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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<ref name=":1">{{Cite book  

  }}</ref> ([[wikipedia:Deductive-nomological_model|deductive-nomological model]]<ref>''<!--52-->DN model of scientific explanation'', <!--53-->also known as ''<!--54-->Hempel's model'', ''Hempel–Oppenheim model'', ''Popper–Hempel model'', <!--55-->or ''<!--56-->covering law model''</ref>). 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 <math>n</math> 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.

To do this we consider the degree of causal relevance <math>(cr)</math> of an event <math>E_1</math> with respect to an event <math>E_2</math> where:

*<math>E_1</math> = patients with bone degeneration of the temporomandibular joint.

*<math>E_2</math> = patients reporting orofacial pain.

*<math>E_3</math> = patients without bone degeneration of the temporomandibular joint.

:<math>P(E_2 \mid E_1)</math> indicates the probability that some people (among <math>n</math> taken into consideration) suffer from Orofacial Pain caused by bone degeneration of the Temporomandibular Joint,

 

:<math>P(E_2 \mid E_3)</math> indicates the probability that other people (always among <math>n</math> taken into consideration) suffer from Orofacial Pain conditioned by something other than bone degeneration of the Temporomandibular Joint.

 

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

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

:*<math>cr=1</math> indicating that the only cause of orofacial pain is bone degeneration of the TMJ,

:*<math>cr=-1</math> which indicates that the cause of orofacial pain is never bone degeneration of the TMJ but is something else,

:*<math>cr=0</math> 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)

:*<math>cr>0</math> indicating that the cause of orofacial pain is more likely to be bone degeneration of the TMJ,

:*<math>cr<0</math> which indicates that the cause of orofacial pain is more likely not bone degeneration of the TMJ.

===Second Clinical Approach===

''(hover over the images)''

File:Spasmo emimasticatorio.jpg|'''<!--81-->Figure 1:''' <!--82-->Patient reporting "Orofacial pain in the right hemilateral"

File:Spasmo emimasticatorio ATM.jpg|'''<!--83-->Figure 2:''' <!--84-->Patient's TMJ Stratigraphy showing signs of condylar flattening and osteophyte

File:Atm1 sclerodermia.jpg|'''<!--85-->Figure 3:''' <!--86-->Computed Tomography of the TMJ

File:Spasmo emimasticatorio assiografia.jpg|'''<!--87-->Figure 4:''' <!--88-->Axiography of the patient showing a flattening of the chewing pattern on the right condyle

File:EMG2.jpg|'''<!--89-->Figure 5:''' <!--90-->EMG Interferential Pattern. Overlapping upper traces corresponding to the right masseter, lower to the left masseter.

So be it then <math>P(D)</math> the probability of finding, in the sample of our <math>n</math> people, individuals who present the elements belonging to the aforementioned set <math>D=\{\delta_1,\delta_2,...,\delta_n\}</math>

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 <math>n</math> the number of people we have to conduct the analyses upon, if we divide (based on certain conditions as explained below) this group into <math>k</math> subsets <math>C_i</math> with <math>i=1,2,\dots,k</math>, a cluster is created that is called a "partition set" <math>\pi</math>:

:<math>\pi = \{C_1, C_2,\dots,C_k \}  \qquad \qquad \text{with} \qquad \qquad C_i \subset n , </math>

where with the symbolism <math>C_i \subset n </math> it indicates that the subclass <math>C_i</math> is contained in <math>n</math>.

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

#For each subclass <math>C_i</math> the condition must apply <math>rc=P(D \mid C_i)- P(D )\neq 0, </math> ie the probability of finding in the subgroup <math>C_i</math> a person who has the symptoms, clinical signs and elements belonging to the set <math>D=\{\delta_1,\delta_2,...,\delta_n\}</math>. A causally relevant partition of this type is said to be '''homogeneous'''.

#Each subset <math>C_i</math> 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 <math>n</math>, 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 <math>D=\{\delta_1,.....\delta_n\}</math> 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 <math>p(D)=0.003</math> showing clinical signs <math>\delta_1</math> and <math>\delta_4

</math>. 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.<ref name=":2">{{Cite book  

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:

|where

|where

|where

|where

 

====Clinical situations====

These conditional probabilities demonstrate that each of the partition's four subclasses is causally relevant to patient data <math>D=\{\delta_1,.....\delta_n\}</math> in the population sample <math>PO</math>. Given the aforementioned partition of the reference class, we have the following clinical situations:

*Mary Poppins <math>\in</math> degeneration of the temporomandibular joint <math>\cap</math> Temporomandibular Disorders

*Mary Poppins <math>\in</math> degeneration of the temporomandibular joint <math>\cap</math> no Temporomandibular Disorders

*Mary Poppins <math>\in</math> no degeneration of the temporomandibular joint <math>\cap</math> Temporomandibular Disorders

*Mary Poppins <math>\in</math> no degeneration of the temporomandibular joint <math>\cap</math> 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 <math>D=\{\delta_1,.....\delta_n\}</math>.

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

*an individual: <math>a</math>

*its initial data set <math>D=\{\delta_1,.....\delta_n\}</math>

*a population sample <math>n</math> to which it belongs,

*a base probability <math>P(D)=0,003</math>

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 <math>D=\{\delta_1,.....\delta_n\}</math> is then a couple of the following logical forms (Analysandum / Analysans<ref>{{Cite book  

  }}</ref>):

*'''Analysandum''' <math>  = \{P(D),a\}</math>: is a logical form that contains two parameters: ''probability'' <math>P(D)</math> to select a person who has the symptoms and elements belonging to the set <math>D=\{\delta_1,\delta_2,...,\delta_n\}</math>, and the ''generic individual'' <math>a</math> who is prone to those symptoms.

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

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

<math>P(D| Deg.TMJ  \cap TMDs)=0.95</math> - 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 <math>D=\{\delta_1,.....\delta_n\}</math>