RECURSIVE
BAYESIAN NETS FOR PREDICTION, EXPLANATION
AND CONTROL IN CANCER SCIENCE
A Position Paper
Lorenzo Casini, Phyllis McKay Illari, Federica Russo and Jon Williamson
Philosophy, University of Kent, Canterbury, U.K.
Keywords:
Bayesian network, Recursive Bayesian network, Prediction, Explanation, Control, Mechanism, Causation,
Causality, Cancer, DNA damage response.
Abstract:
The Recursive Bayesian Net formalism was originally developed for modelling nested causal relationships. In
this paper we argue that the formalism can also be applied to modelling the hierarchical structure of physical
mechanisms. The resulting network contains quantitative information about probabilities, as well as qualitative
information about mechanistic structure and causal relations. Since information about probabilities, mecha-
nisms and causal relations are vital for prediction, explanation and control respectively, a recursive Bayesian
net can be applied to all these tasks.
We show how a Recursive Bayesian Net can be used to model mechanisms in cancer science. The highest level
of the proposed model will contain variables at the clinical level, while a middle level will map the structure
of the DNA damage response mechanism and the lowest level will contain information about gene expression.
1 INTRODUCTION
Bayesian networks were originally developed to
model probabilistic and causal relationships (Pearl,
1988). In the last two decades Bayesian nets have
become the model of choice for making quantitative
predictions and for deciding which variables to inter-
vene on in order to control variables of interest. Thus
a Bayesian net can be used to answer questions such
as: given that patient has has treatment t what is the
probability P(r|t) that their cancer will recur in the
next 5 years? and: on which variables should we in-
tervene in order to minimise the probability of recur-
rence? Causal information is important here because
it is only worth intervening on the causes of recur-
rence, not on other variables which might be indica-
tors or evidence of recurrence.
The causal structure modelled by a Bayesian net
can also help answer certain simple explanatory ques-
tions, such as: what was the chain of events that led
up to the recurrence of the patient’s cancer? But
often we want to be able to offer explanations, not
in this backwards, ætiological sense, but in a down-
wards, mechanistic sense. In order to answer how did
the patient’s cancer recur? we may need to specify
the lower-level activities of the relevant cancer mech-
anism and the corresponding cancer response mecha-
nisms. To answer such explanatory questions a model
needs to represent the relevant mechanisms, including
their hierarchical organisation.
Bayesian nets have been extended to model hier-
archy in a number of ways. For example, recursive
Bayesian multinets model context-specific indepen-
dence relationships and decisions (Pe
˜
na et al., 2002),
recursive relational Bayesian networks model rela-
tional structure and more complex dependence re-
lationships (Jaeger, 2001), object-oriented Bayesian
networks can simplify the structure of large and com-
plex Bayesian nets (Koller and Pfeffer, 1997), hierar-
chical Bayesian networks offer a very general means
of modelling arbitrary lower-level structure (Gyftodi-
mos and Flach, 2002) and recursive Bayesian net-
works were developed to model nested causal rela-
tionships (Williamson and Gabbay, 2005). In this pa-
per we shall show how recursive Bayesian networks
can also be used to model mechanisms, thus provid-
ing an integrated modelling formalism for prediction,
explanation and control. This is important from the
AI perspective of providing models that can be used
to answer a variety of queries in decision support sys-
tems, but also from the bioinformatics perspective of
233
Casini L., McKay Illari P., Russo F. and Williamson J. (2010).
RECURSIVE BAYESIAN NETS FOR PREDICTION, EXPLANATION AND CONTROL IN CANCER SCIENCE - A Position Paper.
In Proceedings of the First International Conference on Bioinformatics, pages 233-238
DOI: 10.5220/0002744902330238
Copyright
c
 SciTePress

providing models that can integrate a variety of data
sources with the more qualitative knowledge of the
basic science involved.
In §2 we introduce the recursive Bayesian network
formalism and show how it can be used to model
mechanisms. In §3 we explain the current under-
standing of mechanisms in the philosophical litera-
ture, introduce the relevant cancer science mecha-
nisms and show how the cancer science mechanisms
fit the philosophical characterisation of mechanisms.
In §4 we introduce the variables under consideration
and the data sources to be used to construct the model.
We summarise and outline future research in §5.
2 RECURSIVE BAYESIAN NETS
Recursive Bayesian networks (RBNs) were origi-
nally developed in (Williamson and Gabbay, 2005) to
model nested causal relationships such as [smoking
causing cancer] causes tobacco advertising restric-
tions which prevent smoking which is a cause of can-
cer. In this section we introduce the RBN formalism
in the context of modelling mechanisms rather than
nested causality.
A Bayesian net consists of a finite set V =
{V
1
, . . . , V
n
} of variables, each of which takes finitely
many possible values, together with a directed acyclic
graph (dag) whose nodes are the variables in V , and
the probability distribution P(V
i
|Par
i
) of each vari-
able V
i
conditional on its parents Par
i
in the dag.
These are linked by the Markov Condition which
says that each variable is probabilistically indepen-
dent of its non-descendants, conditional on its par-
ents, written V
i
⊥⊥ ND
i
| Par
i
. A Bayesian net deter-
mines a joint probability distribution over its nodes
via P(v
1
···v
n
) =
∏
n
i=1
P(v
i
|par
i
) where v
i
is an as-
signment V
i
= x of a value to V
i
and par
i
is the assign-
ment of values to its parents induced by the assign-
ment v = v
1
···v
n
. In a causally-interpreted Bayesian
net or causal net, the arrows in the dag are interpreted
as direct causal relations (Williamson, 2005) and the
net can be used to infer the effects of interventions as
well as make probabilistic predictions (Pearl, 2000);
in this case the Markov Condition is called the Causal
Markov Condition.
A recursive Bayesian net is a Bayesian net de-
fined over a finite set V of variables whose values may
themselves be RBNs. A variable is called a network
variable if one of its possible values is an RBN and
a simple variable otherwise. A standard Bayesian net
is an RBN whose variables are all simple. An RBN X
that occurs as the value of a network variable in RBN
Y is said to be at a lower level than Y ; variables in Y
are the direct superiors of variables in X while vari-
ables in the same net are peers. If an RBN contains
no infinite descending chains—i.e., if each decending
chain of nets terminates in a standard Bayesian net—
then it is well-founded. We restrict our attention to
well-founded RBNs here.
To take a very simple example, consider an RBN
on V = {A, B}, where A is kind of tumour which takes
two possible values 0 and 1 while B is survival after
5 years which takes two possible values yes and no.
The corresponding Bayesian net is:
µ´
¶³
A
-
µ´
¶³
B
P(A), P(B|A)
Suppose B is a simple variable but that A is a net-
work variable, with each of its two values denoting
a lower-level (standard) Bayesian network that rep-
resents connections between gene expression levels
in the corresponding kind of tumour. When A is as-
signed value 0 we have a net a
0
with no dependence
between gene expression levels C and D:
µ´
¶³
C
µ´
¶³
D
P
a
0
(C), P
a
0
(D)
On the other hand, under assignment a
1
, C and D are
dependent:
µ´
¶³
C
-
µ´
¶³
D
P
a
1
(C), P
a
1
(D|C)
Since these two lower-level nets are standard
Bayesian nets the RBN is well-founded and fully de-
scribed by the three nets.
Since an RBN is a Bayesian net, the Markov Con-
dition is imposed. But RBNs are subject to a further
condition, the Recursive Markov Condition, which
says that each variable is probabilistically indepen-
dent of those variables that are neither its inferiors
nor peers, conditional on its direct superiors, written
V
i
⊥⊥ NIP
i
| DSup
i
. Let V = {V
1
, . . . , V
m
} (m ≥ n)
be the set of variables of an RBN closed under the
inferiority relation: i.e., V contains the variables in
V , their direct inferiors, their direct inferiors, and so
on. Let N = {V
i
1
, . . . , V
i
k
} ⊆ V be the network vari-
ables in V . For each assignment n = v
i
1
, . . . , v
i
k
of
values to the network variables we can construct a
standard Bayesian net, the flattening of the RBN with
respect to n, denoted by n
↓
, by taking as nodes the
simple variables in V plus the assignments v
i
1
, . . . , v
i
k
to the network variables (these can be thought of as
BIOINFORMATICS 2010 - International Conference on Bioinformatics
234

variables that can only take one possible value, i.e.,
constants), and including an arrow from one vari-
able to another if the former is a parent or direct
superior of the latter in the original RBN. The con-
ditional probability distributions are constrained by
those in the original RBN: P(V
i
|Par
i
∪ DSup
i
) must
match the P
v
i
j
(V
i
|Par
i
) given in the RBN. If each
variable has at most one direct superior in the RBN
then this will uniquely determine the required distri-
bution P(V
i
|Par
i
∪ DSup
i
); in other cases we follow
(Williamson and Gabbay, 2005, §5) and take the dis-
tribution to be that, from all those that satisfy the con-
straints, which has maximum entropy. The Markov
Condition holds in the flattening because the Markov
Condition and Recursive Markov Condition hold in
the RBN.
In our example, for assignment a
0
of network vari-
able A we have the flattening a
↓
0
:
µ´
¶³
a
0
-
?
H
H
H
H
Hj
µ´
¶³
B
µ´
¶³
C
µ´
¶³
D
with probability distributions P(a
0
) = 1, P(B|a
0
) de-
termined by the top level of the RBN and with
P(d
1
|a
0
) = P
a
0
(d
1
) and similarly for d
0
, c
0
and c
1
.
The flattening with respect to assignment a
1
is:
µ´
¶³
a
1
-
?
H
H
H
H
Hj
µ´
¶³
B
µ´
¶³
C
-
µ´
¶³
D
Again P(d
1
|c
1
a
1
) = P
a
1
(d
1
|c
1
) etc. In each case
the required conditional distributions are fully deter-
mined by the distributions given in the original RBN.
As long as certain consistency requirements are
satisfied (Williamson and Gabbay, 2005, §4), the flat-
tenings suffice to determine a joint probability dis-
tribution over the variables in V via P(v
1
···v
m
) =
∏
m
i=1
P(v
i
|par
i
dsup
i
) where the probabilities on the
right-hand side are determined by a flattening induced
by v
1
···v
m
.
With a joint distribution the model can be
used for prediction. For example, the proba-
bility that D is expressed at level 1 and that
the patient will survive 5 years is P(b
1
d
1
) =
P(a
0
b
1
d
1
) + P(a
1
b
1
d
1
) = P(b
1
|a
0
)P(a
0
)P
a
0
(d
1
) +
P(b
1
|a
1
)P(a
1
)(P
a
1
(d
1
|c
1
)P
a
1
(c
1
) + P
a
1
(d
1
|c
0
)P
a
1
(c
0
)).
More than that, if at each level the arrows in the
RBN can be interpreted causally then the model
can be used for control and ætiological explanation:
one might cite cancer type 0 as the reason a patient
survived 5 years. If the inter-level relations match
that of mechanistic composition then the model can
be used for mechanistic explanation. Thus the gene
expression levels C = 0 and D = 1 and the link
between the two might explain survival. And one can
use an RBN to reason across levels: by intervening on
expression level D one might increase the probability
of survival.
3 MECHANISMS IN
PHILOSOPHY OF SCIENCE
AND IN CANCER SCIENCE
The main thesis of this paper is that recursive
Bayesian are legitimate descriptions of physical
mechanisms, capable of modelling interesting aspects
of those mechanisms. The aim of this formal model is
to combine a qualitative description of the hierarchi-
cal and causal organization of a multi-field and multi-
level cancer mechanism with quantitative information
about the strengths of the causal and hierarchical con-
nections. In this section we examine the philosophical
progress on the question of what a physical mecha-
nism is, and describe how this validates our thesis.
The current dominant philosophical characteriza-
tion of a mechanism is: ‘Mechanisms are entities and
activities organized such that they are productive of
regular changes from start or set-up to finish or termi-
nation conditions’ (Machamer et al., 2000, p. 3). For
alternative accounts see: (Glennan, 2002, p. S344);
(Bechtel and Abrahamsen, 2005, p.423).
When considering the mechanisms of cancer, the
entities are the objects involved, such as people, tu-
mours, cells, DNA molecules, and so on. Activities
are just the things these objects do, such as survive,
grow, replicate, damage and be damaged, repair and
be repaired, and so on. So the mechanisms of cancer
can be described in terms of entities and activities:
people surviving or not, tumours growing or shrink-
ing, and DNA being damaged and being repaired.
Organization is a crucial feature of such mecha-
nisms: ‘The organization of these entities and activ-
ities determines the ways in which they produce the
phenomenon’ (Machamer et al., 2000, p. 3). See also
(Bechtel and Abrahamsen, 2005, p.435) and (Darden,
2002, p. S355).
This is again true of cancer, with the complex or-
ganization of the human body, cells in the tumour, and
all the multiple cellular mechanisms, vital to the pro-
duction of cancer.
It is now recognised that such mechanisms are
usually hierarchical. As (Machamer et al., 2000,
RECURSIVE BAYESIAN NETS FOR PREDICTION, EXPLANATION AND CONTROL IN CANCER SCIENCE - A
Position Paper
235

p. 13) write: ‘Mechanisms occur in nested hierar-
chies and the descriptions of mechanisms in neuro-
biology and molecular biology are frequently multi-
level. . . . lower level entities, properties, and activities
are components in mechanisms that produce higher
level phenomena’. That is, mechanisms involve inter-
nal structure, often several levels of internal structure.
The mechanisms of cancer involve just such hi-
erarchical levels. There is the level of the person,
their socioeconomic background, and their lifestyle
choices like diet, exercise and smoking. Then there is
the level of the tumour, particularly its growth rates,
containment, blood supply and so on. Then the cells
in the tumour have particular properties. Within the
cell itself, we are increasingly able to distinguish be-
tween the levels of gene expression, RNA and pro-
teins. All these levels affect the process of the cancer,
and the patient’s prognosis.
It is also recognised that explaining some phe-
nomenon requires finding the mechanism responsible
for that phenomenon using empirical work from mul-
tiple scientific fields. Craver discusses this with re-
gard to neuroscience: ‘The central idea is that neu-
roscience is unified not by the reduction of all phe-
nomena to a fundamental level, but rather by using
results from different fields to constrain a multilevel
mechanistic explanation’—see (Craver, 2007, p. 231).
See also (Russo, 2008) on social mechanisms, and
(McKay-Illari and Williamson, 2009) on explanation
for the spread of HIV. We can call these multi-field
mechanisms as they are modelled on the basis of dif-
ferent kinds of data, for instance molecular, genetic,
chemical, and environmental.
Treating cancer involves integrating information
from multiple fields, each studying one of the hierar-
chical levels we have described. Socioeconomic, clin-
ical, genomic, transcriptomic and proteomic data are
all known to be relevant to therapy choice and prog-
nosis.
Recursive Bayesian nets are legitimate descrip-
tions of physical mechanisms, provided that intra-
and inter-level relations are compatible with avail-
able theoretical knowledge. If so, RBN intra-level
(causal) relations among peer variables stand for
mechanisms and RBN inter-level relations stand for
decompositions of network variables into constituent
sub-mechanisms. Accordingly, simple and network
variables can model entities (or sets of these entities)
in their various states; and RBN causal relations can
model interactions and influences among these enti-
ties, i.e., activities.
Current philosophical work on biological mech-
anisms tends to cover purely qualitative aspects
of mechanisms ((Russo, 2008) discusses, however,
quantitative modelling of social mechanisms). Our
work allows the possibility of adding a quantitative
dimension: the probabilities within an RBN quan-
tify the strengths of the causal connections and lead
to a joint probability distribution over all the vari-
ables in the network. Such a quantitative descrip-
tion of the mechanism is vital for the task of predic-
tion, which requires determining the outcomes that
are most probable given available evidence. Since
causal information is required for control and mech-
anistic information for explanation, the RBN formal-
ism offers the prospect of multiple uses—prediction,
explanation and control—as well as the capacity to
integrated different kinds of evidence and evidence
from different fields.
4 CANCER APPLICATION:
VARIABLES AND DATA
SOURCES
In the last decade the human genome project and tech-
nological breakthroughs such as those in microarray
technology and mass spectrometry-based proteomics
have had a big impact on cancer research. They have
led to a vast increase in available data, from genomics,
transcriptomics, proteomics and metabolomics. Tra-
ditional diagnostics is under strain, and there is in-
creasing need for biomedical decision support tools.
Bayesian nets are an obvious choice for such tools,
but the prospect of being able to include hierarchy in
such Bayesian nets is exciting for cancer research. Hi-
erarchy is important to cancer research since it is still
unclear how the DNA, RNA, protein and metabolic
levels interact to produce cancer and affect prognosis.
Our formalism can be applied to build a model
using TCGA data (clinical patient data) and NCI-60
(cell line data) integrating the following variables:
Top level: Clinical. The top level includes the fol-
lowing recursive variables. The first recursive vari-
able kind of tumour. (Since most studies are focused
on single tumour types, only information on sub-types
will be available in a single study.) Each different sub-
type of tumour is a value of this variable, each value
corresponds to a lower-level Bayesian network of de-
pendencies between gene expression levels. If data is
available, a second important recursive variable will
be metastasis. Metastasis present/absent will corre-
spond to lower-level nets of gene expression levels.
Finally, it is proposed that the model will also include
standard simple variables at the clinical level such as
therapy, age, and survival in months.
BIOINFORMATICS 2010 - International Conference on Bioinformatics
236

Mid-level: DNA Damage Response. A first recur-
sive variable will DNA damage response: success
and failure of the DNA damage response mechanism
will each be modelled by different lower-level DNA
damage response nets, which consider the expres-
sion levels of various DNA damage response genes
as a surrogate for DNA damage. A second recursive
variable will be type of DNA damage, with values
for single-strand damage, double-strand damage, and
damage from alkylating agents, each corresponding
to a lower-level gene expression net. There will prob-
ably be no direct measure of this, so a surrogate will
have to be used. There will also be simple variables
for therapy choice, assembly of repair agents, repair
success and apoptosis. Apoptosis surrogates could
take the form of expression levels of the Caspase-3
and Caspase-9.
Low-level. Gene expression data is now readily
available, and methylation status of the genes has
been collected in TCGA.
Quantitatively representing both causal and hier-
archical structure in a single model in this way allows
extra methods of manipulating vast amounts of as-yet
poorly-understood data.
5 CONCLUSIONS
We have introduced the recursive Bayesian network
formalism, extending it from the modelling of nested
causal relationships to the modelling of mechanisms.
We have discussed exactly how such networks can be
used to model mechanisms, thus providing an inte-
grated modelling formalism for explanation, predic-
tion and control. Further formal work is needed on
how to perform inference in the net.
We have discussed how this formalism can be ap-
plied to modelling cancer mechanisms, where hierar-
chy is ubiquitous, and vast amounts of data are in-
creasingly available. This model will be built, tested
and validated in the ensuing programme of research.
We have also shown how this kind of model can
add to the philosophical literature on mechanisms by
integrating a quantitative description of the interac-
tion between variables with the philosophically more
familiar structural description of hierarchical relations
between activities and entities.
There is promise for future theoretical work. By
treating hierarchical structure in a formally equivalent
way to causal structure, this formalism might allow
us to extend known methods for extracting unknown
causal structure from data to extracting unknown hi-
erarchical structure. This is an exciting possibility for
studying cancer, where both causal and hierarchical
structure are still to be discovered in the areas opened
up by new technology in the last decade.
ACKNOWLEDGEMENTS
We are grateful to the Leverhulme Trust and the
British Academy for supporting this research. We are
also grateful to Amos Folarin, May Yong and Sylvia
Nagl of the UCL Cancer Institute for valuable discus-
sions.
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