Symmetric role of prior and `integrated likelihood'

Since we cannot go into indefinite and sterile discussions on all the possible priors that we might use (remember that if we collect and analyze data is to improve our knowledge, often used to make practical decision in a finite time scale!) it is important to understand a bit deeper their role in the inference. This can be done factorizing Eq. (), written here in compact notation as

$$f(\ldots) = f(p,n_I,n_{NI},n_{P_I},n_{P_{NI}},\pi_1,\pi_2,n_P,n_s,r_1,s_1,r_2,s_2)\,,$$ (81)

into two parts: one that only contains $f_0(p)$ and the other containing the remaining factors of the `chain', indicated here as $f_{\emptyset}(\ldots)$:

$$f(\ldots) = f_{\emptyset}(\ldots)\cdot f_0(p)\,.$$ (82)

The unnormalized pdf of $p$, conditioned by data and parameters, can be then rewritten (see Appendix A) as

$$f(p\,\vert\,n_P,n_s,r_1,s_1,r_2,s_2) \propto \left[ \sum_{n_I}\sum_{n_{NI}}\sum_{n_{P_I}}\sum_{n_{P_{NI}}} \in... ...tyset}(\ldots)\,\mbox{d}\pi_1 \mbox{d}\pi_2\right]\! \cdot f_0(p) \ \ \ \mbox{}$$ (83)

$$\propto {\cal L}(p\,;\,n_P,n_s,r_1,s_1,r_2,s_2)\cdot f_0(p)\,,$$ (84)

in which we have indicated with the usual symbol used for the (`integrated') likelihood (in which constant factors are irrelevant) the part which multiplies $f_0(p)$. It is then rather evident the role of ${\cal L}$ in `reshaping' $f_0(p)$.⁵³In the particular case in which $f_0(p)=1$ the inference is simply given by

$$f(p\,\vert\,n_P,n_s,r_1,s_1,r_2,s_2,f_0(p)=1) \propto {\cal L}(p\,;\,n_P,n_s,r_1,s_1,r_2,s_2)$$ (85)

(“the inference is determined by the likelihood”).

If, instead, the prior is not flat, then it does reshape the posterior obtained by ${\cal L}$ alone. Therefore there are two alternative ways to see the contributions of ${\cal L}$ and $f_0(p)$: each one reshapes the other. In particular

• in the regions of $p$ set to zero by either function the posterior vanishes;
• the function which is more narrow around its maximum `wins' against the smoother one.

Therefore, for the case shown in Fig. , obtained by a flat prior, the `density of $p$' is nothing but the shape of ${\cal L}(p;n_P,n_s,r_1,s_1,r_2,s_2)$. If $f_0(p)$ is constant, or varies slowly, in the range $[0.02,0.17]$ it provides null or little effect. If, instead, it is very peaked around 0.15 (e.g. with a standard deviation of $\approx 0.01$) it dominates the inference.

But what is more interesting is that the reshape by $f_0(p)$ can be done in a second step.⁵⁴This is the importance of choosing a flat prior (and not just a question of laziness): the data analysis expert could then present a result of the kind of Fig.  to an epidemiologist who could then reshape her priors (or, equivalently, reshape the curve provided by the data analyst with her priors). But she could also have such a strong prior on the variable under study, that she could reject tout court the result, blaming the data analysis expert that there must be something wrong in the analysis or in the data - see Sec. .