Probabilistic Logic Learning
Luc De Raedt
Institut für Informatik, Albert-Ludwigs-University,
Georges-Koehler-Allee, Building 079, D-79110
Freiburg, Germany
deraedt@informatik.uni-freiburg.de
ABSTRACT
The past few years have witnessed an significant interest in
probabilistic logic learning, i.e. in research lying at the intersection
of probabilistic reasoning, logical representations,
and machine learning. A rich variety of different formalisms
and learning techniques have been developed. This paper
provides an introductory survey and overview of the stateof-the-art
in probabilistic logic learning through the identification
of a number of important probabilistic, logical and
learning concepts.
Keywords
data mining, machine learning, inductive logic programming,
multi-relational data mining, uncertainty, probabilistic
reasoning
1. INTRODUCTION
One of the central open questions in data mining and artificial
intelligence, concerns probabilistic logic learning, i.e.
the integration of relational or logical representations, probabilistic
reasoning mechanisms with machine learning and
data mining principles. In the past few years, this question
has received a lot of attention and various different approaches
have been developed, cf. [82; 41; 88; 66; 73; 45; 60;
61; 57; 81; 52; 77; 2; 55; 87]. This paper provides an introductory
survey and overview of those developments that lie
at the intersection of logical (or relational) representations,
probabilistic reasoning and learning, cf. Figure 1. While doing
so, we identify some of the key concepts and techniques
underlying probabilistic logic learning. This, in turn, we
hope, should allow the reader to appreciate the differences
and commonalities between the various approaches and at
the same time provide insight into some of the remaining
challenges in probabilistic logic learning.
Before starting our survey, let us specify what we mean by
probabilistic logic learning. The term probabilistic in our
context refers to the use of probabilistic representations and
reasoning mechanisms grounded in probability theory, such
as Bayesian networks [78; 10; 47], hidden Markov models [85;
84; 22] and stochastic grammars [22; 64]. Such representations
have been successfully used across a wide range of applications
and have resulted in a number of robust models for
reasoning about uncertainty. Application areas include genetics,
computer vision, speech recognition and understand-
Kristian Kersting
Institut für Informatik, Albert-Ludwigs-University,
Georges-Koehler-Allee, Building 079, D-79110
Freiburg, Germany
kersting@informatik.uni-freiburg.de
Probability Logic
Learning
Probabilistic
Logic
Learning
Figure 1: Probabilistic Logic Learning as the intersection of
Probability, Logic, and Learning.
ing, diagnostic and troubleshooting, information retrieval,
software debugging, data mining and user modelling.
The term logic in the present overview refers to first order
logical and relational representations such as those studied
within the field of computational logic. The primary advantage
of using first order logic or relational representations
is that it allows one to elegantly represent complex situations
involving a variety of objects as well as relations between
the objects. Consider e.g. the problem of building a
logical troubleshooting system of a set of computers. Computers
are complex objects which are themselves composed
of several complex objects. Furthermore, assume that even
though the overall structure of the computers can be the
same, their individual components can be highly different.
Such situations can be elegantly modelled using relational
or first order logic representations. E.g., we could use the
relation ram(r1, c1) to specify that computer c1 has a RAM
component of type r1, and cpu(p2, c1) and cpu(p3, c1) to denote
that c1 is a multi-processor machine with two cpu’s p2
and p3. In addition to these base (or extensional) relations,
one can also add virtual (or intensional) relations to describe
generic knowledge about the domain. E.g., consider the rule
multiprocessor(C) :− cpu(P1, C), cpu(P2, C), P1 �= P2
which defines the concept of a multi-processor machine (as
a machine that possesses two different cpu’s). Representing
the same information employing a propositional logic would
require one to specify for each machine a separate model.
The term learning in the context of probabilistic logic refers
to deriving the different aspects of the probabilistic logic
on the basis of data. Typically, one distinguishes various
learning algorithms on the basis of the given data (fully or
partially observable variables) or on the aspect being learned
(the parameters of the probabilistic representation or its logical
structure). The motivation for learning is that it is of-
SIGKDD Explorations. Volume 2, Issue 2 - page 1
ten easier to obtain data for a given application domain and
learn the model than to build the model using traditional
knowledge engineering techniques.
So, probabilistic logic learning aims at combining its three
underlying constituents: learning and probabilistic reasoning
within first order logic representations. The recent interest
in probabilistic logic learning may be explained by the
growing body of work that has addressed pairwise intersections
of these domains:
- Indeed, various techniques for probabilistic learning such
as gradient-based methods, the family of EM algorithms
or Markov Chain Monte Carlo methods have been developed
and exhaustively investigated in different communities,
such as in the Uncertainty in AI community for
Bayesian networks and in the Computational Linguistic
community for Hidden Markov Models. These techniques
are not only theoretically sound, they have also resulted
in entirely new technologies for, and revolutionary novel
products in computer vision, speech recognition, medical
diagnostics, troubleshooting systems, etc. Overviews of
and introductions to probabilistic learning can be found
in [20; 84; 44; 8; 48; 22; 64].
- Inductive Logic Programming and multi-relational data
mining studied logic learning, i.e. learning and data mining
within first order logical or relational representations.
Inductive Logic Programming has significantly broadened
the application domain of data mining especially in bioand
chemo-informatics (cf. the other papers in this volume)
and now represent some of the best-known examples
of scientific discovery by AI systems in the literature.
Overviews of inductive logic learning and multi-relational
data mining can be found in this volume and in [69; 23].
- Early work on incorporating probabilities into logic programs
was done by Clark and McCabe [9] and by
Shapiro [97]. Subsequently, researchers such as Nilsson
[75], Halpern [43], Ng and Subrahmanian [71], and
Poole [82] have studied probabilistic logics from a knowledge
representational perspective. The aim of this research
is more a probabilistic characterization of logic than
suitable representations for learning.
Finally, in the past few years there have been some important
attempts that address the full problem of probabilistic
logic learning [59; 30; 67; 31; 14; 53; 51; 92; 68].
The goal of this paper is to provide an introduction and
overview to these works. In doing so, we restrict our attention
to those developments that address the intersection of
probabilistic logic learning, cf. Figure 1. Consequently, we
will not discuss interesting work lying on boundaries such
as Flach and Lachiche’s work on Naïve Bayes for structured
terms [27; 62] and Craven and Slattery’s work on combining
Naïve Bayes with a relational rule learner [11]. Giving an
overview of the pairwise intersections (let alone the underlying
domains) would lead us too far (as it would require a
book instead of a paper) and is therefore beyond the scope
of this paper. Because probabilistic logic learning involves
three domains, one can approach this topic from various
directions. In this paper, motivated by the topic of this
special issue, we start from a multi-relational and inductive
logic programming perspective. The terminology and
notation employed throughout this paper is based on computational
logic rather than on relational databases because
alarm :− burglary, earthquake.
johncalls :− alarm.
marycalls :− alarm.
Figure 2: The alarm program.
the relational model is not expressive enough to model the
logical component of many of the probabilistic logical representations
(e.g., [88; 66; 90; 52]). Nevertheless, we have
attempted to be self-contained and to reduce the required
logical machinery and background as much as possible.
The paper is organized as follows: Sections 2 and 3 introduce
the key concepts underlying computational logic and probabilistic
reasoning formalisms, respectively; Section 4 then
studies first-order probabilistic logics, i.e., how probabilistic
reasoning and computational logic can be combined; afterwards,
Section 5 then surveys various approaches to learning
within those probabilistic logics, and, finally, in Section 6,
we conclude.
2. INTRODUCTION TO LOGIC
As logic programs will be used throughout this paper as
the underlying knowledge representation framework, we
now briefly introduce their key concepts. Afterwards, we
show how state-of-the-art probabilistic representations employ
these concepts.
2.1 Logic
The choice of logic programs as the underlying representation
is motivated by its expressive power and generality.
Indeed, logic programs allow us to represent relational
databases, grammars and also programs. This will turn out
to be useful when discussing commonalities and differences
in the probabilistic relational or logical representations later
on. In addition, logic programs are well-understood, have a
clear semantics and are wide-spread.
Three example logic programs are given in Figures 2, 3
and 4.b. The first one specifies some regularities in the
alarm program (inspired on Judea Pearl’s famous example
for Bayesian networks). The genetic program is a simple
model of Mendel’s law applied to the color of plants. Finally,
the grammar example encodes a context-free grammar. The
corresponding type of logic program is known under the term
”definite clause grammar”.
Let us now introduce some terminology. The
predicates in the above examples include
alarm/0, burglary/0, earthquake/0, . . . , pc/2, mc/2, pt/2,
mother/2, father/2, s/2, vp/2, . . . Their arity (i.e., the number
of arguments) has been listed explicitly. Predicates with
arity 0, such as alarm/0 are sometimes called propositions.
Furthermore, 1, 1m, 1mf, . . . , g, green, . . . , dog, barks, . . . are
constants and P, M, X, C, S, T, . . . are variables. Constants and
variables are also terms. In addition, there exist structured
terms such as [barks], shorthand for [ ](barks, nil), which
contains the list functor [ ]/2, and the terms barks and
nil (the empty list). Constants are often considered as
functors of arity 0. Atoms are predicate symbols followed
by the necessary number of terms, e.g., v([barks|T], T) and
pt(P, white) . We are now able to define the key concept of
a definite clause. Definite clauses are formulas of the form
A :− B1, . . . , Bm where A and the Bi are atoms and where all
variables are understood to be universally quantified. E.g.,
SIGKDD Explorations. Volume 2, Issue 2 - page 2
mc(P, X) :− mother(P, M), mc(M, X), pc(M, Y).
mc(P, X) :− mother(P, M), mc(M, Y), pc(M, X).
pc(P, X) :− father(P, F), mc(F, X), pc(F, Y).
pc(P, X) :− father(P, F), mc(F, Y), pc(F, X).
mc(1mf, r). mother(1, 1m).
pc(1mf, g). father(1, 1f).
mc(1mm, g). mother(1m, 1mm).
pc(1mm, r). father(1m, 1mf).
mc(1f, g).
pc(1f, g).
pt(P, green) :− mc(P, g), pc(P, g).
pt(P, red) :− mc(P, r).
pt(P, red) :− pc(P, r).
S → NP, VP.
VP → V.
Figure 3: The genetic program.
NP → Det, Noun.
Det → [the].
Noun → [dog].
V → [barks].
s(S, T) :− np(S, S1), vp(S1, T).
vp(S, T) :− v(S, T).
np(S, T) :− det(S, S1), noun(S1, T).
det([the|T], T).
noun([dog|T], T).
v([barks|T], T).
(a) (b)
Figure 4: Context-free grammars: (a) A context-free grammar
and (b) the corresponding grammar program.
the definite clause pt(P, green) :− mc(P, g), pc(P, g) can be
read as “the phenotype of P is green if the m-chromosome
of P contains the gene g and the p-chromosome of P
contains the gene g. Because we will restrict attention to
definite clauses only in the present paper, we will omit the
term “definite” as long as no ambiguities arise. We call
pt(P, green) the head of the clause, and mc(P, g), pc(P, g) the
body. Clauses with an empty body, such as v([barks|T], T)
are called facts. A clause program (or logic program for
short) consists of a set of clauses. For a more thorough
introduction to logic programming we refer to [63; 26].
Within logic, two types of semantics can be distinguished:
a model-theoretic and a proof-theoretic one. Because this
distinction will also be useful from a probabilistic point of
view, let us briefly introduce these two concepts.
2.2 A Model Theoretic View
Consider the alarm program in Figure 2. In this example,
there are five propositions that occur, i.e. {alarm,
earthquake, marycalls, johncalls, burglary}. This is the
so-called Herbrand base HB. It consists of all facts that
one can construct with the predicate, constant and function
symbols in the program. The Herbrand base – in a sense
– specifies the set of all possible worlds described by the
program. Indeed, for the alarm program, there are 2 5 = 32
possible assignments of truth-values to the Herbrand base.
These assignments are sometimes called (Herbrand) interpretations
and they each describe a world.
From a model-theoretic point of view, a logic program restricts
the set of possible worlds. Indeed, if we interpret
the clause alarm :− burglary, earthquake as an implication,
then the interpretations in which both burglary and
earthquake are true, but alarm is not, do not satisfy this
clause. The idea is that if we accept the clause as an axiom,
then such worlds are impossible. More formally, we have
that a (Herbrand) interpretation I satisfies a clause c if and
only if for all substitutions θ such that body(c)θ ⊆ I also
head(c)θ ∈ I. The interpretation I is called a model of c if it
satisfies c. A substitution θ = {V1/t1, . . . , Vn/tn}, e.g. θ1 =
{P/1mm}, is an assignment of terms ti to variables Vi. Applying
a substitution θ to a term, atom or clause e yields the instantiated
term, atom, or clause eθ where all occurrences of
the variables Vi are simultaneously replaced by the term ti.
Consider e.g. the clause c pt(P, green) :− mc(P, g), pc(P, g)
and the substitution θ1. Applying θ1 to c yields the instantiated
clause cθ1 pt(1mm, green) :− mc(1mm, g), pc(1mm, g).
Furthermore, the reader may want to verify that the interpretation
{mc(1m, g), pc(1m, g)} is not a model for the clause
c. Finally, let us remark that an interpretation satisfies a
set of clauses if it satisfies all the clauses in the set.
The most important point about model theory – for our
purposes – is that it specifies which worlds (i.e., interpretations)
are possible with respect to (i.e., satisfy) a given
logical theory.
2.3 A Proof Theoretic View
Another way of looking at logic programs comes from proof
theory. From this perspective, a logic program can be used
to prove that certain atoms, goals (see below) or clauses
are logically entailed by the program. Consider, e.g., the
genetic program in Figure 3. In this program, the fact
pt(P, C) will be provable for any P and C whenever C is one
of the possible colors for the plant P according to the axioms
encoded in the program.
Proofs are typically constructed using SLD-resolution procedure,
which we now briefly introduce. More formally,
given a goal :− G1, ..., Gn and a clause G :− L1, ..., Lm such
that G1θ = Gθ, applying resolution results in the goal
:− L1θ, ..., Lmθ, G2θ, ..., Gnθ. A (successful) resolution refutation
of a goal :− G is then a sequence of resolution steps that
leads to the empty goal :− . A failure does not end in the
empty goal. For instance, in the genetic example, one can
prove that pt(1f, green) is true, because of the following
resolution derivation
:− pt(1f, green).
:− mc(1f, g), pc(1f, g).
:− pc(1f, g).
:−
Resolution is employed by many theorem provers (such as
Prolog). Indeed, when given the goal :− pt(1f, green), Prolog
would compute the above resolution derivation and answer
yes. Furthermore, when given the goal :− pt(1f, X),
Prolog would answer with the substitutions that make the
goal true, i.e., {X/green}. The set of possible resolution
steps for a given goal is captured in an SLD-tree. Two examples,
one in the genetic domain, and one in the grammar
domain are shown in Figure 5. At this point it is important
to see the connection between an SLD-derivation for
the grammar domain and a traditional derivation using the
context free grammar. Indeed, there is a one to one mapping
between these two concepts for our grammar.
SIGKDD Explorations. Volume 2, Issue 2 - page 3
:- s([the,dog,barks],[])
S/[the,dog,barks]
:- np([the,dog,barks],S1),vp(S1,[]).
:- det([the,dog,barks],S1’),noun(S1’,S1),vp(S1,[])
:- mother(1mm,M),
mc(M,r),
pc(M,Y).
:- father(1mm,F),
mc(F,r),
pc(F,Y).
S1’/[dog,barks]
:- noun([dog,barks],S1),vp(S1,[])
S1/[barks]
:- vp([barks,[])
:- v([barks,[])
:-
:- pt(1mm,red).
:- mc(1mm,r). :- pc(1mm,r).
:- mother(1mm,M),
mc(M,Y),
pc(M,r).
:- father(1mm,F),
mc(F,Y),
pc(F,r)).
Figure 5: Examples of SLD trees. Root nodes are shaded.
The upper tree is the SLD tree of querying the grammar
program with :− s([the, dog, barks], []). The lower tree
is the SLD tree of querying the genetic program with
:− pt(1mm, red). In both cases, there is only one refutation,
i.e. successful derivation.
There are thus two ways of viewing a logic program.
The first view, illustrated by the alarm program, is the
model-theoretic view. It determines what the possible
worlds (i.e. interpretations) are that satisfy the program.
The second view, illustrated by the genetic program, is the
proof-theoretic view. It determines what the possible proofs
are with respect to a given goal. Of course, these two views
are intimately connected. Indeed, for clause programs, we
have that a (ground) fact has a resolution refutation if and
only if it belongs to all (Herbrand) models of the program.
3. PROBABILISTIC LOGICS
The model theoretic and proof theoretic views of logic are
also useful from a probabilistic perspective. Indeed, many
of the present probabilistic representations can be regarded
as defining probabilities on possible worlds (e.g., Bayesian
networks) or on proofs (e.g., stochastic context free grammars).
As these probabilistic representations form the basis
for the probabilistic logics, we will now provide a short
overview to these. We will use P to denote a probability
distribution, e.g. P(X), and P to denote a probability
:-
burglary
alarm
earthquake
johncalls marycalls
Figure 6: The graphical structure of the alarm Bayesian
network.
value, e.g. P (X = x).
3.1 Probabilities on Possible Worlds
The most popular formalism for defining probabilities on
possible worlds is that of Bayesian networks. As an example
of a Bayesian network, consider Judea Pearl’s famous
alarm network graphically illustrated in Figure 6.
Formally speaking, a Bayesian network [78; 10; 47] is an
augmented, directed acyclic graph, where each node corresponds
to a random variable Xi (e.g., {alarm, earthquake,
marycalls, johncalls, burglary}) and each edge indicates
a direct influence among the random variables. We will
denote the set of parents of a node Xi by Pa(Xi), e.g.,
Pa(alarm) = {earthquake, burglary}. A Bayesian network
specifies a joint probability distribution P(X1, . . . , Xn) over
a fixed, finite set {X1, . . . , Xn} of random variables. As we
will – for simplicity – assume that the random variables are
all boolean, i.e., they can have the domain {true, false} 1 ,
this actually amounts to specifying a probability distribution
on the set of all possible interpretations. Indeed, in our
alarm example, the Bayesian network defines a probability
distribution over truth-assignments to {alarm, earthquake,
marycalls, johncalls, burglary}.
A Bayesian network stipulates the following conditional independency
assumption:
Each node Xi in the graph is conditionally independent
of any subset A of nodes that are not
descendants of Xi given a joint state of Pa(Xi),
i.e. P(Xi | A, Pa(Xi)) = P(Xi | Pa(Xi)).
For example, marycalls is conditionally independent of
earthquake and of burglarygiven a joint state of alarm. Because
of the conditional independence assumption, we can
write down the joint probability density as
P(X1, . . . , Xn) =
n�
P(Xi | Pa(Xi))
i=1
by applying the independency assumption to the chain
rule expression of the joint probability distribution. Therefore,
we associate with each node Xi of the graph the
conditional probability distribution P(Xi | Pa(Xi)),
denoted as cpd(Xi). The conditional probability
distributions in our alarm examples would include
P(alarm | earthquake, burglary), P(earthquake), etc.
They are illustrated in Table 1.
So, Bayesian networks define a probability distribution on
the possible worlds, i.e., the interpretations. Furthermore,
1 In general, the domains of a random variable can be continuous
or discrete as well, cf. [25].
SIGKDD Explorations. Volume 2, Issue 2 - page 4
P(burglary)
(0.001, 0.999)
P(earthquake)
(0.002, 0.998)
burglary earthquake P(alarm)
true true (0.95, 0.05)
true false (0.94, 0.06)
false true (0.29, 0.71)
false false (0.001, 0.999)
alarm P(johncalls)
true (0.90, 0.10)
false (0.05, 0.95)
alarm P(marycalls)
true (0.70, 0.30)
false (0.01, 0.99)
Table 1: The conditional probability distributions associated
with the nodes in the alarm network, cf. Figure 6. The
distributions are specified over {true, false}. When viewing
the child-parents pairs as clauses, the clauses in the alarm
program, cf. Figure 2, specify some regularities of the alarm
network. The corresponding conditional probability distributions
can be associated with the clauses.
the resulting probabilities induce a probability distribution
at the level of propositions and propositional formulae. Indeed,
reconsider the alarm example, and assume that we
are interested in the probability of P(alarm = true). This
probability can be obtained by marginalizing over the other
variables, i.e.,
P (a = t) = �
P (a = t, b = x1, e = x2, m = x3, j = x4)
x i∈{t,f}
where we have abbreviated the variables and states appropriately.
The probability of a propositional formula can be
obtained similarly, it is simply the sum of the probabilities
of the interpretations that are a model for the formula.
The key limitation of Bayesian networks is that they lack
the notion of an object. The possible worlds considered are
propositional. Using Bayesian networks, it is impossible to
model relations between objects, i.e., relational or logical
worlds. This problem is similar to that with traditional
propositional mining and learning systems.
3.2 Probabilities on Proofs
Because of the closeness of proofs of logical formulae
and derivations in grammar, we will start by discussing
stochastic grammars, see e.g. [1; 64]. Consider therefore
the stochastic context free grammar in Figure 7 (adapted
from [1]), where we have omitted many of the terminal symbols
for brevity. This context-free grammar is stochastic
in that every grammar rule has a probability associated to
it. Moreover, for every non-terminal symbol T the sum of
the probabilities of the rules with T on the left hand side
equals 1.0, e.g., for NP: 0.2 + 0.8 = 1.0. Stochastic contextfree
grammars define probabilities on derivations (or, when
mapped onto logic programs, on proofs). Indeed, one of the
proofs (with associated probabilities) of the sentence ”Rice
flies like sand” goes as follows:
:− S. 1.0
:− NP VP. 1.0
:− N N VP. 0.2
:− Rice N VP. 0.2 · 0.01
:− Rice flies VP. 0.2 · 0.01 · 0.01
(1.0) S → NP, VP.
(0.2) NP→ N, N.
(0.8) NP→ N.
(0.6) VP→ V, NP.
(0.4) VP→ V, PP.
(1.0) PP→ P, NP.
(0.01) N → [Rice].
(0.01) N → [flies].
. . .
(0.02) V → [like].
(0.02) V → [flies].
. . .
(0.05) P → [like].
. . .
Figure 7: An example of a stochastic context-free grammar.
The numbers associated with the grammar rules denote
probability values. Note that [Rice] does not denote a
variable but the string Rice.
:− Rice flies V NP. 0.2 · 0.01 · 0.01 · 0.6
:− Rice flies like NP. 0.2 · 0.01 · 0.01 · 0.6 · 0.02
:− Rice flies like N. 0.2 · 0.01 · 0.01 · 0.6 · 0.02 · 0.8
:− Rice flies like sand. 0.2 · 0.01 · 0.01 · 0.6 · 0.02 · 0.8 · 0.1
Each time a rule is applied in a proof, the probability of the
rule is multiplied with the overall probability. It is easy to
verify that this induces a probability distribution over all
possible proofs or successful derivations for the same nonterminal
symbols. Failures (if any) are only due to nonmatching
terminals which can be considered to have 0.0 as
probability label. Furthermore, the probabilities over the
proofs induce a probability over the accepted sentences for
each of the non-terminals. Indeed, consider again the sentence
”Rice flies like sand” for which there are two proofs,
one with probability p1 (as illustrated above) and another
one with probability p2. The sum of these probabilities specifies
the probability that a random sentence generated by
the non-terminal symbol S is indeed ”Rice flies like sand”.
These distributions are useful for various purposes. For instance,
in natural language processing, one may be interested
in either the most likely parse tree (or derivation) when
dealing with ambiguity, or the total probability that a particular
sentence is derived. Also, in bioinformatics, there
are numerous applications of stochastic grammars, most notably
of (hidden) Markov models [84], which are essentially
stochastic regular grammars. These are e.g. used to determine
how likely it is that a given protein belongs to some
fold [22].
From a computational and logical perspective, the key limitation
of stochastic context free grammars concerns their
expressive power. Indeed, context free grammars (and
therefore stochastic context free grammars) are not Turingequivalent,
i.e. they cannot be used as a programming language.
Thus it is useful to lift the underlying grammar representation
to that of logic programs, a Turing equivalent
language, cf. [66].
SIGKDD Explorations. Volume 2, Issue 2 - page 5
4. FIRST ORDER PROBABILISTIC
LOGICS
By now, everything is in place to introduce the key representational
frameworks that combine probabilistic reasoning
with logical or relational representations. Furthermore, in
the spirit of the above presentation, we can distinguish them
according to whether they define probabilities on interpretations
or on proofs.
4.1 Probabilistic Logical Models
The first class of representations extends Bayesian networks
with abilities to define probabilities on first order logical or
relational interpretations, cf. [82; 41; 73; 45; 57; 81; 31; 52].
Important predecessors of these logical Bayesian networks
discussed below, include the work by Breese, Charniak,
Bacchus, Goldman, Poole and Wellmann [40; 6; 4; 7; 39].
These predecessors were largely based on the idea of knowledge
based model construction (KBMC). In knowledge based
model construction, a knowledge base (sometimes in the
form of an annotated logic program) is used to describe sets
of probabilistic models. A query then results in constructing
a particular model that can be used to answer the query.
Many of these approaches can represent Bayesian networks.
However, they are not direct upgrades of Bayesian networks
in the same way that the work by e.g. Haddawy [41] is. This
may also explain why – to the authors’ knowledge – Bayesian
network learning techniques have not been applied to these
representations.
A central contribution in combining clausal logic with
Bayesian networks is due to Peter Haddawy [41]. It is based
on the observation that the structure of a Bayesian network
can essentially be represented as a set of (propositional)
clauses, where there will be one clause for each node in the
graph, cf. the alarm program and Bayesian network. This
observation allows one to upgrade Bayesian networks toward
first order logic by upgrading the corresponding propositional
clauses towards definite clauses. These clauses will
specify the so-called intensional part I, i.e. the overall regularities.
In addition, we will need an extensional part E,
which specifies the specific objects and situations under consideration.
The distinction between extensional and intensional
is akin to that made in database theory and computational
logic. E.g., in the genetic program, the extension
corresponds to the facts and the intension to the non-fact
clauses. Given an extensional part E and an intensional part
I, the central ideas are then
- to view the atoms in the Herbrand interpretation as random
variables 2 ; they will constitute the nodes in the induced
Bayesian network,
- to view the instantiated clauses H :− B1, . . . , Bm as defining
the probabilistic dependencies, i.e. the Bi will correspond
to parents of the node H; therefore, a conditional probability
distribution cpd needs to be associated to each clause.
Before continuing to discuss some problems and extensions,
let us first illustrate these ideas on our genetic example. In
doing, so let us slightly redefine the predicates pc/2, mc/2
2 More specifically, it only makes sense to talk about the
atoms that are logically entailed by the program, i.e. that
are true in all possible Herbrand interpretations, that is the
atoms belonging to the so-called least Herbrand model.
mc(P) :− mother(P, M), mc(M), pc(M).
pc(P) :− father(P, F), mc(F), pc(F).
mc(1mf). mother(1, 1m).
pc(1mf). father(1, 1f).
mc(1mm). mother(1m, 1mm).
pc(1mm). father(1m, 1mf).
mc(1f).
pc(1f).
pt(P) :− mc(P), pc(P).
Figure 8: The truncated genetic program.
pc(1mf)
mc(1mf)
father(1m,1mf)
mother(1m,1mm)
pc(1mm)
mc(1mm)
pt(1mf)
pc(1m)
mc(1m)
mother(1,1m)
father(1,1f)
pc(1f)
mc(1f)
pt(1mm)
pt(1m)
mc(1)
pc(1)
pt(1f)
pt(1)
Figure 9: The structure of the Bayesian network induced
by the truncated genetic program, see Figure 8. All nodes
nodes (with a non-empty set of parents) for the same predicate
share the same associated cpds.
and pt/2. We will now turn these two-place predicates into
predicates of arity 1, the idea being that pt(P) will be (in
state) true if and only if the phenotype of the person P is
green. The other predicates are dealt with in a similar way.
Together with the facts for mother/2 and father/2 (listed
in Figure 3), the truncated facts for mc/1 and pc/1 result in
the truncated genetic program as shown in Figure 8. This
program then induces a Bayesian network with the graphical
structure shown in Figure 9. All nodes (with a non-empty
set of parents) for the same predicate (i.e., mc/1, pc/1 and
pt/1) share the same local probability model. So, the cpd’s
for mc(1m) and mc(1mm) are the same.
There are some problems with this approach. Let us dicuss
the most notable ones. First, the Bayesian network obtained
is required to be acyclic. This is required because the semantics
of cyclic Bayesian networks are not well-defined.
Therefore, virtually all approaches require acyclicity or enforce
other restrictions that guarantee acyclicity.
Second, the domains of the random variables may not be
boolean, i.e., they may have more than two possible values,
or even be continuous. There are two ways to realize
this. First, one could simply allow the domains of the random
variables to be arbitrary, cf. [41; 52]. One side-effect is
that boolean or logical random variables get mixed up with
SIGKDD Explorations. Volume 2, Issue 2 - page 6
other ones. This may sometimes cause some confusion. Second,
one could simply add an extra argument to the atoms,
cf. [57; 73]. E.g., in the genetic domain, this would correspond
to the encoding used in Figure 3. The disadvantage
of this approach is that one needs to add extra constraints
that would state that the phenotype must be either red or
white, cf. [73]. This in turn results in a more complicated
inference engine.
Finally, it might be the case that there exist multiple ground
clauses with the same head. Indeed, consider the program
consisting of the following two clauses:
p :− q, r.
p :− s, t.
The first clause states that Pa(p) = {q, r}, the second one
that Pa(p) = {s, t}. This problem can also arise with
a program consisting of a single clause. Indeed, consider
p(X) :− q(X, Y), for which there may be multiple (true) instantiations
of q(X, Y) for a single X. Various solutions which
might even be combined have been developed for dealing
with this situation. First, one could introduce so-called
combination rules, cf. [73; 45; 52]. Combination rules are
functions that would – in the above case – take the corresponding
conditional probability distributions P(p | q, r)
and P(p | s, t) as inputs and would produce the desired
P(p | q, r, s, t) as output. Classical examples of combining
rules are noisy-or and noisy-and. For illustration purposes,
consider the clauses
cold. fever :− cold.
flu. fever :− flu.
malaria. fever :− malaria.
together with noisy-or as combining rule. The Bayesian
network induced is
cold
flu
malaria
fever
where noisy or ((cpd(fever :− flu), cpd(fever :− cold),
cpd(fever :− malaria)) is associated with fever, cf. [86,
page 444]. Then fever is false only if all parents are independently
false. The probability of fever being true then
depends on the parents which are true. More precisely,
P (fever = true) = 1 −
�
P (X = true)
X∈{cold,flu,malaria}
which is true
Second, one could employ an aggregate function as done in
probabilistic relational models (PRMs) [57; 81]. Aggregate
functions are well-known from databases; examples include
min, max, sum, average, count, etc. To illustrate their use
in the present context, reconsider the clause p(X) :− q(X, Y)
and consider that for {X = a} there are multiple substitutions,
let us say {Y = b, Y = c}, such that q(a, Y) holds. Then
{q(a, b), q(a, c)} are the two random variables that influence
p(a). Furthermore, these random variables will have values,
say v1 and v2. An aggregate function τ could then map
this set of values {v1, v2} onto a single value v. Let us use
the notation τ(q(a, Y)) to denote this value v. The random
variable p(a) thus depends on τ(q(a, Y)), which is of course
also a random variable. If multiple atoms appeared in the
clause, e.g., p(X) :− q(X, Y), r(X, Z), then each of the atoms
would be aggregated independently. The latter clause thus
represents the probabilistic dependence of p(X) on τ1(q(X, Y))
and τ2(r(X, Z)), i.e., the cpd associated with the clauses actually
specifies for each value in τ1(q(X, Y)) × τ2(r(X, Z)) a
distribution for p(X). The effect of aggregate functions is
that they reduce the information in a natural way. This is
advantageous because it reduces the size of the conditional
probability distributions in a natural way, but – on the other
hand – it may also result in a loss of information [46].
Above we introduced the principles underlying logical extensions
of Bayesian networks. Let us now briefly survey
some of the key representational frameworks that have been
developed around these principles.
As mentioned before, the first approach to directly upgrade
Bayesian networks to relational representations, seems due
to Peter Haddawy [41]. In his initial approach, the random
variables correspond to ground atoms. Furthermore, the
need for combining rules was alleviated by requiring that for
each random variable, there was at most one instantiation
of a clause having the random variable as the head.
In later work, Haddawy now together with Ngo [73], used
combination rules to deal with the latter problem, and also,
the values of the random variables were now written as
an extra argument of the atoms, which necessitates the
need to add constraints, cf. above. The language, called
probabilistic-logic programs (PLPs), was further extended
in the knowledge based model construction tradition by distinguishing
between logical and probabilistic conditions in
clauses. The former then specified the (logical) conditions
under which the probabilistic dependencies held. This increased
the expressive power and was certainly useful for
specifying probabilistic models by hand, but on the other
hand, it also significantly extended and therefore complicated
the description language. These complications may
also explain why – to the authors’ knowledge – no (structural)
learning results are known for this formalism. The
framework has been applied within medical domains [42;
72].
Building on Ngo and Haddawy’s work, Kersting and
De Raedt [52] introduced the framework of Bayesian logic
programs (BLPs). This framework can be viewed as a simplification
of that by Ngo and Haddawy in that BLPs employ
a minimal set of primitives such that BLPs have both definite
clause logic (i.e. pure Prolog) and Bayesian networks
as a direct special case. This in turn facilitates the learning,
cf. [53; 51; 54]. Bayesian logic programs again view the
atoms as the random variables. More precisely, the atoms
of the (least) Herbrand model constitute the set of random
variables of the induced Bayesian network. BLPs deal with
continuous random variables and employ a simpler form of
combination rule.
Finally, there is the quite popular formalism of probabilistic
relational models 3 (PRMs) [57; 81; 31; 33]. This formalism
3 Several other extensions of Bayesian network have been
developed [60; 58; 61; 80; 81] by Pfeffer et al. before introducing
probabilistic relational models. These extensions
have been termed frame-based and object-oriented, in which
the relational and logical issues are less direct. Therefore,
we base our discussion of probabilistic relational models on
SIGKDD Explorations. Volume 2, Issue 2 - page 7
does not use definite clause logic (i.e. pure Prolog) as the underlying
representational framework, but rather the entityrelationship
model. In PRMs, the idea is that the information
about one entity type is stored in one relation. E.g.,
in the genetic illustration, this could be realized using the
relation person(Person, MC, PC, PT). So each ground atom
can store information about multiple attributes and the dependencies
are defined at the level of these attributes. Two
types of dependencies are allowed: direct dependencies from
other attributes (for the same entity, e.g., the blood type of
a person could depend on the two chromosomes of that person),
or dependencies via so-called slot chains. Slot-chains
are binary relations (or binary projections of other relations)
that relate the attributes of one entity to that of other ones.
E.g., in the genetic example, the MC attribute of one person
would depend via the slot-chain mother(Person, Mother) on
the attributes MC and PC of the entity corresponding to the
Mother. Furthermore, in the basic PRM setting the relations
among entities (such as Mother or Father in the genetic example)
are assumed to be deterministic and given, cf. [33].
This implies that it would be impossible to specify that the
probability is 0.70 that Jef is the father of Mary. Nevertheless,
some partial solutions for dealing with probabilistic
relations have been developed, cf. [34; 31]. The genetic example
shows that – at the logical level – PRMs essentially
define via single clauses probabilistic dependencies between
the attributes of various entities. It also shows that probabilistic
dependencies between relations are harder to represent
and require special treatment. Within the more logical
approaches sketched above, these issues are dealt with in a
uniform manner because random variables are represented
by ground atoms. Indeed, the more logically oriented version
of the genetic example, introduced above, shows how
the person/4 relation in PRMs would be split-up (or ”normalized”)
into atoms that directly correspond to the random
variables. On the other hand, the advantages of PRMs are
that they (1) possess an elegant graphical representation,
see Figure 10, (2) may be more efficient to implement and
learn, and (3) can test the acyclicity requirement more easily.
Other differences with the above mentioned logical notation
include the use of aggregate functions to deal with the
multiple instantiations problem and the use of a purely relational
language, which does not allow for functors (needed
for representing infinite structures), but see [79]. PRMs have
been successfully applied to various problems such as relational
clustering [98], hypertext classification [37], selectivity
estimation within databases [38], and modelling identity
uncertainty [76], and led to impressive results within computational
biology [95; 93; 94].
Recently, Getoor et al. [31; 35; 32] have introduced stochastic
relational models (SRMs). At the syntactic level SRMs
are almost identical to PRMs but SRMs possess a different
semantics, which allow PRMs to answer statistical queries
such as ‘What is the probability that a randomly chosen
patient is elderly and has a contact who is a coworker?”
The key idea underlying SRMs is that probability distributions
are defined and used at an abstract level determined
by the database schema. At this level, one is not interested
in specific entities (such as specific patients, contacts
or co-workers) but rather of their properties (such as the
probability that a randomly chosen patient is elderly, or a
the presentation by Getoor et al. in [23].
Figure 10: Graphical representation of a probabilistic relational
model of the genetic domain.
randomly chosen contact is a coworker). Whereas PRMs
define dependencies among different entities via slot-chains,
SRMs define them via chains of joins. Many of the other
issues SRMs are quite similar to those in PRMs, which explains
why we will not provide any further details on SRMs
in this paper. For more information, see [31; 35; 32], who
present efficient algorithms to construct the desired Bayesian
network, to estimate (in particular the join) parameters and
to learn the dependency structure from a given database.
SRMs have also been used for selectivity estimation within
databases [31].
4.2 Probabilistic Proofs
To define probabilities on proofs, there are at least two different
options. They correspond to the approaches taken in
stochastic logic programs [24; 66; 12; 13; 15] and PRISMs
respectively [88; 90; 50; 89].
Let us start by introducing stochastic logic programs
(SLPs). Stochastic logic programs are a direct upgrade of
stochastic context free grammars. This is realized by associating
to each clause a probability value. Furthermore, as
in the case of stochastic context free grammars, the sum 4 of
the probabilities associated to clauses of the same predicate
equals 1.0. To illustrate stochastic logic programs, reconsider
the derivation of ”Rice flies like sand” but now with the
definite clause grammar (with the same associated probabilities).
Both give the same probabilities to the same (ground)
queries. However, stochastic context free grammars constitute
very simple stochastic logic programs. Failures are
only due to non-matching terminals. In general SLPs, (intensional)
clauses may cause already failures. Consider the
genetic example in Figure 3 and assume that we have associated
uniform probabilities to all clauses for the same
predicate. Under these assumption one could e.g. show
that the probability of the single proof of :− mc(1m, r) is
0.2 · 0.5 · 0.2 · 0.2 = 0.004. However, computing the probabilities
of all other ground atoms over mc/2 shows that they
do not sum to 1.0 but to 0.624 (see below) because some of
the refutations fail already at the clause level. Stochastic
logic programs account for failures by normalization. The
normalization constant of an atom over some predicate mc/2
is the sum of probabilities of all refutations of the most gen-
4 More general definitions of stochastic logic programs where
e.g. the probability values do not sum to 1.0 are discussed
by Cussens [12; 14]. Because learning of stochastic logic
programs is considered only for ‘normalized’ programs we
restrict our attention to them.
SIGKDD Explorations. Volume 2, Issue 2 - page 8
eral goal over mc(X, Y), i.e. the sum of the probabilities of
mc(1mf, r), mc(1mm, r), mc(1f, r), mc(1, r), and of mc(1m, r):
0.2
+ 0.2
+ 0.2
+ 2 · (0.2 · 0.5 · 0.2 · 0.2)
+ 2 · [0.5 · 4 · (0.2 · 0.5 · 0.2 · 0.2)]
= 0.624
Thus, the normalized probability of :− mc(1m, r) is
(2 · 0.004)/0.624 = 0.012820513. More details on inference
(including approximative inference) can be found in [13; 15].
Stochastic logic programs have been used to specify declarative
priors for probabilistic models [3].
Whereas stochastic logic programs attach probability labels
to all clauses, PRISMs and the earlier representation by
David Poole [82] attach probability labels to facts. Indeed,
let a logic program consist of the set of facts E (this could
be considered the extensional part) and the set of proper
clauses I (the intension). The frameworks by Poole and
Sato would then associate a probability label to each of the
facts in E. E.g., in the genetic example (where we take
the truncated encoding of Figure 8), one could associate a
probability of 1.0 to the facts for mother and father and in
addition there could be probabilities associated to the facts
for mc/1 and pc/1, e.g.,
(1.0) mc(1mf).
(1.0) pc(1mf).
(0.7) mc(1mm).
(0.7) pc(1mm).
(1.0) mc(1f).
(1.0) pc(1f).
A labelled fact p : f then has the meaning that there is a
probability of p that the fact f is true. This induces a probability
at the level of proofs. Indeed, consider now the atom
pt(1mm). The proof of pt(1mm) only succeeds when both
mc(1mm) and pc(1mm) are true. These two facts are only true
with a probability of 0.7 · 0.7 = 0.49. Therefore, the probability
that the proof of pt(1mm) succeeds is 0.49. Because
probability values are associated with facts, no normalization
is needed as opposed to stochastic logic programs.
At this point the reader has probably observed that the
probability distribution for proofs induces a probability distribution
for the other atoms and even for the interpretations.
Indeed, because there is only one possible proof for
pt(1mm), the probability for this atom to be true is also
0.49. If there would be more proofs, then one would have
to sum the probabilities. This in turn defines a probability
distribution at the level of interpretations and so the formalisms
by Sato and Poole, can also be interpreted from a
model-theoretic perspective. Sato has called this a distributional
semantics, cf. [88], stressing the declarative character
of PRISMs. However, the key difference with the Bayesian
network approaches presented above is that the former approaches
rely more on logical inference methods, whereas
the latter rely more on probabilistic inference mechanisms.
In a sense, the assumptions (on which the proofs rely) are
probabilistic, whereas in Bayesian networks the inference
mechanism is. This in turn explains the strong link with
0.7
department
0.1
0.3
0.3
course
0.2
0.8
0.3
0.1
0.1
lecturer
Figure 11: The graph structure of a Markov model for web
navigation.
abductive inference methods, cf. [82]. Finally, let us remark
that we discussed simplified versions of Poole’s and Sato’s
frameworks. Originally, they have been introduced to deal
with situations where some of the facts are mutually exclusive.
E.g., if three genes would be involved (say a, b and
c), then the corresponding facts for mc/2 and pc/2 would
be mutually exclusive. This could be specified using logical
constraints such as
false :− mc(X, a), mc(X, b).
false :− mc(X, a), mc(X, c).
false :− mc(X, b), mc(X, c).
mc(X, a); mc(X, b); mc(X, b) :− person(X).
A more compact notation used by Sato and Poole is
disjoint[p1: mc(X, a),p2: mc(X, b),p3: mc(X, c)] where pi denotes
a probability value.
4.3 Intermediate Representations
The above introduced frameworks integrate probabilistic
models with very expressive relational representations or
even logic programs. This however comes at a computational
cost. Therefore, some recent approaches such as relational
Markov models (RMMs) [2], hidden tree Markov
models (HTMMs) [28; 21], and logical hidden Markov models
(LOHMMs) [55] have tried to upgrade some well understood
probabilistic representations, such as (Hidden)
Markov Models [84]. These approaches can also be viewed
as downgrading one of the more expressive probabilistic logics
presented above. Whereas the resulting approaches are
typically less expressive than the above introduced ones, the
advantages are that (1) they are closely related to the underlying
representations, (2) they are – in principle – more
efficient, and (3) that the learning algorithms for the underlying
representations can almost directly be applied.
Let us illustrate this idea on Markov models (where we follow
the ideas underlying RMMs and LOHMMs using an
example by Anderson et al. [2]). A Markov model is essentially
a finite state automaton with probabilities associated
to each of the edges instead of symbols from an alphabet.
Alternatively, it can be regarded as stochastic regular grammar
5 . A (toy) example for web navigation (within a given
academic site) is given in Figure 11. The model denotes the
probabilities that if one is at a web page of type X the next
web page will be of type Y . E.g., in the graph listed above,
the probability of moving from department to lecturer is
0.3. It should be clear that this model is too simple to be
useful for web user modelling because the abstraction level
is too high. One alternative would be to build a model with
5 This is a special case of a stochastic context free grammar
in which all rules have exactly one symbol on the right hand.
SIGKDD Explorations. Volume 2, Issue 2 - page 9
one node for each of the web pages. This model would however
be so huge that it would hardly be usable (let alone
learnable). Therefore, a better way, supported by relational
or logical Markov models is to use proper relational or logical
atoms to represent the state instead of propositional
symbols. A sequence of possible states could then be e.g.
dept(cs) → course(cs, dm) → lecturer(cs, pedro) →
course(cs, stats) → . . .
The sequence of pages represented would include the web
page of the department of cs, the dm course, its lecturer
pedro and the course stats taught by the same lecturer.
This type of relational and logical sequence could be modelled
using the following type of (grammar) rules
(0.7) dept(D) → course(D, C).
(0.2) dept(D) → lecturer(D, L).
. . .
(0.3) course(D, C) → lecturer(D, L).
(0.3) course(D, C) → dept(D).
(0.3) coures(D, C) → course(D ′ , C ′ ).
. . .
(0.1) lecturer(X, L) → course(X, C).
. . .
Now, starting from any ground state, e.g., dept(cs), one
can determine the possible transitions as well as their probabilities.
However, the above logical Markov model is only
specifying the probability of a transition to an abstract state
(i.e., a state that is not ground - it contains variables);
in our example, the abstract states are course(cs, C) and
lecturer(cs, L). To be able to determine the corresponding
real states, one also needs to know (1) the domains of the
various arguments (in our examples, the set of all lecturers
and the set of all courses, say dom(L) and dom(C)), and (2)
a probability distribution on these domains (i.e., PL and
PC) that specifies the probability of selecting a particular
instance from these domains, e.g. the probability of selecting
pedro as a lecturer would be given by PL(pedro). Given
these domains and their corresponding distributions, one
can now instantiate the abstract transitions. E.g., starting
from dept(cs) there would be a probability of 0.7·PL(pedro)
of going to lecturer(cs, pedro). This illustrates the key
ideas underlying these representations: proof steps correspond
to time steps. Nevertheless, various differences among
these approaches exist:
• Relational Markov Models [2] do not allow for variables
and unification. Instead they employ a taxonomy
to define the domains and probability estimation
trees [83] to specify the domain distributions. Anderson
et al. [83] present impressive experimental results
on adaptive web navigation.
• Logical Hidden Markov Models [55] do allow for variables
and unification and they also upgrade hidden
Markov models (instead of the simpler Markov models
presented above). Kersting et al. [55] present a bioinformatics
application on predicting the fold class of
proteins.
• Hidden tree Markov models [21] and other hidden
Markov models over data structures [28] focus on modelling
probability distributions defined over spaces of
trees and graphs. They do not use logical concepts
such as predicates. Instead, they adapt automatatheoretical
concepts such as deterministic transduction
to generically define the dependency structure among
random variables. Diligenti et al. present an application
on document image classification in [21].
Further possible choices for these approaches exist. One
of these is concerned with the level at which to specify
the domain probabilities. One can specify them locally,
i.e., for each ”predicate” independently, or globally.
Secondly, how should one combine these probability
distributions to account for larger substitutions,
e.g., X = cs, Y = pedro in lecturer(X, Y )? Either,
one could take a naïve approach by assuming that the
domains are independent (similar to naïve Bayes), i.e.,
P (X = cs, Y = pedro) = P (X = cs) · P (Y = pedro), or employ
more advanced models such as Bayesian networks or
other probabilistic models. More experience and research is
necessary in order to fully understand and appreciate the
different options, opportunities and limitations of these recent
intermediate approaches.
Finally, let us remark that the intermediate representations
presented in this section should be viewed as belonging to
the proof-theoretic approach as probability distributions are
defined at the level of derivations and (partial) proofs respectively.
5. LEARNING PROBABILISTIC LOGICS
Learning denotes the process by which one adapts (probabilistic)
models on the basis of data. When learning within
probabilistic logics (as well as within the more traditional
probabilistic representations), one typically distinguishes
two tasks. The first task is concerned with the probabilistic
part, i.e., the question where do the numbers come from; it is
the so-called parameter estimation problem. In this task, it
is assumed that the structure (i.e., the underlying logic program)
of the representation is fixed and known, and the parameters
(i.e., the probability labels) have to be estimated.
In the second task, the so-called structure learning or model
selection, one assumes that neither the logic program nor
the parameters are fixed but have to be learned.
Again, it will turn out to be useful to distinguish the modelbased
from the proof-based representations. It will turn out
that — from the inductive logic programming perspective —
the distinctions are akin to those between learning from entailment
and learning from interpretations, cf. [17]. We will
therefore start this section by discussing the types of data
employed for learning. Because one also needs to estimate
the parameters when learning the structure of a probabilistic
model, we will introduce parameter estimation before
structure learning.
5.1 The Evidence
The model-based and the proof-based representations employ
different forms of data which we will now sketch.
5.1.1 Model Theoretic
An example or data case used for learning in a Bayesian
network setting consists of the random variables specified
in the network and a possibly partial assignment of values
to states 6 . To illustrate this, reconsider our earlier alarm
6 In this paper, we will assume that all random variables are
SIGKDD Explorations. Volume 2, Issue 2 - page 10
example. A possible example for this network could be
{burglary = false, earthquake = true, alarm = ?
johncalls = ?, marycalls = true}
This example partially describes a particular situation that
arose in alarm. Judea was called by Mary after an earthquake
took place. Judea neither knew whether the alarm
went off nor whether John tried to call him. Therefore,
when working with logical or relational upgrades of Bayesian
networks, it is natural to work with examples in the form
of interpretations. Indeed, the atoms in the interpretations
specify the random variables, and their values specify the
state they are in. E.g., in the genetic illustration, one could
have the following example:
{pc(1f) = true, mc(1f) = true, pc(1m) = true, mc(1m) = ?,
father(1, 1f) = true, mother(1, 1m) = true, pc(1) = ?,
mc(1) = ?, pt(1) = true} .
Remark that — from a logical perspective — there are two
constraints on this type of examples. First, the random
variables specified by a single example can be considered
a Herbrand interpretation, and this interpretation should
be a model of the unknown target program. This is akin
to the learning from interpretations setting from the field of
inductive logic programming. Second, the Bayesian network
structure induced by the target program for this particular
example should be acyclic. These two properties will turn
out to be important when learning the structure.
5.1.2 Proof Theoretic
To obtain insight into the nature of the examples for the
proof theoretic setting, it is a good idea to look at the nature
of the data when learning stochastic context free grammars.
When learning stochastic context free grammars, the
examples given are strings that should be accepted (with an
unknown probability) by the target grammar. E.g., when
learning a stochastic context free grammar in a natural language
domain for English, the examples would be English
sentences. Furthermore, these sentences should be derivable
by the target grammar. When working with definite clause
grammars (as in Figure 4 (a)) sentences would be represented
by facts. E.g., the sentence for the non-terminal
symbol S would be represented as s([the, dog, barks], []).
Thus, within the proof oriented setting of probabilistic logics,
the evidence will correspond to facts (or even clauses).
Furthermore, these examples e should be logically entailed
by the target program T , i.e., T |= e. This is akin to the traditional
learning from entailment setting in inductive logic
programming, cf. [69].
5.1.3 Intermediate Representations
Intermediate representations, such as (hidden) Markov
models, are typically learned from examples that correspond
to sequences of states that arose in particular
situations. E.g., the Markov model for the web
site domain, could be learned from examples such as
department, lecturer, course, lecturer, . . ., or in the first
order case by sequences such as the one specified in Sec-
known, i.e., that no latent variables, whose states are never
observed have to be introduced by the learning algorithm.
tion 4.3:
{dept(cs) → course(cs, dm) → lecturer(cs, pedro) →
course(cs, stats) → . . .}
Such sequences correspond to a kind of (partial) trace (or
derivation) of the underlying proof system. Learning from
traces has also been considered in the field of inductive logic
programming, see [96; 5] for two well-known examples where
traces have been used to induce logic programs. Traces impose
a strong logical constraint on the underlying program.
Indeed, for each two subsequent states s1 and s2 in a trace
(for a logical Markov model), it is required that there exists
a clause b → h and a substitution θ such that s1 = bθ and
s2 = hθ.
5.2 Parameter Estimation
Parameter estimation methods start from a set of examples
E and a given logic program L of a probabilistic logical
model, and aims at computing the values of the parameters
λ ∗ that best explain the data. So, λ is a set
of parameters (i.e. the quantitative part of the model)
and can be represented as a vector. To measure the
extent to which a model fits the data, one usually employs
the likelihood of the data. The likelihood of the
data is the probability of the observed data as a function
of the unknown parameters with respect to the current
model, i.e. P (E|L, λ). Thus, maximum likelihood estimation
(MLE) aims at finding λ ∗ = argmax λ P (E|L, λ).
As argmax λ P (E|L, λ) = argmax λ log P (E|L, λ), one often
works with the log-likelihood log P (E|L, λ), which is easier
to manipulate. Often, one assumes that the examples
are independently sampled from identical distributions, i.e.
log P (E|L, λ) = �
e∈E log P (e|L, λ). Variants or extensions
of the MLE criterion presented are sometimes used as well.
They include: a Bayesian approach which would take into
account parameter priors or the minimum description length
(MDL) principle that is often used for structure learning, cf.
Section 5.3.
To illustrate MLE, let us consider tossing a coin. The coin
can either be heads or tails. Assume that the evidence is
E = {{toss = head}, {toss = tail}, {toss = tail}}
(respectively {{head}, {tail}, {tail}} in the proof-based
notation), and that our current quantitative part is
P (toss = head) = p. Then
log P (E|L, λ) = �
log P (e|L, λ)
e∈E
= 1 · log p + 2 · log(1 − p)
We can maximize the log-likelihood by setting the partial
derivative w.r.t. to p equal to 0.0:
∂ log P (E|L, λ)
∂p
= 1 2
− = 0
p 1 − p
Solving this give p = 1
, which is the “intuitive” estimate for
3
p namely the proportion of heads which have been seen in
the examples. More formally,
p =
count(toss = head)
count(toss = head) + count(toss = tail)
where count(·) denotes the frequency of an event. Thus,
MLE reduces to frequency counting.
SIGKDD Explorations. Volume 2, Issue 2 - page 11
However, in the presence of missing data, the maximum
likelihood estimate typically cannot be written in closed
form. It is a numerical optimization problem, and all known
algorithms involve nonlinear optimization The most commonly
adapted technique for probabilistic logic learning is
the Expectation-Maximization (EM) algorithm [20; 65] 7 .
EM is based on the observation that learning would be easy
(i.e., correspond to frequency counting), if the values of all
the random variables were known. Therefore, it first estimates
these values, then uses these to maximize the likelihood,
and then iterates. More specifically, EM assumes
that the parameters have been initialized (e.g., at random)
and then iteratively perform the following two steps until
convergence 8 :
- E-Step: On the basis of the observed data and the present
parameters of the model, compute a distribution over all
possible completions of each partially observed data case.
- M-Step: Using each completion as a fully-observed data
case weighted by its probability, compute the updated parameter
values using (weighted) frequency counting.
The frequencies over the completions are called the expected
counts. Let us illustrate EM using the evidence
E = {{toss = head}, {toss = ?}, {toss = tail}}
and assume that in the current model P (toss = head) = p.
The completions of the data case then are
1 : {toss = head}, 1 : {toss = tail},
p : {toss = head}, 1 − p : {toss = tail} .
The updated maximum likelihood estimation of the probability
that the coin will be head is then 1+p
. Assuming
3
p = 0.8, this gives 0.6, and iterating yields 0.53, 0.511, . . .
which converges to 0.5.
Let us now discuss how EM has been applied within the
various probabilistic logics introduced earlier. For the sake
of simplicity, we will only state the key ideas and will not
address any advanced issues such as efficiency concerns.
5.2.1 Model Theoretic
Parameter estimations for probabilistic-logic programs [59],
probabilistic relational models [30; 31; 33], and Bayesian
logic programs [51; 54] all follow the same principle: The
given data and the current model induce a Bayesian network
explaining each data cases. Then, the parameters of
the induced Bayesian network are estimated using standard
Bayesian network parameter estimation methods, such as
those discussed in Heckerman’s tutorial [44]. For instance
the examples in Section 5.1.1 together with the genetic
program would induce the Bayesian network shown in Figure
9. All nodes of the Bayesian network, which do not
occur in the evidence but are introduced by the program,
7 Some approaches employ gradient-based approaches instead
of EM, e.g., Muggleton [67; 68] for structural learning
of SLPs, and Kersting and De Raedt [51; 54] developed both
EM-based and gradient-based approaches for parameter estimation
of BLPs.
8 This is only the underlying idea of the EM. More formally,
the E-step consists of computing the expectation of
the likelihood given the old parameters and the observed
data. Then, the M-step consists of maximizing the expected
likelihood w.r.t. the parameters.
are not observed, i.e. they are assigned a question mark.
Thus the evidence corresponds to a partial joint state of the
resulting Bayesian network. It is completed by the E-step,
which relies on traditional Bayesian network inference and
which is used to determine the distribution of the values for
the unobserved states. The improved estimates of the parameters
in a node (i.e., the values in the cpds) are then
computed in the M-step, by dividing the expected counts of
corresponding node-parents and parents joint states. The
only difference with standard Bayesian network parameter
estimation is that parameters for different nodes in the network
— those corresponding to different ground instances of
the same clause — are forced to be identical. This technique
is akin to that in recurrent neural network [99] or dynamic
Bayesian networks [19]. It however requires a unique assignment
of parent nodes to clauses. Aggregate functions
guarantee this for PRMs as the aggregated examples constitute
the states of the nodes in the induced Bayesian network.
Within PLPs and BLPs, uniqueness is enforced by
using decomposable combining rules [59; 51]. The effect
of a decomposable combining rule can be represented using
extra nodes in the induced Bayesian network. Most combining
rules commonly employed in Bayesian networks such as
noisy or, noisy and, or linear regression are decomposable.
5.2.2 Proof Theoretic
Whereas model-based approaches complete the data in
terms of a Bayesian network, proof-theoretic approaches
complete the data based on refutations and failures.
For stochastic logic programs, Cussens [14] introduced the
failure-adjusted maximization (FAM) algorithm. In FAM,
the logical part of the SLP is fixed and given, and the parameters
have to be learned on the basis of the evidence.
The evidence or the examples are atoms for a predicate p/m.
They are assumed to be generated from the target SLP according
to the probability distribution that it induces over
p/m. This implies that the examples are already logically
entailed by the SLP. The parameters are estimated by computing
for each example the SLD tree and treating each path
from root to leaf in the SLD tree as one possible completion.
For instance the evidence pt(1mm, red) yields five possible
completions, namely the five paths of the lower SLD tree
in Figure 5, of which four are failed and one constitutes a
proof 9 . Secondly, the completions are weighted with the
product of probabilities presently associated with clauses
used in the corresponding derivations, cf. Section 4.2. Finally,
the improved estimates for each clause are obtained
by dividing the clause’s expected counts by the sum of the
expected counts of clauses for the same predicate.
The EM algorithm for PRISM programs [50; 91; 49; 92]
is similar in spirit. The main differences are that (1) no
probability values are associated to (intensional) clauses,
(2) failed derivations have a probability of zero, and (3)
the disjoint representation introduced in Section 4.2 is
assumed. Given an example, all explanations S of the
example are computed. Similar to abductive reasoning,
an explanation of an example is a set of facts S that allows
the example to be proven. For instance, {pc(1mm, r)}
is the only explanation for pt(1mm, red), i.e. the failure
derivation in Figure 5 is neglected. Thus, the explana-
9 The approach is called failure-adjusted maximization because
both failed and successful derivations are taken into
account.
SIGKDD Explorations. Volume 2, Issue 2 - page 12
tions constitute the completions of the data, and they are
weighted with their probability P (S) as specified by the current
parameters. The maximum likelihood estimation of the
probability value associated with a fact, say pc(1mm, r), is
then the fact’s expected count over all explanations where
it is true divided by the expected counts of all corresponding
mutually exclusive facts, e.g. given the declaration
disjoint[p1: pc(1mm, r),p2: pc(1mm, g)], pc(1mm, r) and
pc(1mm, g) are mutually exclusive. To speed up computations,
Sato and Kameya employ well-known tabulation techniques
from the field of computational logic.
As already argued above, the intermediate representations,
such as LOHMMs and RMMs can be viewed as belonging
to the proof-theoretic approach. Therefore, we will omit a
detailed discussion of parameter estimation for these methods.
5.3 Structure Learning
In structure learning, one starts from a set of examples E,
a language bias L that determines the set of possible hypotheses,
and aims at computing a hypothesis H ∗ ∈ L such
that
1. H ∗ logically covers the examples E, i.e., cover(H ∗ , E),
and
2. the hypothesis H ∗ is optimal w.r.t. some scoring function
score, i.e., H ∗ = argmax H∈L = score(H, E).
The hypotheses H are of the form (L, λ) where L is the logical
part L and λ the vector of parameter values as in Section
5.2. The coverage criterion used will typically depend on the
type of data and representation formalisms considered, cf.
Section 5.1, where we have discussed three different types
of data for our three types of representation formalisms. In
addition, as mentioned in Section 5.2, various scoring functions
can be employed as well. A simple example would
be score(H, E) = P (E | H) = P (E | L, λ), i.e., the likelihood
criterion. The type of data, the coverage criterion,
the language bias and the scoring function are the central
components of any structure learning method. However, in
addition, various enhancements or variants can be considered
as well. These include a possible initial hypothesis that
could serve as a starting point for guiding the search and the
incorporation of background knowledge. Background knowledge
could consist of a set of given and fixed clauses (possibly
with attached probability labels) that would provide useful
information about the problem domain. The use of an initial
hypothesis is common in many probabilistic learning as
well as in theory revision approaches [100] to inductive logic
programming. Background knowledge is typically employed
in multi-relational data mining and inductive logic programming,
cf. [69]. Further distinctions could be made according
to whether the data are given initially and fixed or whether
they are gathered incrementally. Such distinctions and enhancements
are quite similar to those made in logic learning
and probabilistic learning. Therefore, we will not provide
more details on these aspects as doing so would lead us too
far. Instead, we will concentrate on the generalized problem
sketched above.
Nearly all (score-based) approaches to structure learning
perform a heuristic search through the space of possible hypotheses
as determined by L. Typically, hill-climbing or
beam-search is applied until the candidate hypothesis (1)
burglary.
earthquake.
alarm.
burglary.
earthquake.
alarm :− burglary.
burglary.
earthquake.
alarm :− burglary, earthquake.
Figure 12: Refinement operators during structure learning
of Bayesian networks. We can add a proposition to the body
of a clause (respectively fact) or delete it from the body.
pt(X) :− mc(X) ,pc(Y).
pt(X) :− mc(X) ,pc(Y).
mc(X).
pt(X) :− mc(X) ,pc(Y).
mc(X)
pc(X).
pt(X) :− pc(Y).
mc(X).
pt(X) :− mc(X) ,pc(Y).
mc(X) :− pc(X).
Figure 13: First-order refinement operators during structure
learning. Atoms can be added to or deleted from the body
of a clause. Facts can be added to and deleted from the
program.
satisfies cover(H, E) and (2) the score(H, E) is no longer
improving. The steps in the search-space are made through
the application of so-called refinement operators [96; 74],
which make perform small modifications to a hypothesis.
From a logical perspective, these refinement operators typically
realize elementary generalization and specialization
steps (usually, under θ-subsumption). In probabilistic learning,
certainly when working with graphical models, larger
steps are sometimes taken, see also below. Let us now illustrate
how these refinement operators work. Assume the
current candidate hypothesis H consists of the set of clauses
{c1, . . . , cn}. Then the following results of applying a (specialization)
refinement operator ρ on the overall hypothesis
H can be imagined. Notice that ρ operates at the hypothesis
or theory level and also that is employs a refinement
operator ρc at clause level 10 :
• ρ(H) = (H \ {ci}) ∪ ρc(ci) where ci ∈ H.
• ρ(H) = H ∪ {f} where f is a fact with f �∈ H.
• ρc(h :− b1, . . . , bm) = {h :− b1, . . . , bm, b | b is an atom}.
In addition, one typically also employs a dual generalization
operator. This operator would generalize hypotheses
by deleting atoms in clauses and facts from hypotheses. The
refinement operator is illustrated in Figure 12 for propositional
clauses and in Figure 13 for first order ones.
By now, we are able to describe some of the key approaches
to structure learning in probabilistic logics.
Bayesian networks directly correspond to propositional
clauses. In the typical structure learning approach, both
a propositional generalization and specialization operator
are applied. In one step, atoms are either added to or
deleted from clauses. Furthermore, there is exactly one
clause for each of proposition. Special care is taken that
the resulting network is acyclic [44]. The state-of-the-art
10 At this point, note that we employ a simplified and idealized
refinement operator. Some essential features, such
as substitutions are not taken into account for reasons of
simplicity, cf. [69; 74].
SIGKDD Explorations. Volume 2, Issue 2 - page 13
learning algorithm is the structural EM (SEM) [29]. It
adapts the standard EM algorithm for structure learning.
The key idea is that the expected counts are not computed
anew for every structure that is proposed, but only
after several iterations. This leads to an improved efficiency.
For PRMs, SRMs and BLPs, each application of a refinement
operator at the hypothesis level acts as a kind of
macro at the level of the induced Bayesian network. Indeed,
when a literal is added to the logical hypothesis, this corresponds
to adding multiple edges to the induced Bayesian
network on the data. Consequently, Bayesian network
learning techniques such as the SEM have been adopted.
Probabilistic relational models can be represented by
clauses such as p(X) : − τ1(q(X, Y)), τ2(r(X, Z)), where the
τi denote different aggregate functions and where there
is at most one clause defining a predicate (such as p/1).
Structure learning with probabilistic relational models now
occurs by refining this type of clauses [30; 36; 34; 33; 31] 11 .
Aggregated literals such as τ1(q(X, Y)) can be added to or
deleted from clauses. At this point, the graphical notation
for probabilistic relational models (illustrated in Figure 10)
is convenient for visualizing this process. Adding or deleting
aggregated literals to or from a clause directly corresponds
to the same operation on the arcs. This in turn clarifies
the connection to traditional Bayesian network learning.
As in Bayesian networks, special care is taken to guarantee
that regardless of the extension used the probabilistic relational
model will be acyclic. Bayesian logic programs
are represented by regular clauses, on which the typical
refinement operators from inductive logic programming can
be applied [53; 54]. However, in Bayesian logic programs,
one must take into account the covers constraint. More
formally, it is required that the examples are models of
the Bayesian logic programs, i.e. cover(H, e) = true if
and only if e is model of H. This requirement is needed
because — as argued in Section 4.1 — the set of random
variables defined by a Bayesian logic program corresponds
to a Herbrand model. This is akin to the learning from
interpretations setting in inductive logic programming [18].
The requirement is enforced when learning the structure
of Bayesian logic programs by starting from an initial
Bayesian logic programs that satisfies this requirement
(such a hypothesis can be computed using the clausal
discovery engine in [18]) and from then on only considering
refinements that do not result in a violation. In addition,
acyclicity is enforced by checking for each refinement and
each example that the induced Bayesian network is acyclic.
The framework of stochastic logic programs is a representative
of the proof-theoretic probabilistic logics. It
differs from the frameworks previously discussed in this
section in that the learning problem is not related to
that of Bayesian networks. Nevertheless, stochastic
logic programs are similar to Bayesian logic programs
in that they are represented by regular clauses on which
the typical refinement operators apply. However, whereas
Bayesian logic programs are learned from interpretations,
stochastic logic programs are learned from entailment.
Indeed, the examples in stochastic logic programs are facts
11 Please note that the original work on PRMs does not
present the learning problem in terms of clauses.
and these facts must be logically entailed by the target
(stochastic) logic program. Thus the coverage requirement
states that H |= e for stochastic logic program. Solving
the general structure learning problem for stochastic logic
programs thus involves applying a refinement operator
at the theory level (i.e. considering multiple predicates)
under entailment. This problem has been studied within
the field of inductive logic programming under the term
theory revision [96; 16; 100] and is known as a very hard
problem. This may explain why this most general form
of structure learning in stochastic logic programs (and
the related PRISMs) has not yet been addressed. So far,
the only contributions to structure learning of stochastic
logic programs are restricted to learning missing clauses
for a single predicate. In [70; 67], Muggleton introduced a
two-phase approach that separates the structure learning
aspects from the parameter estimation phase. In a more
recent approach [68] Muggleton presents an initial attempt
to integrate both phases for single predicate learning.
To learn models within the intermediate representations, one
can take advantage of their simplified underlying logic program.
Logical and relational Markov models can be
represented by transition rules of the form B → H where B
and H are atoms representing the abstract states. One could
— in principle — consider each possible abstract transition
B → H and estimate the corresponding transition probability
directly from the data. However, the number of such
transitions is way too large for this approach to work in
practice. Therefore, other approaches — that only select
the most informative transitions — are needed. For RMMs,
this is realized by 1) employing one abstract transition for
each predicate pair, and 2) by assigning a probability estimation
tree [83], a kind of decision tree, to each such predicate.
The probability estimation tree will assign to any given concrete
state in the body part of the transition a probability
distribution over the possible resulting states. So, probability
estimation trees encode a kind of abstraction hierarchy.
Kersting et al. [56] propose a different approach, which is
essentially a structural, generalized EM algorithm using refinement
operators for the LOHMM formalism.
6. CONCLUSIONS
An overview and survey of the new and exciting area of probabilistic
logic learning has been presented. It combines principles
of probabilistic reasoning, logical representations and
statistical learning into a coherent whole. The techniques
of probabilistic logic learning were analyzed starting from a
logical (or inductive logic programming) perspective. This
turned out to be quite useful for obtaining an appreciation
of the differences and similarities among the various frameworks
and formalisms that have been contributed to date.
In particular, the distinction between a model- and a prooftheoretic
view was used for clarifying the relation among the
logical upgrades of Bayesian networks (such as PLPs,PRMs,
BLPs, etc.) and grammars (such as PRISMS and SLPs).
This distinction is not only relevant at the representational
level but also for learning. It turned out that one learns from
interpretations in the model-theoretic approaches, from entailment
in the proof-theoretic ones, and from traces in the
intermediate ones (such as RMMs and LOHMMs). Furthermore,
principles of both statistical learning and inductive
logic programming (or multi-relational data mining) are
SIGKDD Explorations. Volume 2, Issue 2 - page 14
employed for learning the parameters and structure of the
probabilistic logics considered. The authors hope that this
survey provides a useful perspective on probabilistic logic
learning that may inspire the reader to contribute to this
challenging and exciting research area.
Acknowledgements
This work benefited from the European Union IST project
IST-2001-33053 (Application of Probabilistic Inductive
Logic Programming – APRIL). The authors thank Lise
Getoor for providing the graphical representation of the
PRM in Figure 10.
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