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Probabilistic-Causal Analysis

From these premises, it is evident that clinical diagnosis employs the so-called hypothetical-deductive method, known as DN (deductive-nomological model). However, this is not realistic, as the medical knowledge used in clinical decision-making rarely contains causal deterministic laws that allow for causal explanations and, consequently, the formulation of clinical diagnoses, among other things in the specialist context. Let us re-examine the case of Mary Poppins, this time adopting a probabilistic-causal approach.

Consider a number $n$ of individuals, including those who report Orofacial Pain, who generally have bone degeneration of the Temporomandibular Joint. However, there might also be other apparently unrelated causes. We must mathematically translate the 'relevance' of these causal uncertainties in determining a diagnosis.

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The casual relevance

To do this we consider the degree of causal relevance $(cr)$ of an event $E_{1}$ with respect to an event $E_{2}$ where:

• $E_{1}$ = patients with bone degeneration of the temporomandibular joint.

• $E_{2}$ = patients reporting orofacial pain.

• $E_{3}$ = patients without bone degeneration of the temporomandibular joint.

We will use the conditional probability $P(A\mid B)$, that is the probability that the $A$ event occurs only after the event $B$ has already occurred.

With these premises the causal relevance $cr$ of the sample $n$ of patients is:

$cr=P(E_{2}\mid E_{1})-P(E_{2}\mid E_{3})$

where

$P(E_{2}\mid E_{1})$ indicates the probability that some people (among $n$ taken into consideration) suffer from Orofacial Pain caused by bone degeneration of the Temporomandibular Joint,

while

$P(E_{2}\mid E_{3})$ indicates the probability that other people (always among $n$ taken into consideration) suffer from Orofacial Pain conditioned by something other than bone degeneration of the Temporomandibular Joint.

Since all probability suggest that $P(A\mid B)$ is a value between $0$ and $1$, the parameter $(cr)$ will be a number that is between $-1$ and $1$.

The meanings that we can give to this number are as follows:

• we have the extreme cases (which in reality never occur) which are:

• $cr=1$ indicating that the only cause of orofacial pain is bone degeneration of the TMJ,
• $cr=-1$ which indicates that the cause of orofacial pain is never bone degeneration of the TMJ but is something else,
• $cr=0$ indicating that the probability that orofacial pain is caused by bone degeneration of the TMJ or otherwise is exactly the same,

• and the intermediate cases (which are the realistic ones)

• $cr>0$ indicating that the cause of orofacial pain is more likely to be bone degeneration of the TMJ,
• $cr<0$ which indicates that the cause of orofacial pain is more likely not bone degeneration of the TMJ.

Second Clinical Approach

(hover over the images)

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  Figure 1: Patient reporting "Orofacial pain in the right hemilateral"

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  Figure 2: Patient's TMJ Stratigraphy showing signs of condylar flattening and osteophyte

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  Figure 3: Computed Tomography of the TMJ

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  Figure 4: Axiography of the patient showing a flattening of the chewing pattern on the right condyle

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  Figure 5: EMG Interferential Pattern. Overlapping upper traces corresponding to the right masseter, lower to the left masseter.

So be it then $P(D)$ the probability of finding, in the sample of our $n$ people, individuals who present the elements belonging to the aforementioned set $D=\{\delta _{1},\delta _{2},...,\delta _{n}\}$

In order to take advantage of the information provided by this dataset, the concept of partition of causal relevance is introduced:

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:

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

• «A homogeneous partition provides what we are used to calling Differential Diagnosis.»

Clinical situations

These conditional probabilities demonstrate that each of the partition's four subclasses is causally relevant to patient data $D=\{\delta _{1},.....\delta _{n}\}$ in the population sample $PO$. Given the aforementioned partition of the reference class, we have the following clinical situations:

• Mary Poppins $\in$ degeneration of the temporomandibular joint $\cap$ Temporomandibular Disorders

• Mary Poppins $\in$ degeneration of the temporomandibular joint $\cap$ no Temporomandibular Disorders

• Mary Poppins $\in$ no degeneration of the temporomandibular joint $\cap$ Temporomandibular Disorders

• Mary Poppins $\in$ no degeneration of the temporomandibular joint $\cap$ no Temporomandibular Disorders

To arrive at the final diagnosis above, we conducted a probabilistic-causal analysis of Mary Poppins' health status whose initial data were $D=\{\delta _{1},.....\delta _{n}\}$.

In general, we can refer to a logical process in which we examine the following elements:

• an individual: $a$
• its initial data set $D=\{\delta _{1},.....\delta _{n}\}$
• a population sample $n$ to which it belongs,
• a base probability $P(D)=0,003$

At this point we should introduce too specialized arguments that would take the reader off the topic but that have an high epistemic importance for which we will try to extract the most described logical thread of the Analysandum/Analysans concept.

The probabilistic-causal analysis of $D=\{\delta _{1},.....\delta _{n}\}$ is then a couple of the following logical forms (Analysandum / Analysans):

• Analysandum $=\{P(D),a\}$: is a logical form that contains two parameters: probability $P(D)$ to select a person who has the symptoms and elements belonging to the set $D=\{\delta _{1},\delta _{2},...,\delta _{n}\}$, and the generic individual $a$ who is prone to those symptoms.

• Analysan $=\{\pi ,a,KB\}$: is a logical form that contains three parameters: the partition $\pi$, the generic individual $a$ belonging to the population sample $n$ and $KB$ (Knowledge Base) which includes a set of $n>1$ statements of conditioned probability.

For example, it can be concluded that the definitive diagnosis is the following:

$P(D|Deg.TMJ\cap TMDs)=0.95$ - this means that our Mary Poppins is 95% affected by TMDs, since she has a degeneration of the Temporomandibular Joint in addition to the positive data $D=\{\delta _{1},.....\delta _{n}\}$