vert\,r=0,r_b)} = {\cal R}(r;n_c,r_b)$ (14.10)

(valid for all possible a priori $ r$ values). $ {\cal R}$ has the probabilistic interpretation of relative belief updating ratio, or the geometrical interpretation of shape distortion function of the probability density function. $ {\cal R}$ goes to 1 for $ r\rightarrow 0$, i.e. in the asymptotic region in which the experimental sensitivity is lost: As long as it is 1, the shape of the p.d.f. (and therefore the relative probabilities in that region) remains unchanged. Instead, in the limit $ {\cal R}\rightarrow 0$ (for large $ r$) the final p.d.f. vanishes, i.e. the beliefs go to zero no matter how strong they were before. In the case of the Poisson process we are considering, the relative belief updating factor becomes

$\displaystyle {\cal R}(r;n_c,r_b,T) = e^{-r\,T}\left(1+\frac{r}{r_b}\right)^{n_c}\,,$ (14.11)

with the condition14.1 $ r_b>0$ if $ n_c>0$.

Figura: Relative belief updating ratio $ {\cal R}$ for the Poisson intensity parameter $ r$ for the cases of Fig. [*].
\begin{figure}\begin{center}
\epsfig{file=fig/rasloglog.eps,clip=,width=0.8\linewidth}\end{center}\end{figure}

These curves transmit the result of the experiment immediately and intuitively: Moreover there are some technical advantages in reporting the $ {\cal R}$ function as a result of a search experiment.


next up previous contents
Next: Propagazioni di incertezza, approssimazioni Up: Effetti sistematici e di Previous: Effetti sistematici e di   Indice
Giulio D'Agostini 2001-04-02