Mathematical Background of Medical Diagnosis
There are several ways to formalize the medical diagnosis process using mathematical models; but one of the most general theories used to describe foundation of medical diagnosis is the set-covering theory. This theory can be stated as the following mathematical formulation:
Let 
be a set of symptoms and let 
be a set of diseases. Let also 
be a binary relation associating each disease in 
with symptoms in 
that are caused by that disease. 
can then be considered to be a knowledge base. Associate with each symptom 
, a set of diseases 
that cause the symptom, and associate with each disease 
, a set of symptoms 
that it causes, called the profile for that disease. Let also 
, 
for sets 
and 
. Therefore, the task of diagnosis can be stated as the search for a set of diseases that, ‘explain’ a set of symptoms exhibited by the patient. By viewing this problem in the context of set theory, a disease 
explains a symptom 
if it causes it, i.e. 
. A differential diagnosis can thus be obtained for a set 
of symptoms, by selecting all diseases that explain a symptom in 
, i.e. the set 
. For example, given the knowledge base:
the set of all diagnoses 
for the set 
are:
This set is the least refined explanation that we can produce using the knowledge base, and in general is not a particularly helpful one. A set of disease 
is said to be an explanation of a diagnostic problem with observed symptoms 
if 
and it satisfies some additional criteria. Various criteria, in particular so-called criteria of parsimony, are in use. The basic idea is that among the various diagnoses of a set of symptoms, those that satisfy certain criteria of parsimony are more likely than others. Some of these criteria are:
-
Minimal cardinality: a diagnosis is an explanation if and only if it contains the minimum number of elements among all diagnoses;
-
Irredundancy: a diagnosis is an explanation if and only if no proper subset of is a diagnosis;
-
Relevance: a diagnosis is an explanation if and only if ;
-
Most probable diagnosis: a diagnosis is an explanation if and only if
According to above criteria, the explanation of our example will reduce to:
