Draft:2510

The partition of causal relevance

Always be $n$ the number of people we have to conduct the analyses upon, if we divide (based on certain conditions as explained below) this group into $k$ subsets $C_{i}$ with $i=1,2,\dots ,k$, a cluster is created that is called a "partition set"

$\pi$:

$\pi =\{C_{1},C_{2},\dots ,C_{k}\}\qquad \qquad {\text{with}}\qquad \qquad C_{i}\subset n,$

where with the symbolism $C_{i}\subset n$ it indicates that the subclass $C_{i}$ is contained in $n$.

The partition $\pi$, in order for it to be defined as a partition of causal relevance, must have these properties:

1. For each subclass $C_{i}$ the condition must apply $rc=P(D\mid C_{i})-P(D)\neq 0,$ ie the probability of finding in the subgroup $C_{i}$ a person who has the symptoms, clinical signs and elements belonging to the set $D=\{\delta _{1},\delta _{2},...,\delta _{n}\}$. A causally relevant partition of this type is said to be homogeneous.
2. Each subset $C_{i}$ must be 'elementary', i.e. it must not be further divided into other subsets, because if these existed they would have no causal relevance.

Now let us assume, for example, that the population sample $n$, to which our good patient Mary Poppins belongs, is a category of subjects aged 20 to 70. We also assume that in this population we have those who present the elements belonging to the data set $D=\{\delta _{1},.....\delta _{n}\}$ which correspond to the laboratory tests mentioned above and precisa in 'The logic of classical language'.

Let us suppose that in a sample of 10,000 subjects from 20 to 70 we will have an incidence of 30 subjects $p(D)=0.003$ showing clinical signs $\delta _{1}$ and $\delta _{4}$. We preferred to use these reports for the demonstration of the probabilistic process because in the literature the data regarding clinical signs and symptoms for Temporomandibular Disorders have too wide a variation as well as too high an incidence in our opinion.

An example of a partition with presumed probability in which TMJ degeneration (Deg.TMJ) occurs in conjunction with Temporomandibular Disorders (TMDs) would be the following:

$P(D|Deg.TMJ\cap TMDs)=0.95\qquad \qquad \;$		<translate> where</translate>			$\mathbb {C} _{1}\equiv Deg.TMJ\cap TMDs$
$P(D|Deg.TMJ\cap noTMDs)=0.3\qquad \qquad \quad$		<translate> where</translate>			$C_{2}\equiv Deg.TMJ\cap noTMDs$
$P(D|noDeg.TMJ\cap TMDs)=0.199\qquad \qquad \;$		<translate> where</translate>			$C_{3}\equiv noDeg.TMJ\cap TMDs$
$P(D|noDeg.TMJ\cap noTMDs)=0.001\qquad \qquad \;$		<translate> where</translate>			$C_{4}\equiv noDeg.TMJ\cap noTMDs$

Bibliography & references