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Probabilistic-causal analysis

From these premises it is clear that the clinical diagnosis is made using the so-called hypothetical-deductive method referred to as DN (deductive-nomological model). But this is not realistic, since the medical knowledge used in clinical decision-making hardly contains causal deterministic laws to allow causal explanations and, hence, to formulate clinical diagnoses, among other things in the specialist context. Let us try to analyse again the case of our Mary Poppins, this time trying a probabilistic-causal approach.

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

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)

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cr>0} indicating that the cause of orofacial pain is more likely to be bone degeneration of the TMJ,
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

• 

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

• 

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

• 

  Figure 3: Computed Tomography of the TMJ

• 

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

• 

  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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} people, individuals who present the elements belonging to the aforementioned set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} subsets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} with $i=1,2,\dots ,k$, a cluster is created that is called a "partition set" $\pi$:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} is contained in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} the condition must apply Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,\delta_2,...,\delta_n\}} . A causally relevant partition of this type is said to be homogeneous.
2. Each subset Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(D| Deg.TMJ \cap TMDs)=0.95 \qquad \qquad \; }		where			$\mathbb {C} _{1}\equiv Deg.TMJ\cap TMDs$
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(D| Deg.TMJ \cap noTMDs)=0.3 \qquad \qquad \quad }		where			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_2\equiv Deg.TMJ \cap noTMDs}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(D| no Deg.TMJ \cap TMDs)=0.199 \qquad \qquad \; }		where			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_3\equiv no Deg.TMJ \cap TMDs}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(D| noDeg.TMJ \cap noTMDs)=0.001 \qquad \qquad \;}		where			Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,.....\delta_n\}} in the population sample Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle PO} . Given the aforementioned partition of the reference class, we have the following clinical situations:

• Mary Poppins Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} degeneration of the temporomandibular joint Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cap} Temporomandibular Disorders

• Mary Poppins Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} degeneration of the temporomandibular joint Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cap} no Temporomandibular Disorders

• Mary Poppins Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} no degeneration of the temporomandibular joint Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cap} Temporomandibular Disorders

• Mary Poppins Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \in} no degeneration of the temporomandibular joint Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,.....\delta_n\}} .

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

• an individual: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a}
• its initial data set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,.....\delta_n\}}
• a population sample Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} to which it belongs,
• a base probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,.....\delta_n\}} is then a couple of the following logical forms (Analysandum / Analysans):

• Analysandum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \{P(D),a\}} : is a logical form that contains two parameters: probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(D)} to select a person who has the symptoms and elements belonging to the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D=\{\delta_1,\delta_2,...,\delta_n\}} , and the generic individual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} who is prone to those symptoms.

• Analysan Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \{\pi,a,KB\}} : is a logical form that contains three parameters: the partition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} , the generic individual Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $D=\{\delta _{1},.....\delta _{n}\}$