Generalised Reichenbachian common cause systems

Abstract

The principle of the common cause claims that if an improbable coincidence has occurred, there must exist a common cause. This is generally taken to mean that positive correlations between non-causally related events should disappear when conditioning on the action of some underlying common cause. The extended interpretation of the principle, by contrast, urges that common causes should be called for in order to explain positive deviations between the estimated correlation of two events and the expected value of their correlation. The aim of this paper is to provide the extended reading of the principle with a general probabilistic model, capturing the simultaneous action of a system of multiple common causes. To this end, two distinct models are elaborated, and the necessary and sufficient conditions for their existence are determined.

Appendix: Proof of Lemma 5

Let $(\varOmega , p)$ be a classical probability space with $\sigma$-algebra of random events $\varOmega$ and probability measure p. Moreover, let $A,B\in \varOmega$ satisfy (17) and (62).

To start with, let us observe that (43) is in fact a system of $n-1$ equations, namely one for each value of $i = 1, \ldots , n-1$. Therefore, (39)–(43) jointly comprise a system of $4 + (n - 1) = n + 3$ equations in 4n variables. This means that each HR-admissible set for $\delta \left( {A},{B}\right)$ is determined by a set of $4n - (n+3) = 3n- 3$ parameters, for every $n\ge 2$. To establish the existence of such set, we accordingly need to prove that such parameters exist. To this purpose, let numbers a, b and $\varepsilon$ be understood as per (44)–(46). Proof will proceed by induction on n.

Let us begin by assuming $n = 2$ as our inductive basis. This has the effect of transforming (39)–(43), respectively, into:

$$\begin{aligned}&\displaystyle a_{2} = \frac{a - c_{1} a_{1}}{1-c_{1}} \end{aligned}$$

(89)

$$\begin{aligned}&\displaystyle b_{2} = \frac{b- c_{1} b_{1}}{1-c_{1}} \end{aligned}$$

(90)

$$\begin{aligned}&\displaystyle c_{2} = 1- c_{1} \end{aligned}$$

(91)

$$\begin{aligned}&\displaystyle d_{2} = \varepsilon + \frac{[a - a_{1}c_{1}][b - b_{1}c_{1}]}{[ 1 - c_{1}]^{2}} \end{aligned}$$

(92)

$$\begin{aligned}&\displaystyle d_{1} - a_{1}b_{1} = \varepsilon . \end{aligned}$$

(93)

Since a, b and $\varepsilon$ are known by hypothesis, choosing numbers $c_{1}$, $a_{1}$ and $b_{1}$ will therefore suffice to fix the values of all $4n = 8$ variables in the system. In particular, (89)–(92) immediately produce:

$$\begin{aligned} lim_{c_{1}\rightarrow 0} \ a_{2}= & {} a \end{aligned}$$

(94)

$$\begin{aligned} lim_{c_{1}\rightarrow 0} \ b_{2}= & {} b \end{aligned}$$

(95)

$$\begin{aligned} lim_{c_{1}\rightarrow 0} \ c_{2}= & {} 1 \end{aligned}$$

(96)

$$\begin{aligned} lim_{c_{1}\rightarrow 0} \ d_{2}= & {} \varepsilon + ab, \end{aligned}$$

(97)

while on the other hand (17) directly requires that

$$\begin{aligned} 1> a,b > 0, \end{aligned}$$

(98)

as it would be easy to verify. Taken together this ensures that, as $c_{1}$ is taken sufficiently close to zero:

$$\begin{aligned} 1\ge & {} a_{2}, b_{2} \ge 0 \end{aligned}$$

(99)

$$\begin{aligned} 1\ge & {} c_{1}, c_{2} \ge 0, \end{aligned}$$

(100)

while (62) and (17) imply that

$$\begin{aligned} 1 \ge d_{2} \ge 0. \end{aligned}$$

(101)

To determine the remaining numbers, we further need to set $a_{1}$ and $b_{1}$. In this case, our choice will depend on the value of $\varepsilon$, as follows:

$$\begin{aligned} \varepsilon\ge & {} 0 \phantom {12 pt} {\left\{ \begin{array}{ll} a> a_{1} \ge 0\\ b > b_{1} \ge 0 \end{array}\right. } \end{aligned}$$

(102)

$$\begin{aligned} \varepsilon< & {} 0 \phantom {12 pt} {\left\{ \begin{array}{ll} 1 \ge a_{1}> a \\ 1 \ge b_{1} > b \end{array}\right. } \end{aligned}$$

(103)

Either option is allowed by (98), and either will ensure that

$$\begin{aligned} 1 \ge d_{i} \ge 0, \end{aligned}$$

(104)

as it would be straightforward to check with the aid of (43), (17) and (62). Thanks to Lemma 3, this is enough to establish that some set $\left\{ {a_{i}, b_{i}, c_{i}, d_{i}}\right\} _{{i = 1}}^{{n}}$ of quasi-admissible numbers exist for $\delta \left( {A},{B}\right)$ if $n = 2$. To further show that such set is HR-admissible for $\delta \left( {A},{B}\right)$, we only need to observe that (34) can be obtained from both (102) and (103), owing to (28)–(30).

Let us now assume, as our inductive hypothesis, that some set $\left\{ {a_{i}, b_{i}, c_{i}, d_{i}}\right\} _{{i = 1}}^{{m}}$ is HR-admissible for $\delta \left( {A},{B}\right)$, where $n = m > 2$. To prove that a HR-admissible set for $\delta \left( {A},{B}\right)$ also exists if $n = m +1$, let us consider the set

$$\begin{aligned} \left\{ {a_{j}, b{j}, c_{j}, a_{m-1}, b_{m-1}}\right\} _{{j = 1}}^{{m-2}}\subset \left\{ {a_{i}, b_{i}, c_{i}, d_{i}}\right\} _{{i = 1}}^{{m}}, \end{aligned}$$

and let us choose numbers $a'_{m} , b'_{m}, c'_{m-1}, c'_{m}$ such that:

$$\begin{aligned}&\displaystyle a_{j}> a'_{m} > 0 \phantom {12 pt} ( j = 1, \ldots , m-1) \end{aligned}$$

(105)

$$\begin{aligned}&\displaystyle b_{j}> b'_{m} > 0 \phantom {12 pt} ( j = 1, \ldots , m-1) \end{aligned}$$

(106)

$$\begin{aligned}&\displaystyle c_{m}, c_{m-1}> c'_{m-1} > 0 \end{aligned}$$

(107)

$$\begin{aligned}&\displaystyle 1> c'_{m} > 0. \end{aligned}$$

(108)

Given (39)–(43), the set

$$\begin{aligned} \left\{ {a_{j}, b{j}, c_{j}, a_{m-1}, b_{m-1}}\right\} _{{j = 1}}^{{m-2}}\cup \left\{ {a'_{m}, b'_{m}, c'_{m-1}, c'_{m}}\right\} _{{}}^{{}} \end{aligned}$$

of $3n-3$ parameters will then suffice to determine $4(m+1)$ numbers:

$$\begin{aligned} \left\{ {a_{j}, b_{j}, c_{j}, d_{j}, a_{m-1}, b_{m-1}, c'_{m-1}, d_{m-1}, a'_{m}, b'_{m}, c'_{m}, d'_{m}, a_{m+1}, b_{m+1}, c_{m+1}, d_{m+1}}\right\} _{{j = 1}}^{{m-2}}. \end{aligned}$$

Because numbers $\left\{ {a_{j}, b_{j}, c_{j}, d_{j}, a_{m-1}, b_{m-1}, c'_{m-1}, d_{m-1}, a'_{m}, b'_{m}, c'_{m},}\right\} _{{j = 1}}^{{m-1}}$ satisfy conditions (31)–(33) and (34) by hypothesis, all we need to show is that said constraints are also satisfied by the remaining numbers $\left\{ {d'_{m}, a_{m+1}, b_{m+1}, c_{m+1}, d_{m+1}}\right\} _{{}}^{{}}$. To this purpose, let us first notice that (32) must be true of $d'_{m}$ by virtue of (43) and (105)–(106). Next, thanks to (39)–(42), it will be sufficient to suppose that

$$\begin{aligned}&a'_{m} \rightarrow 0 \end{aligned}$$

(109)

$$\begin{aligned}&b'_{m} \rightarrow 0 \end{aligned}$$

(110)

$$\begin{aligned}&c'_{m} \rightarrow c_{m-1}-c'_{m-1} \end{aligned}$$

(111)

to obtain

$$\begin{aligned}&lim_{a'_{m} \rightarrow 0, \ c'_{m} \rightarrow c_{m-1}-c'_{m-1}} a_{m+1} = a_{m} \end{aligned}$$

(112)

$$\begin{aligned}&lim_{b'_{m} \rightarrow 0, \ c'_{m} \rightarrow c_{m-1}-c'_{m-1}} b_{m+1} = b_{m} \end{aligned}$$

(113)

$$\begin{aligned}&lim_{c'_{m} \rightarrow c_{m-1}-c'_{m-1}} c_{m+1} = c_{m} \end{aligned}$$

(114)

$$\begin{aligned}&lim_{a'_{m} \rightarrow 0, \ b'_{m} \rightarrow 0, \ c'_{m} \rightarrow c_{m-1}-c'_{m-1}} d_{m+1} = d_{m} \end{aligned}$$

(115)

which we already know, by our inductive hypothesis, to satisfy (31)–(33). Moreover, (105) and (106), along with the inductive assumption whereby

$$\begin{aligned} {[}a_{m} - a_{i}][b_{m}-b_{i}] > 0 \phantom {12 pt} (i = 1, \ldots , m-1), \end{aligned}$$

(116)

ensures that

$$\begin{aligned} {[}a_{m+1} - a_{i}][b_{m+1}-b_{i}] > 0 \phantom {12 pt} (i = 1, \ldots , m), \end{aligned}$$

(117)

which together with our inductive hypothesis suffices to establish (34). Due to Lemma 3 and Definition 8, the set of $4(m+1)$ numbers so determined is therefore HR-admissible for $\delta \left( {A},{B}\right)$. Lemma 5 is thus demonstrated by induction.