Modelling and Reasoning in Biomedical Applications with Qualitative Conditional Logic
Jonas Haldimann , Anna Osiak, Christoph Beierle Knowledge Based Systems FernUniversität in Hagen, Hagen, Germany
Abstract
Three case studies showing how biomedical knowledge can be modelled with conditionals
Evaluation of the examples with different established inference methods
Background
Conditional Logic
Syntax
Propositional language \(\mathcal{L}\)
A conditional: \((A \mid B) \) with \(A, B \in \mathcal{L}\).
Set of all conditionals: \((\mathcal{L} \mid \mathcal{L}) = \{ (A \mid B) \mid A, B \in \mathcal{L} \}\).
A conditional knowledge base \( \mathcal{R} \subseteq (\mathcal{L} \mid \mathcal{L}) \): A finite set of conditionals.
Semantic
Universe \( \Omega \): Set of all models (for the underlying signature)
E.g., for \(\Sigma = \{a, b\}\) we have \( \Omega_\Sigma = \{ab, a\bar b, \bar a b, \bar a \bar b \} \)
A ranking function: \( \kappa: \Omega \rightarrow \mathbb{N}_0\) with \(\kappa^{-1}(0) \neq \emptyset\)
The most plausible models have rank 0. Less plausible models have higher ranks.
Rank of a formula: \(\kappa(A) = \min_{\substack{\omega \in \Omega \\ \omega \models A}} \kappa(\omega)\)
A ranking function \(\kappa\) fulfills \((A \mid B) \), denoted \(\kappa \models (A \mid B) \) iff \(\kappa(AB) < \kappa(A\bar B)\).
A ranking function \(\kappa\) fulfills a conditional knowledge base \(\mathcal{R}\) iff \(\kappa\) fulfills every conditional in \(\mathcal{R}\).
Inference operators
P-entailment
A knowledge base \(\mathcal{R}\) p-entails \((B \mid A)\) iff every ranking function that fulfills \(\mathcal{R}\) fulfills \(B \mid A\)
System Z
\((B \mid A)\) is a system Z inference if the uniquely determined Pareto-minimal ranking function that fulfills \(\mathcal{R}\) fulfills \((B \mid A)\).
C-inference
\((B \mid A)\) is a sceptical c-inference if every c-representation of \(\mathcal{R}\) fulfills \((B \mid A)\).
\((B \mid A)\) is a credulous c-inference if at least one c-representation of \(\mathcal{R}\) fulfills \((B \mid A)\).
\((B \mid A)\) is a sceptical c-inference if at least one c-representation of \(\mathcal{R}\) fulfills \((B \mid A)\) and no c-representation of \(\mathcal{R}\) does not accept \((B \mid A)\).
C-inference can be refined by considering only minimal c-representations.
Modelled Scenarios
Mammals
Scenario
Some information about mammals we would like to model:
Mammals cam be divided into three major groups.
Most mammals are placentals, i.e., their embryos are nourished by a placenta. Placentals' offspring are born alive (viviparous).
Marsupials' are viviparous, but they usually do not develop a (complex) placenta.
Monotremes lay eggs. Hence, they do not have a placenta.
Knowledge Base
We use the signature \(\Sigma = \{m, v, c, e, k\}\) where
m is true if the animal is a mammal,
v if it is viviparous,
c if it has a placenta,
e if it is a marsupial, and
k if it is a monotreme.
The resulting knowledge base:
\((v \mid m)\)
Mammals are usually viviparous.
\((c \mid m)\)
Mammals usually have a placenta.
\((m \mid e)\)
Marsupials are mammals.
\((\neg c \mid e)\)
Marsupials usually do not have placentas.
\((m \mid k)\)
Monotremes are mammals.
\((\neg v \neg c \mid k)\)
Monotremes are neither viviparous nor have a placenta.
Evaluation
Query
Inf. mode
c-inference
p-entailment
system Z
expert opinion
all
cw min
sum min
ind min
\((v \mid e)\)
sk.
yes
yes
yes
yes
no
no
yes
ws.
yes
yes
yes
yes
cr.
yes
yes
yes
yes
\((c \mid e)\)
sk.
no
no
no
no
no
no
no
ws.
no
no
no
no
cr.
no
no
no
no
Malaria Tropica
Scenario
We want to model the following information about malaria infections.
Malaria is caused by an infection with Plasmodium falciparum. But not everyone who is infected gets sick.
Patients with the sickle cell allele (a genetic mutation) do not get sick.
Infected patients without the sickle cell allele usually get sick.
Patients usually do not have the sickle cell allele.
Patients with a chemoprophylaxe usually do not get sick, except if infected with a resistant pathogen.
Knowledge Base
We use the signature \(\Sigma = \{m, s, p, r\}\) where
m is true if the patient get sick with malaria,
s if he has the sickle cell allele,
p if he got a chemoprophylaxe, and
r if he is infected with a resistant malaria pathogen.
We assume that all considered patients are infected with the malaria pathogen.
\((\neg s \mid \top)\)
Patients usually do not have the sickle cell allele.
\((m \mid \neg s)\)
Infected patients without the sickle cell allele usually get sick.
\((\neg m \mid s)\)
Infected patients with the sickle cell allele usually do not get sick.
\((\neg m \mid p)\)
Infected patients with a chemoprophylaxis usually do not get sick.
\((m \mid pr)\)
Patients with a chemoprophylaxis that are infected with a resistant malaria pathogen usually get sick.
Evaluation
Query
Inf. mode
c-inference
p-entailment
system Z
expert opinion
all
cw min
sum min
ind min
\((m \mid \top)\)
sk.
yes
yes
yes
yes
yes
yes
yes
ws.
yes
yes
yes
yes
cr.
yes
yes
yes
yes
\((m \mid rp)\)
sk.
yes
yes
yes
yes
yes
yes
yes
ws.
yes
yes
yes
yes
cr.
yes
yes
yes
yes
\((m \mid rsp)\)
sk.
no
no
no
no
no
yes
no
ws.
no
no
no
no
cr.
no
no
no
no
Chronic Myeloid Leukemia
Scenario
Chronic Myeloid Leukemia (CML) is one of the four common forms of leukaemia and is caused by a specific genetic defect.
We want to model the following information.
Most cases of CML are caused by a BCR-ABL translocation.
Patients with CML caused by the BCR-ABL translocation have good long-term survival rates.
Some cases of CML are atypical (aCML). This cases are not caused by a BCR-ABL translocation.
aCML can be treated with a hematopoetic stem cell transplantation (HSCT).
Recipients of a HSCT can in seldom cases suffer from a severe form of the Graft-versus-Host-Disease (GvHD).
Those patients do not have good long-term survival rates.
Knowledge Base
We use the signature \(\Sigma = \{c, a, b, g, m, r\}\) where
c is true if the patient has CML,
a if he has aCML,
b if he has the BCR-ABL translocation,
g if he has good chances to survive the CML,
m if he gets a HSCT, and
r if he suffers from a severe form of GvHD.
\((b \mid c)\)
CML is usually caused by a BCR-ABL translocation.
\((g \mid b)\)
Patients with a BCR-ABL translocation usually have good survival chances.
\((c \mid a)\)
aCML is a form of CML.
\((\neg b \mid a)\)
aCML usually coincides with no BCR-ABL translocation.
\((g \mid m)\)
Patients getting a HSCT usually have good survival chances.
\((\neg g \mid mr)\)
Patients getting a HSCT and suffering from severe GvHD usually have poor survival chances.