Dependence of the rate ratio from a physical quantity

Another interesting question is how to approach the problem of a ratio of rates that depends on the value of another physical quantity. That is we assume a dependence of $\rho$ from $v$ (symbol for a generic variable),

$$\rho = g(v; \mbox{\boldmath$\theta$} _\rho)\,,$$ (132)

with $\theta$$_\rho$ the set of parameters of the functional dependence. The simplest and best understood case is the linear dependence

$$\rho = m\cdot v + c\,,$$ (133)

where $\theta$$_\rho=\{m,c\}$, treated in detail in Ref. [43] where we used the same approach we are adopting here. In analogy to what done in Fig. 1 there, we can extend the Model B of Fig.  to that of Fig. 

Figure: Further extension of the model of Fig.  (neglecting the `complications' of the models of Figs.  and ) to take into account that each value $\rho_j$ might depend on the physical quantity $v_j$, measured as $v_{m_j}$ via the set of parameters $\theta$

(we continue to neglect efficiency and background issues in order to focus to the core of the problem). Moreover, as in Fig. 1 of Ref. [43], we have considered the fact that the physical quantity $v$ is `experimentally observed' as $v_O$. In the simple case of a linear dependence the model is described by the following relations among the variables

$$X_{1_j} \sim (\lambda_{1_j})$$ (134)

$$X_{2_j} \sim (\lambda_{2_j})$$ (135)

$$\lambda_{1_j} = r_{j_i}\cdot T_{1_j}$$ (136)

$$\lambda_{2_j} = r_{j_2}\cdot T_{2_j}$$ (137)

$$r_{1_j} = \rho_j\cdot r_{2_j}$$ (138)

$$\rho_j = m\cdot v_j + c$$ (139)

$$v_{O_j} \sim {\cal N}(v_j, \sigma_{E_j})\,,$$ (140)

in which we have assumed a Gaussian (`normal') error function of $v_{O_j}$ around $v_j$, with standard deviations $\sigma_{E_j}$. But the description of the model provided by the above relations is not complete (besides the complications related to inefficiencies and background, that we continue to neglect). In fact, we miss priors for $v_j$, $r_{2_j}$ and $\theta$$_\rho$, as they have no parent nodes (instead, we continue to consider $T_{1_j}$ and $T_{2_j}$ `exactly known', being their uncertainty usually irrelevant).

The priors which are easier to choose are those of $v_j$, if their values are `well measured', that is if $\sigma_{E_j}$ are small enough. We can then confidently use flat priors, as done e.g. for the `unobserved' $\mu_{y_i}$ of Fig. 1 in Ref. [43].

Also the priors about $\theta$$_\rho$ can be chosen quite vague, paying however some care in order to forbid negative values of $\rho$. Incidentally, having mentioned the simple case of linear dependence, an important sub-case is when $m$ is assumed to be null: the remaining prior on $c$ becomes indeed the prior on $\rho$, and the inference of $c$ corresponds to the inferred value of $\rho$ having taken into account several instances of $X_1$ and $X_2$ - this is indeed the question of the `combination of values of $\rho$' on which we shall comment a bit more in detail in the sequel.

As far as the priors of the rates are concerned, one could think, a bit naively, that the choice of independent flat priors for $r_{2_j}$ could be a reasonable choice. But we need to understand the physical model underlying this choice. In fact, most likely, as the ratio $\rho$ might depends on $v$, the same could be true for $r_2$, but perhaps with a completely different functional dependence. For example $r_1$ and $r_2$ could have a strong dependence on $v$, e.g. they could decrease exponentially, but, nevertheless, their ratio could be independent of $v$, or, at most, could just exhibit a small linear dependence. Therefore we have to add this possibility into the model, which then becomes as in Fig. ,

Figure: Extension of the model of Fig. , but making also $r_2$ depend on $v$ according to a suited law depending on the set of parameters $\theta$$_{r_2}$.

in which we have included a set of parameters $\theta$$_{r_2}$, such that

$$r_2 = h(v; \mbox{\boldmath$\theta$} _{r_2})\,,$$ (141)

and, needless to say, some priors are required for $\theta$$_{r_2}$.

At this point, any further consideration goes beyond the rather general purpose of this paper, because we should enter into details that strongly depend on the physical case. We hope that the reader could at least appreciate the level of awareness that these graphical models provide. The existence of computing tools in which the models can be implemented makes then nowadays possible what decades ago was not even imaginable.