Models and Algorithms for Probabilistic and ... - Semantic Scholar

Models and A l g o r i t h m s for Probabilistic and Bayesian Logic P i e r r e Hansen Ecole des Hautes Etudes Commerciales 5255 avemie Decelles Montreal (Quebec) H3T 1V6, CDN

Brigitte Jaumard Ecole Poly technique de Montreal C.P. 6079, Succ. Centre-Ville Montreal (Quebec) H3C 3A7, CDN

Guy-Blaise Douanya Nguetse Ecole Polytechnique de Montreal, CDN

M a r c u s Poggi de A r a g a o Univ. Estadual de Campinas, Brazil

Abstract An overview is given, w i t h new results, of m a t h ematical models and algorithms for probabilistic logic, probabilistic entailment and various extensions. A n a l y t i c a l and numerical solutions are considered, the former leading to automated generation of theorems in the theory of probabilities. Ways to restore consistency and relationship w i t h Bayesian networks are also studied.

1

Introduction

Numerous models and a l g o r i t h ms have been proposed for reasoning under uncertainty in knowledge-based systems. A m o n g these, models based on logic and the theory of probabilities are, after a period of relative disfavor, a t t r a c t i n g much a t t e n t i o n again. These models differ according to the independence assumptions made about the events or logical sentences under consideration and the amount of i n f o r m a t i o n requested f r o m the expert or decision maker. At one extreme of the s p e c t r u m , well illustrated by the probabilistic logic and probabilistic entailment models of [Nilsson, 1986], no independance assumptions are made and o n l y the available i n f o r m a t i o n is used. Moreover, this i n f o r m a t i o n may be vague, i.e., expressed by p r o b a b i l i t y intervals instead of precise values. In probabilistic logic, the probabilities of being true of m logical sentences are given. It is asked whether these probabilities are consistent or n o t . In probabilistic entailment an a d d i t i o n a l logical sentence is considered and it is asked to f i nd best possible lower and upper bounds on its p r o b a b i l i t y of being true. In b o t h cases the j o i n t p r o b a b i l i t y d i s t r i b u t i o n on the set of possible outcomes is o n l y p a r t i a l l y specified. At the other extreme of the s p e c t r u m , the Bayesian network m o d els (e.g., [Pearl, 1988], [Lauritzen et ai, 1988]) usually make strong independence assumptions and request suff i c i e n t i n f o r m a t i o n for the j o i n t p r o b a b i l i t y d i s t r i b u t i o n to be entirely specified. W h e n those requirements are satisfied the p r o b a b i l i t y of any event m a y be c o m p u t e d , often in moderate t i m e . Moreover, evidence can be eff i c i e n t l y propagated t h r o u g h the network. A t t e m p t s t o combine b o t h methodologies have been made by [Van der Gaag, 1990] [Van der G a a g , 1991] and by [Andersen et ai, 1994].

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The purpose of this paper is to present an overview, w i t h new results, of m a t h e m a t i c a l models and algorithm s for probabilistic logic, probabilistic entailment and their extensions. M o t i v a t i o n stems f r o m the facts t h a t b o t h problems have a long history and are the object of research dispersed among several literatures. T h i s explains recent overly pessimistic statements as to the possibili t y of solving large instances. After s t a t i n g the problems m a t h e m a t i c a l l y, a n a l y t i c al solution is studied. It is shown t h a t one can use Fourier e l i m i n a t i o n or enum e r a t i o n of vertices and extreme rays of polytopes. T h e latter approach leads to a u t o m a t ed generation of theorems in the theory of probabilities. Numerical solution of large instances is then discussed. T h e column generation approach of linear p r o g r a m m i n g , combined w i t h specialized nonlinear 0-1 p r o g r a m m i n g techniques to solve auxiliary subproblems ( c o m p u t a t i o n of the most negative or positive reduced costs) leads to algorithms efficient in practice. Extensions are then examined, i.e., use of p r o b a b i l i t y intervals, c o n d i t i o n a l probabilities, Linear relations between probabilities and q u a l i t a t i v e probabili ties. Moreover, we show t h a t (i) restoration of consistency t h r o u g h m i n i m a l changes in the p r o b a b i l i t y intervals can be handled by the same type of models and (11) e l i m i n a t i o n of inconsistency t h r o u g h m i n i m a l delet i o n of logical sentences can be solved by combinin g colu m n generation w i t h branch-and-bound. T h e Bayesian logic model proposed by [Andersen et a/., 1994] is finally investigated: we show t h a t while this model is one of nonlinear nonconvex p r o g r a m m i n g , many cases to which it applies can in fact be expressed as linear programs.

sum of products, each product involving all logical v a r i ables in direct or complemented f o r m ; (it) associate u n known probabilities to each of these products and ident i f y the resulting sums to the given probabilities; (iii) eliminate in the equations so obtained and in the nonnegativity constraints on the probabilities the variables corresponding to the probabilities of the products. More t h a n a century later, [ H a i l p e r i n , 1965] [Hailperin, 1986] discusses Boole's methods and shows t h a t the above mentioned one is equivalent to Fourier elimination. Moreover, [Hailperin, 1965] expresses ( P S A T ) as the linear p r o g r a m ( l ) - ( 3 ) or ( l ) - ( 4 ) and shows t h a t an analytical expression for the bounds on the probability 7r m +i can be obtained by vertex enumeration of polytopes. To this effect, consider the dual Dmm ( A n a x )

of(l)-(4):

subject to constraints ( l ) - ( 3 ) . Note t h a t [Nilsson, 1986] briefly discusses how to use standard techniques to reduce problems of first-order probabilistic logic to the propositional case. Instead of the names probabilistic logic and probabilistic entailment, [Georgakopoulos et a/., 1988] propose to use the name probabilistic satisfiability, in decision and o p t i m i z a t i o n versions respectively. Indeed, [Nilsson, 1986] proposes useful models b u t not a logici i.e., a system of axioms and a study of inference rules, for reasoning about logic and probabilities. Such a logic extending the results of [Nilsson, 1986] has been explored by [Fagin et a/., 1990]. There are many other proposals in t h a t area. Moreover, the name probabilistic satisfiability stresses the relationship of problem ( l ) - ( 3 ) w i t h the classical satisfiability ( S A T ) problem of propositional logic (which corresponds to the case where is equal to 1 ) . F r o m now o n , we use the name probabilistic satisfiability (PSAT). The ( P S A T ) problem has a long history. The earliest occurrence of b o t h versions appears to be in the classical w o r k of [Boole, 1854] on The Laws of Thought. They are called conditions of possible experience and general problem in the theory of probabihttes respectively. B o t h problems also appear in the subjective approach to probability theory of [de F i n e t t i , 1974], De Finetti' s fundamental theorem tn the theory of probability ([de F i n e t t i , 1974], p. 112) is indeed very close to Boole's general problem. T h e work of [Boole, 1854] on probabilit y attracted l i t t l e a t t e n t i o n u n t i l it was revived in a seminal paper of [ H a i l p e r i n , 1965] and discussed and extended in a subsequent book of the same author on Boole a Logic and Probability [ H a i l p e r i n , 1986]. Several independent rediscoveries of (PSAT) have been made (includin g t h a t of [Nilsson, 1986]).

3

Analytical Solution of PSAT

In his book of 1854 and in several contemporary and subsequent papers, [Boole, 1854] proposes procedures to solve ( P S A T ) approximately or exactly. The most efficient one works as follows: (i) express each sentence as a

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W h i l e results such as the above are easily obtained by direct reasoning, a u t o m a t i o n becomes useful when more sentences are considered as the numbers of conditions and of t e r m s in the bounds increase rapidly.

4

N u m e r i c a l S o l u t i o n of P S A T

( P S A T ) is NP-hard, as it is in NP and contains the NPh a r d p r o b l e m ( S A T ) as a p a r t i c u l a r case ([Georgakopoulos et ai, 1988]). Moreover, the problems ( l ) - ( 3 ) and ( l ) - ( 4 ) have a number of columns exponential in the size of the i n p u t when, as is usually the case, the size (or t o t a l number of variable occurrences) of the sentences S i is bounded by a constant. ( N o t e t h a t this restriction on size is n a t u r a l , as otherwise reading the i n p u t w o u l d require t i m e exponential in the number of variables). So w r i t i n g ( l ) - ( 3 ) or ( l ) - ( 4 ) e x p l i c i t l y already requires exp o n e n t i a l t i m e . T h i s has led [Van der G a a g, 1990] [Van der G a a g , 1991] t o surmise t h a t solution of ( P S A T ) requires exponential t i m e in general and not only in worst case. ( I n fact, m a n y p o l y n o m i a l cases have been i d e n t i fied, see [Georgakopoulos et ai, 1988], [Kavvadias et ai, 1990], [ J a u m a r d et ai, 1991]). [Nilsson, 1986] [Nilsson, 1993] stresses less formally, b u t as strongly, the diffic u l t y o f solving instances o f ( P S A T ) w i t h many variables and suggests Looking for heuristics. [Frisch et at., 1994] propose under t h e name of anytime deduction a heuristic approach to ( P S A T ) based on sequential application of rules g i v i n g smaller and smaller intervals. T h i s has t h e advantage of allowing reasoning to be followed step by step b u t may n o t yield best possible bounds. However, the powerful c o l u m n generation technique of l i n ear p r o g r a m m i n g (see, e . g . , [ C h v a t a l , 1983], chapter 18) can be brought to bear. T h i s was proposed by [Zemel , 1982] for an application of ( P S A T ) to reliability, then for the general case by [Georgakopoulos et ai, 1988], whose work is extended in [Jaumard et a/., 1991], and ([Hooker , 1988], see also [Andersen et ai, 1994]). W h e n solvi n g a linear p r o g r a m by column generation a compact tableau is kept; at each iteratio n the entering column is f o u n d by solving a c o m b i n a t o r i al subproblem and the tableau is u p d a t ed following the rules of the revised s i m plex m e t h o d . F i n d i n g the column w i t h m i n i m u m ( m a x i m u m ) reduced cost at a current iteratio n is equivalent to minimization (maximization) of

which is a nonlinear expression in the variables X j w i t h the operators V, A and -. These operators may be e l i m i nated where x and y are logical variables. M i n i m i z a t i o n ( m a x i m i z a t i o n ) of the resulting nonlinear f u n c t i o n in 0-1 variables can be done a p p r o x i m a t e l y by variable-depth search ([Kavvadias et ai, 1990]) or t a b u search ( [ j a u m a r d et ai, 1991]) a n d exactly by an algebraic m e t h o d ( [ J a u m a r d et ai, 1991], [ C r a m a et ai, 1990]) or by l i n earization ([Hooker , 1988], [Andersen et ai, 1994]). As an exact solutio n is o n l y required when no more c o l u m n w i t h a reduced cost of adequate sign can be f o u n d heuristically, variable-depth and t a b u search are useful even if one wants proved best possible bounds. Heuristics w i l l be used as long as possible and followed by a usually more time-consumin g exact m e t h o d . T h e column generation technique has led to solve large instances of ( P S A T ) , w i t h up to 140 variables and 300 sentences ( [ J a u m a r d et ai, 1991]) in reasonable c o m p u t i n g t i m e . T h e number of columns generated is a very small p r o p o r t i o n of the overall number in the instance (e.g., about 2100 columns for problems w i t h 70 variables, and hence 2 7 0 columns, and 200 sentences).

5

E x t e n s i o n s of P S A T

In a d d i t i o n to uncertainty, expressed by probabilities , expert knowledge often suffers f r o m vagueness. Indeed g i v i n g a single value for the t r u t h p r o b a b i l i t y of a sentence is q u i t e a r b i t r a r y in m a n y situations. Vagueness may be expressed in ( P S A T ) by using p r o b a b i l i t y intervals for t h e t r u t h of sentences S, instead of single values. T h e n the expert is not forced to provide more i n f o r m a t i o n t h a n he has. Generalizing ( P S A T ) in this way was already proposed by [ H a i l p e r i n, 1965]. Constraints (2) are replaced by (14) T h e c o l u m n generation technique for ( P S A T ) described above extends readily to t h i s case, columns correspondi n g to slack or surplus variables being treated separately. T h e increase in c o m p u t i n g t i m e when replacing single p r o b a b i l i t y values by intervals is moderate ( [ J a u m a r d et ai, 1991]). E x p e r t knowledge m a y also be precise in some situations only, which is expressed by using c o n d i t i o n a l probabilities i. Such c o n d i t i o n a l p r o b a b i l ities can be integrated i n t o ( P S A T ) in several ways. As prob

, one can use [ J a u m a r d et ai,

1991] the t w o constraints: (12) where the u i are the dual variables associated w i t h constraints (1) and ( 2 ). Associating the values true w i t h 1 and false w i t h 0, (12) may be r e w r i t t e n (13)

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where with is t r u e and 0 otherwise, ) with is t r u e for X k and 0 otherwise. A more compact expression, obtained by e l i m i n a t i o n of is: (15)

and the problem becomes a hyperbolic (or fractional) p r o g r a m m i n g one. [ H a i l p e r i n , 1986] observes t h a t this problem can be reduced to a linear program w i t h one more variable using a standard technique of [Charnes et a/., 1962]: one minimizes Aap adding to the constraints ABp = 1 and m u l t i p l y i n g r i g h t - h a n d sides by a scaling factor t; once the solution is f o u n d , the probabilities p, are divided by t. A l t e r n a t e l y [Jaumard et ai, 199l] one can apply the l e m m a of [Dinkelbach, 1967] for fractional p r o g r a m m i n g and solve ( l ) - ( 3 ) , (16) by a sequence of linear programs. A g a i n c o l u m n generation techniques apply and c o m p u t i n g t i m e is not much larger t h a n for standard (PSAT) problems of the same size [Jaumard et a/., 1991]. Intervals for conditional p r o b a b i l i ty values can be handled as for usual probabilities. [Fagin et a/., 1990] develop a logic for reasoning about probabilities which extends the results of [Nilsson, 1986]. In particular they consider linear expressions in the probabilities Wj. If some of these are u n k n o w n , such expressions can be handled w i t h i n the column generation approach, again by keeping separate explicit columns. A step further is made by [ C o l e t t i , 1994] who considers q u a l i t a t i v e probabilities or conditional probabilities: their values are u n k n o w n b u t a p a r t i a l order on them is assumed to be given. T h e resulting generalization of ( P S A T ) remains linear for probabilities b u t is a n o n l i n ear nonconvex p r o b l e m for c o n d i t i o n a l probabilities and thus hard to solve. [ C o l e t t i , 1994] presents conditions of consistency for such problems. If a system of sentences is not consistent, which easily happens after a d d i t i o n of rules by different experts, one may seek to restore consistency w i t h m i n i m a l changes. A first c r i t e r i o n is to m i n i m i z e the sum of increases of the p r o b a b i l i t y intervals, possibly weighted to express the degree of confidence of the expert in his evaluations. This leads to the following linear p r o g r a m [ j a u m a r d et a/., 1991]: subject t o :

where in t h e decrease in the lower b o u n d (increase in the upper b o u n d ) on the p r o b a b i l i t y of and are a t t e n u a t i o n factors for these changes. A n o t h e r c r i t e r i o n is to m i n i m i z e the number of sentences to remove in order to restore consistency. T h e

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T h e  p r o b a b i l i t y  of  any  sentence  S i  may  be  c o m p u t e d  in  a  similar  way.  For  other  types  of  operations  answer­ able  by  Bayesian  networks  and  in  p a r t i c u l ar  for  propa­ g a t i o n  of evidence,  see,  e.g.,  [Pearl,  1988],  [Andersen  et  ai,  1994].  T h e  Bayesian  Logic  proposed  by  [Andersen  et  al,  1994]  consists  in  using  the  ( P S A T )  model  to  interpret  p r o b a b i l i t y  statements  associated  w i t h  Bayesian  net­ works  and  then  to  s t u d y  various  generalizations.  To  this  effect  c o n d i t i o n a l  independence  statements  are  encoded  as  a d d i t i o n a l  nonlinear  constraints.  These  constraints  have  the  general  f o r m  (17)  where  A,  B  and  C  are  sets  of  propositional  variables,  w i t h  \A\  =  a,  \B\  =  6,  \C\  —  c and  Ao,  Bo  and  Co  are  sets  of  fixed  atomic  propositions.  [Andersen  et  ai,  1994]  show  t h a t  there  are  ( 2 a —1)2 b (2 C —1)  nonredundant  constraints  among  those  described  by  (17).  F r o m  the  definition  of  c o n d i t i o n a l  p r o b a b i l i t y,  (17)  is  equal  to 

[Andersen  et  ai,  1994]  propose  to  solve  the  extended  ( P S A T )  model  w i t h  constraints  (17)  by  generalized  Ben­ ders  decomposition.  Following  t h a t  approach  the  p r o b ­ lem  is  split  i n t o  a  nonlinear  master  problem  in  the  n  variables  and  a  linear  subproblem  in  the  p  variables,  of  the  ( P S A T )  t y p e .  T h e  subproblem  is  used  to  generate  f r o m  its  d u a l ,  linear  constraints  in  the  n  variables,  called  Benders  cuts,  as  long  as  it  is  infeasible.  These  cuts  are  added  to  the  master  p r o b l e m .  T h e  procedure  stops  af­ ter  a  finite  number  of  steps  when  the  master  problem  is  infeasible  or  the  subproblem  is  feasible.  T h e  master  p r o b l e m  has  the  f o r m  of  a  signomial  geometric  program  for  which  specialized  algorithms  exist.  Such  problems  belong  to  global  o p t i m i z a t i o n  and  only  instances  w i t h  few  variables  can  be  solved  in  reasonable  t i m e .  T h i s  ap­ parently  l i m i t s  the  scope  of  Bayesian  logic,  even  if  the  number  of  π  variables  is  much  smaller  t h a n  the  n u m ­ ber  of p  variables.  Fortunately,  there  are  many  cases  in  which  one  need  only  add  linear  constraints  to  ( P S A T )  to  express  the  independence  assumptions  of  Bayesian  net­ works  and  generalizations  of  t h e m .  T h e o r e m  4  Computing  the  probability  of  a  sentence  S,  in  a  Bayesian  network  can  be  expressed  as  a  (PSAT)  prob­ lem  with  conditional  probabilities.  Proof.  C o n d i t i o n a l  probabilities  for  nodes  given  the  t r u t h  value  of  their  i m m e d i a t e  predecessors  can  be  ex­ pressed  by  (15)  and  m a r g i n a l  probabilities  by  (2).  For  independence  conditions,  let  Bj  denote  the  set  of  atomic  propositions  associated  w i t h  immediate  predecessors  of  V J  and  Aj  a  similar  set  for  non  immediate  predecessors  of  vj.  T h e n  the  conditio n 

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The o p t i m a l value is 0.26863. Clearly the number of sets of non-immediate predecessors of a node may be exponential. However, not all corresponding constraints need be w r i t t e n . [Lauritzen et a/., 1988] explain how to represent independence relations by an undirected graph G' in which all pairs of i m m e d i a te predecessors are j o i n e d and edges are added u n t i l the graph is t r i a n g u l a t e d . T h e n the j o i n t probability d i s t r i b u t i o n can be expressed as a product of m a r g i n al p r o b a b i l i t y d i s t r i b u t i o n s on the m a x i m a l cliques of G ' , adequately scaled. [Van der Gaag, 1990] [Van der Gaag, 199l] proposes to use this property in a decomposition m e t h o d for ( P S A T ) , discussed in a companion paper ([Douanya et al, 1995]). It is shown there that the usual ( P S A T ) model gives the same bounds as the decomposition-based version. Consequently ( P S A T ) takes i m p l i c i t l y i n t o account in the c o m p u t a t i o n of the bounds the c o n d i t i o n a l independence constraints (18) i n volving variables which do not all belong to the same maximal clique. E x a m p l e (continued). A graph G' associated w i t h the example after deletion of v5 and v6 is composed of triangles on v1,v2,V3 and on V2,v3,V4. T h i s shows that when c o m p u t i n g bounds on prob (x2Ix1 x3 one neeed not take e x p l i c i t l y i n t o account the constraints prob , i.e., the four last ones listed above. [Andersen c/ a/., 1994] also explore cases where the number of independence constraints is l i m i t e d . The main interest of Bayesian logic is n o t , however, to propose an alternate m e t h o d for the c o m p u t a t i o n made in Bayesian networks, b u t to consider more general assumptions. E x a m p l e (continued). Assume as done by [Andersen et a/., 1994] t h a t the atomic propositions x5 and x6 are not independent. T h e n , 0.25 < p r o b ( x ) < 0 4 0 . Replacing the line g i v i n g the marginal p r o b a b i l i ty of a:4 by

m i n i m i z i n g and m a x i m i z i n g yields bounds of 0.21786 and 0.28358. Note t h a t when using Benders decomposition the c o m p u t a t i o n of the lower bound required 57 iterations, i.e., solution of 57 ( P S A T ) and 57 signomial geometric p r o g r a m m i n g problems. As discussed in [Andersen et a/., 1994], many other extensions of Bayesian networks can be considered w i t h i n the ( P S A T ) f r a m e w o r k: one can replace single probability values by intervals, add constraints of different types than the c o n d i t i o n a l implications , allow for networks w i t h cycles, etc. N o t all extensions w i l l remain linear. For instance, if in the example, the m a r g i n al probabilities for x5 and X6 are replaced by intervals and the independence assumption is kept a quadratic constraint arises. T h e resulting quadratic programs can be solved in many ways using global o p t i m i z a t i o n techniques. F i n d i n g which are most efficient is an open problem. To conclude, ( P S A T ) appears t o be a flexible and comp u t a t i o n n a l l y t r a c t a b l e m o d e l for reasoning under u n certainty. It has already been extended in many ways,

while remaining linear. Further exploratio n of the problems which may be so expressed and of solution methods for the nonlinear case are attractive topics for f u t u r e research.

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