Advances in decision graphs

Advances in Decision Graphs

Thomas D.Nielsen and Finn V.Jensen

Aalborg University,

Department of Computer Science,

Research Unit of Decisions Support Systems,

Fredrik Bajers Vej7E,

9220Aalborg?,Denmark

Abstract.Frameworks for handling decision problems have been subject to many advances in the last years,both w.r.t.representation languages,solution algorithms and methods for analyzing decision problems.In this paper we outline some of the recent advances by taking outset in the in?uence diagram framework.In partic-ular,we shall focus on advances in representation languages and exact solution algorithms for decision problems with a single decision maker.Moreover,we give a brief outline of recent contributions to methods for performing sensitivity analysis in in?uence diagrams.

1Introduction

Decision graphs refer to a general class of models for representing decision problems.These types of models can be characterized by two components: i)a graphical structure,and ii)numerical information in the form of proba-bilities for representing uncertainty and utilities for representing preferences. The developments in this area were initiated by von Neumann and Mor-genstern(1944)who proposed the decision tree framework,see also(Rai?a 1968).A decision tree provides a direct representation of a decision prob-lem by modeling all decision scenarios explicitly.Although such an explicit representation has its advantage for some decision problems,it is also one of the main weaknesses:the size of the decision tree grows exponentially in the number of variables.This shortcoming motivated the development of the in?uence diagram framework(Howard and Matheson1981),which provides a compact representation of symmetric decision problems with a single decision maker.Initially,these models were solved by converting the in?uence diagram into a corresponding decision tree representation,such that existing decision tree algorithms could be used to solve the decision problem.Unfortunately, this solution technique also su?ers from the complexity problem mentioned above.As an alternative,Shachter(1986)proposed a solution method that works directly on the in?uence diagram model,i.e.,it does not require a sec-ondary structure.With the introduction of this solution algorithm,the use of in?uence diagrams found a more widespread interest and since then there have been several advances in the development of new representation lan-guages(which relax some of the assumptions underlying in?uence diagrams) as well as new methods for solving and analyzing decision problems.

2Thomas D.Nielsen and Finn V.Jensen

In this paper we outline some of these recent advances.Obviously,we are not able to cover the entire area and,in particular,we shall restrict our attention to frameworks for representing decision problems involving a sin-gle decision maker.Note that representation languages and solution methods for decision problems with several decision makers(usually considered in a game setting)have also been proposed,see e.g.(Koller and Milch2001). Moreover,we only deal with frameworks for handling discrete variables al-though in?uence diagrams with mixed variables have also been considered, see e.g.(Madsen and Jensen2003).Additionally,we will focus on exact so-lution algorithms for these models;for an overview of approximate solution methods,the interested reader is referred to(Charnes and Shenoy2002)and the references within.

Finally,we outline a few advances in methods for analyzing decision prob-lems.More speci?cally,we shall consider methods for performing sensitivity analysis in in?uence diagrams although the general task of model analysis also covers other areas such as value of information analysis,see e.g.(Dittmer and Jensen1997;Shachter1999).

2In?uence diagrams

The in?uence diagram(ID)framework(Howard and Matheson1981)serves as an e?cient modeling tool for symmetric decision problems with several decisions and a single decision maker.An in?uence diagram can be seen as a Bayesian network(BN)augmented with decision nodes and value nodes, where value nodes have no descendants.Thus,an in?uence diagram is a directed acyclic graph G=(U,E),where the nodes U can be partitioned into three disjoint subsets;chance nodes U C,decision nodes U D and value nodes U V.In the remainder of this paper we will use the concept of node and variable interchangeably if this does not introduce any inconsistency.We will also assume that no barren nodes are speci?ed by the in?uence diagram since they have no impact on the decisions(Shachter1986);a chance node or a decision node is said to be barren if it has no children,or if all its descendants are barren.Furthermore,in an in?uence diagram we have a total ordering of the decision nodes indicating the order in which the decisions are made(the ordering of the decision nodes is traditionally represented by a directed path which includes all decision nodes).

With each chance variable and decision variable X we associate a?-nite state space sp(X),which denotes the set of possible outcomes/decision options for X.For a set U of chance variables and decision variables we de?ne the state space as sp(U )=×{sp(X)|X∈U },where A×B de-notes the Cartesian product of A and B.The uncertainty associated with each chance variable C is represented by a conditional probability potential P(C|pa(C)):sp({C}∪pa(C))→[0;1],where pa(C)denotes the parents of C

Advances in Decision Graphs3 in the in?uence diagram.The domain of a conditional probability potential φC=P(C|pa(C))is denoted dom(φC)={C}∪pa(C).

The decision maker’s preferences is described by a multi-attribute util-ity potential,and in the remainder of this paper we shall assume that this utility potential is linearly-additive with equal weights,see e.g.(Tatman and Shachter1990);the set of value nodes U V de?nes the set of utility potentials which appear as additive components in the multi-attribute utility potential.1 Each utility potential indicates the local utility for a given con?guration of the variables in its domain.The domain of a utility potentialψV is denoted dom(ψV)=pa(V),where V is the value node associated withψV.Analo-gously to the concepts of variable and node we shall sometimes use the terms value node and utility potential interchangeably.

A realization of an in?uence diagram I is an attachment of potentials to the appropriate variables in I,i.e.,a realization is a set{P(C|pa(C))|C∈U C}∪{ψV(pa(V))|V∈U V}.So,a realization speci?es the quantitative part of the model whereas the in?uence diagram constitutes the qualitative part.

The arcs in an in?uence diagram can be partitioned into three disjoint subsets,corresponding to the type of node they go into.Arcs into value nodes represent functional dependencies by indicating the domain of the associated utility potential.Arcs into chance nodes,termed dependency arcs,represent probabilistic dependencies,whereas arcs into decision nodes,termed infor-mational arcs,imply information precedence;if there is an arc from a node X to a decision node D,then the state of X is known when decision D is made.

Let U C be the set of chance variables and let U D={D1,D2,...,D n}be the set of decision variables.Assuming that the decision variables are ordered by index,the set of informational arcs induces a partitioning of U C into a collection of disjoint subsets C0,C1,...,C n.The set C j denotes the chance variables observed between decision D j and D j+1.Thus the variables in C j occur as immediate predecessors of D j+1.This induces a partial order?on U C∪U D,i.e.,C0?D1?C1?···?D n?C n.

The set of variables known to the decision maker when deciding on D j is called the informational predecessors of D j and is denoted pred(D j).By assuming that the decision maker remembers all previous observations and decisions,we have pred(D i)?pred(D j)(for D i?D j)and in particular, pred(D j)is the variables that occur before D j under?.This property is known as no-forgetting and from this we can assume that an in?uence diagram does not contain any no-forgetting arcs,i.e.,pa(D i)∩pa(D j)=?if D i=D j. Example1(The reactor problem).An electric utility?rm is considering build-ing a reactor,and must decide(B)whether to build an advanced reactor(a), a conventional reactor(c)or no reactor at all(n).If an advanced reactor(A)

4Thomas D.Nielsen and Finn V.Jensen

is built,the pro?t is larger than for a conventional reactor(C)assuming that no accidents occur.However,past experience indicates that an advanced re-actor is more probable of having accidents than a conventional reactor.If the ?rm builds a conventional reactor,the pro?ts are$8B if it is a success(cs) or-$4B if there is a failure(cf).On the other hand,the pro?ts of building an advanced reactor are$12B if it is a success(as),-$6B if there is a limited accident(al)and-$10B if there is a major accident(am).

Before deciding on what reactor to build,the?rm has the option of having a test(T)performed on the components of the advanced reactor.The results (R)of the test(T=t)can be classi?ed as either bad(b),good(g)or excellent (e);the cost of performing the test is$1B.If the test results are bad,the nuclear regulatory commission will not allow an advanced reactor to be built.

An in?uence diagram representation of the reactor problem can be seen in Fig.1.Note that neither the state spaces nor the realization have been speci?ed.

Fig.1.An in?uence diagram representation of the reactor problem.

When evaluating an in?uence diagram we identify a strategy for the deci-sions involved;a strategy can be seen as a prescription of responses to earlier observations and decisions.The evaluation is usually performed according to the maximum expected utility principle,which states that we should always choose an alternative that maximizes the expected utility.

De?nition1.Let I be an in?uence diagram and let U D denote the decision variables in I.A strategy is a set of functions?={δD|D∈U D},whereδD is a policy given by:

δD:sp(pred(D))→sp(D).

A strategy that maximizes the expected utility is termed an optimal strategy, and each policy in an optimal strategy is termed an optimal policy.

Advances in Decision Graphs 5

In general,the optimal policy for a decision variable D k is given by:2

δD k (C 0,D 1,...,D k ?1,C k ?1)=(2)

arg max D k C k P (C k |C 0,D 1,...,C k ?1,D k )ρD k +1,where ρD k +1= V ∈U V ψV if k =n ;otherwise ρD k +1is the maximum ex-pected utility potential for decision D k +1:

ρD k +1(C 0,D 1,...,D k ,C k )=max D k +1 C k +1

P (C k +1|C 0,D 1,...,C k ,D k +1)ρD k +2.As the domain of a policy function grows exponentially with the number of variables in the past,it is important to weed out variables irrelevant for the decision.

De?nition 2.Let I be an in?uence diagram and let D be a decision variable in I .The variable X ∈pred(D )is said to be required for D if there exists a realization of I ,a con?guration ˉy over dom(δD )\{X },and two states x 1and x 2of X s.t.δD (x 1,ˉy )=δD (x 2,ˉy ).

A way of determining the variables required for D would be to analyze δD .However,then we would not have avoided the computational problem.Instead,methods for structural analysis of relevance have been constructed,see e.g.(Shachter 1999;Nielsen and Jensen 1999;Lauritzen and Nilsson 2001).Common for these methods is that they start o?by determining the required past for the last decision D .When this is done,D is replaced by a chance variable with D ’s required past as parents,and the methods then recursively work their way backwards in the temporal order.

To analyze relevance for the last decision D ,let U be a utility node which is a descendant of D .A variable X ∈pred(D )is then required for D ,if X is not d-separated from U given pred(D )\{X }.3This is illustrated in Fig.2a,which is used for analyzing the required past for D 2;the analysis for D 1is performed on the network in Fig.2b.

Finally,observe that Equation 2conveys that in order to determine an op-timal policy for a decision variable,we have to perform a series of alternating max-marginalizations and sum-marginalizations to eliminate the variables.The order in which the variables are eliminated must respect the partial or-der induced by the in?uence diagram.Thus we de?ne a legal elimination ordering as a bijection α:U C ∪U D ?{1,2,...,|U C ∪U D |},where X ?Y

6Thomas D.Nielsen and Finn V.Jensen

Fig.2.Figure a:D1is not required for D2because D1is d-separated from U2and U3given pred(D2)\{D1}.Figure b:The in?uence diagram used for analyzing the required past for D1(A is required for D1due to U2and U3).

impliesα(X)<α(Y).Note that a legal elimination ordering is not necessar-ily unique,since the chance variables in the sets C j can be commuted.Even so,any two legal elimination orderings result in the same optimal strategy since the decision variables are totally ordered and sum-operations commute; the total ordering of the decision variables ensures that the relative elim-ination order for any pair of variables of opposite type is invariant under the legal elimination orderings(this is needed since a max-operation and a sum-operation do not commute in general).

3Modeling decision problems

3.1Non-sequential decision scenarios

Consider the classical situation:we would like to buy a used car,but only if the car is in a satisfactory state.We cannot observe the state C directly but we can get information I by just looking at the car and we may-or may not-put an e?ort into additional tests T A and T B before we decide to buy the car(we assume that the tests may be performed in any order). Although this decision problem can be represented as an in?uence diagram, the representation is awkward(see Fig.3a).Instead,we may wish to represent the decision problem more directly.This is done in Fig.3b which is also called a partial in?uence diagram(PID).

In general,a partial in?uence diagram is like an in?uence diagram,but without the requirement of a linear ordering of the decisions(and observa-tions).For instance,the partial order of observations and decisions induced by the PID in Fig.3b is given by I?T A?O A?Buy and I?T B?O B?Buy.

As the temporal order is only partially speci?ed,the question is whether it matters,i.e.,whether the PID is well-de?ned.That is,will the expected utility of an optimal strategy be a?ected by the order in which the observations and decisions are taken.This means that we would in principle need to investigate all linear orders extending the partial order(such linear orders are called admissible).As there is nothing gained by delaying a(cost free)observation,

Advances in Decision Graphs7

Fig.3.Figure(a)shows an in?uence diagram representing two tests and a buy option.The test nodes have three options,t A,t B and no?test.The O nodes have?ve states,pos A,pos B,neg A,neg B,no?test.The arc O1→O2indicates that repeating a test will give identical results.Figure(b)gives a more direct representation of the test and buy example by not specifying a temporal ordering of T A and T B.

we introduce the convention that an observation is performed whenever it can be done.We say that an observation is free when all its preceding decisions have been made,and we say that the last of these decisions releases the observation.In Fig.3b,the decision T A releases O A,and T B releases O B. De?nition3.Let

De?nition4.A PID I is well-de?ned if for any realization of I it holds that any admissible in?uence diagram over I yields the same expected utility for an optimal strategy.

Theorem1.(Nielsen and Jensen1999)Let I be a PID.Then i)and ii)are equivalent.

i)Let<1and<2be any two strictly admissible orders,and let D be any

decision variable.Then the set of chance variables in the required past of

D in<1is identical to the set of chance variables in the required past of

D in<2.

ii)I is well-de?ned.

To investigate whether a PID is well-de?ned it is su?cient to investigate all admissible in?uence diagrams.Furthermore,as two neighboring decisions (and observations)can be permuted without a?ecting the expected utility, we need not care about such permutations in the ordering.

8Thomas D.Nielsen and Finn V.Jensen

For example,to investigate the PID in Fig.3b we should investigate the two strictly admissible orders illustrated in Fig.4.As the test nodes have di?erent required pasts in the two orders,the PID is not well-de?ned.

Fig.4.A directed graph representing the possible admissible orders of the graph in Fig.3b.

If a PID is not well-de?ned,the system may return it to the modeler requesting further speci?cation.This request may be followed by indications of how the partial order may be extended,for instance according to the number of?ll-ins making the PID well-de?ned,see(Nielsen2002).

3.2Unconstrained in?uence diagrams

An ill-de?ned PID may also be considered as an optimization problem:what strategy with respect to the order of decisions and observations should be followed in order to maximize the expected utility?Notice that in this case, the answer is not a single strictly admissible order.

For example,consider the following story.The beautiful princess in the kingdom Lovania has a wooer.It is rather convenient for the king as he considers retirement.Furthermore,in case he starts a war with the neighbor king,he needs a good general.As customary,the king shall confront the wooer with three tasks.One of the tasks(T1)shall be either to kill a unicorn or a dragon.Another task(T2)will be to spend a night in the royal tomb or in the haunted castle tower.The third type of task(T3)is to swim across the river or to climb the highest mountain in the kingdom.

The king can decide to retire(Rt)or to start a war(Wr)at any time. However,he cannot start a war after retirement,and he cannot give his daughter to the wooer before he has been confronted with all three tasks.

To represent this type of decision problem we extend the language of PIDs so that chance nodes which may be observed are speci?ed directly.We call them observables(depicted by double circles),and the language is called unconstrained in?uence diagrams(UIDs).A UID for the kings problem is given in Fig.5.

As the next step in a strategy(decision or observation variable)may be dependent on the past,a strategy for a UID is not one unique strictly admissible order together with policies for the decision variables.It is rather a DAG over decision nodes and observables.The set of strictly admissible

Advances in Decision Graphs

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orders is organized in a so-called normal form S-DAG:a directed acyclic graph where each path from source to sink is a strictly admissible order,and where all strictly admissible orders are represented.

A normal form S-DAG may represent any optimal strategy for the cor-responding S-DAG.Jensen and Vomlelov′a(2002)give a precise de?nition of normal form S-DAGs together with an algorithm for constructing them.A normal form S-DAG for the kings problem can be seen in Fig.6,and Fig.4 shows a normal form S-DAG for a UID corresponding to the PID in Fig.3b.

3.3Limited memory in?uence diagrams

The frameworks presented in the previous sections all rely on the no-forgetting assumption,i.e.,at any point in time the decision maker remembers all pre-vious observations and decisions.Unfortunately,this assumption may not always be valid,and,more importantly,it may make the computational com-plexity impractical as the required past for a decision may become intractably large.A way to restrict the size of the required past is to use information blocking.That is,by introducing variables which,when observed,d-separates some of the past from the present decision(Jensen2001).Alternatively,we can explicitly pinpoint which variables are remembered when taking a par-ticular decision,thereby dropping the no-forgetting assumption.The latter approach is pursued in the limited memory in?uence diagrams(LIMIDs)by Lauritzen and Nilsson(2001),where informational arcs are used to indicated which variables are known when taking a certain decision.

Example2(The North sea?shing problem).Every year,the European Union undertakes very delicate political and biological negotiations to determine a volume of?shing for most kinds of?sh in the North sea.Oversimpli?ed,you can say that each year we have a test for the volume of?sh,and based on this

10Thomas D.Nielsen and Finn V.

Jensen

Fig.

6.

A

normal form S-DAG for

the kings problem.

Advances in Decision Graphs11 test the volume of allowable catch is decided;this decision also has an impact on the volume of?sh for next year.A LIMID representation for a three year strategy is illustrated in Fig.7,where we only remember the last decision and observation(these two variables are therefore also the only variables in the policy for a given decision variable).

Fig.7.A LIMID representation of the North sea?shing problem,where only the last decision and observation are remembered.

Alternatively,by representing the North sea?shing problem using an in-?uence diagram with the no-forgetting assumption,we?nd that the required past for decision F V i constitute the entire past for that decision.Hence,the policies grows exponentially large;this analysis can be performed using the method described in Section2.

3.4Asymmetric decision problems

The frameworks described above have mainly been developed for representing symmetric decision problems with a single decision maker.However,another important type of decision problem is the class of so-called asymmetric deci-sion problems;these decision problems cannot be represented e?ciently using e.g.in?uence diagrams or valuation networks(Shenoy1992).There is cur-rently no complete consensus about the de?nition of an asymmetric decision problem although most authors agree that a decision problem is asymmetric if the number of scenarios is less than the cardinality of the Cartesian prod-uct of the state spaces of all chance and decision variables,see e.g.(Qi et al. 1994;Bielza and Shenoy1999;Shenoy2000;Nielsen and Jensen2000).For example,the decision problem described in Example1is asymmetric as can be seen by unfolding the in?uence diagram into a decision tree.

Various frameworks have been proposed as alternatives to the in?uence diagram when dealing with asymmetric decision problems.Covaliu and Oliver (1995)extend the in?uence diagram with another diagram,called a sequential decision diagram,see Fig.8.The sequential decision diagram is used for describing the asymmetric structure of the problem,as complementary to the in?uence diagram which is used for specifying the probability model;the functional and numerical information from these two diagrams are combined in a so-called formulation table similar to that of Kirkwood(1993).

12Thomas D.Nielsen and Finn V.Jensen

Fig.8.A sequential decision diagram representation of the reactor problem.

Smith et al.(1993)introduce the notion of distribution trees within the framework of in?uence diagrams,see Fig.9.The use of distribution trees allows the possible outcomes of an observation to be speci?ed,as well as the legitimate decision options for a decision variable,see also(Qi et al.1994).

Fig.9.The distribution tree for Result.

However,as the distribution trees are not part of the in?uence diagram, the structure of the decision problem cannot be deduced directly from the graphical structure.Moreover,the sequence of decisions and observations is predetermined,i.e.,previous observations and decisions cannot in?uence the temporal order of future observations and decisions.Finally,distribution trees have a tendency of creating large conditionals during the evaluation,since they encode both numeric information and information about asymmetry. To overcome this problem,Shenoy(2000)presents the asymmetric valuation network as an extension of the valuation network for modeling symmetric de-cision problems.The asymmetric valuation network uses so-called indicator functions to encode asymmetry,thereby separating it from the numeric in-formation(see Fig.10).However,asymmetry is still not represented directly in the model and,as in(Smith et al.1993),the sequence of observations and decisions is predetermined.Further details and comparisons of these methods can be found in(Bielza and Shenoy1999).

Nielsen and Jensen(2000)propose another framework(called asymmetric in?uence diagrams)which extends the in?uence diagram representation by introducing guards at the graphical level:guards on arcs and nodes specify

Advances in Decision Graphs 13

Fig.10.An asymmetric valuation network representation of the reactor problem.The triangles ρ,αand χare probability valuation,whereas δ1and δ2are indicator valuations.Observe that information precedence is encoded by the directed paths involving chance and decision nodes.

conditions for when the associated arcs and nodes are part of the decision problem.Moreover,restrictive functions (associated with decision variables)are introduced to allow the state space of a decision variable to depend on variables in its past;the domain of a restrictive function is indicated by dashed arcs into the associated decision node (see Fig.11).Having guards associated with informational arcs

also supports the speci?cation of decision problems where the temporal order of a set of variables depends on previous observations and decisions.

Fig.11.An asymmetric in?uence diagram representation of the reactor problem.The domain of the restrictive function associated with Buy is speci?ed by the dashed arc.

The use of guards has also been pursued by Demirer and Shenoy (2001)who propose a framework,called sequential valuation networks ,which is a combination of the sequential decision diagram and the asymmetric valuation network.Basically,this framework takes outset in the sequential decision diagram and augments this representation with valuation functions as found in the asymmetric valuation network.

Although several frameworks have been proposed for modeling asymmet-ric decision problems,there is currently no complete speci?cation and com-

14Thomas D.Nielsen and Finn V.Jensen

parison of their strengths and weaknesses.4Additionally,the expressive power of the frameworks is not clear,i.e.,what types of decision problems can be modeled in the various frameworks without introducing redundant informa-tion such as arti?cial states or duplicated variables.

4Evaluating decision problems

4.1Lazy evaluation of(partial)in?uence diagrams

An in?uence diagram is solved through dynamic programming using Equa-tion2.That is,we start by determining an optimal policy for the last decision, and then move backwards in the temporal order to determine a policy for the other decisions.When a policy for the last decision is found,the utility func-tions are substituted by a utility function representing the expected utility of an optimal choice for that decision.

This procedure has been translated to a variable elimination procedure, where variables are eliminated in reverse admissible order(Shenoy1992; Jensen et al.1994;Cowell1994;Ndilikilikesha1994).Taking advantage of lazy propagation,the elimination procedure is described as follows(Mad-sen and Jensen1999):The method keeps two sets of potentials:Φ,a set of probability potentials;Ψ,a set of utility potentials.When a variable X is eliminated,the potential sets are modi?ed in the following way:5

1.SetΦX:={φ∈Φ|X∈dom(φ)}andΨX:={ψ∈Ψ|X∈dom(ψ)}.

2.If X is a chance variable,then

φX:= X ΦX andψX:= X ΦX( ΨX).

If X is a decision variable,then

φX:=max X ΦX andψX:=max X ΦX( ΨX).

3.LetΦ:=(Φ\ΦX)∪{φX}andΨ:=(Ψ\ΨX)∪{ψX

4Note that a comparison of sequential decision diagrams,asymmetric valuation networks and extended in?uence diagrams can be found in(Bielza and Shenoy 1999).

5Note that the operation max XΦX simply corresponds to removing X from the domain ofΦX sinceΦX is a constant function over X.

Advances in Decision Graphs15 4.2Evaluation of unconstrained in?uence diagrams

When a normal form S-DAG for a UID has been established,it is solved in almost the same manner as in?uence diagrams.That is,variables are eliminated in reverse temporal order.

When a branching point is met,the elimination is branched out,and you work with particular potential sets for each branch.When paths meet they have to meet in a chance variable,and the variable immediately after must be a decision variable in all branches.So,assume that two branches meet in A from D1and D2,respectively.Now,the sets of probability po-tentials are the same(they represent sum-marginalizations of the same vari-ables in various orders),but the utility potentials may be di?erent.The uni-?ed utility potentials are determined through maximization.For simplicity, let U1(B,A)and U2(B,A)be the potentials.Then the potential for A is max(U1(B,A),U2(B,A)).

4.3Evaluation of LIMIDs

The evaluation of a LIMID is based on an iterative improvement of the poli-cies for the decision variables,and is closely connected to the method of policy iteration for Markov decision processes.

As a starting point,the decision variables are assigned random policies, i.e.,policies which specify probability distributions over the state spaces of the associated decision variables given their parents;recall that the variables known to the decision maker when deciding on a decision D are the parents of the corresponding node.If a random policy speci?es a unique alternative for each con?guration of pa(D),then the policy is called a pure policy;a pure policy corresponds to the traditional notion of a policy if pa(D)contains the variables being required for D in the corresponding ID representation.

Given the initial policies of the decision variables,the LIMID is converted into a junction tree.However,as opposed to the construction of a strong junction tree,informational arcs are not removed before moralization and the triangulation need not respect any partial or total ordering of the nodes. The junction tree is then initialized by associating each probability potential and utility potential to a clique which can accommodate it.Afterwards all potentials assigned to a clique are combined thereby following the approach of Jensen et al.(1994).Note that in the“lazy”version of the algorithm the latter step should not have been performed.

Based on the initialized junction tree structure,the evaluation algorithm proceeds by iteratively improving the policy for each decision variable.This is performed by making a collect propagation(in the usual fashion)to the clique containing the decision variable in question.The current policy is then substituted by another policy which maximizes the expected utility for that decision;the algorithm converges after having updated the policy for each

16Thomas D.Nielsen and Finn V.Jensen

decision variable.Obviously,this method only provides an approximate solu-tion,and for lower and upper bounds of the expected utility of the approxi-mation the reader is referred to(Nilsson and H¨o hle2001).

4.4Evaluation of asymmetric decision problems

Existing methods for solving asymmetric decision problems can roughly be characterized based on how asymmetry is represented in the associated frame-works.

For those frameworks which use secondary structures to represent asym-metry,such as distribution trees and indicator functions,the solution algo-rithms incorporate the information about asymmetry into the solution algo-rithm.That is,information about asymmetry is directly combined with the numerical information during the solution phase,see e.g.(Smith,Holtzman, and Matheson1993)and(Shenoy2000).For example,the algorithm for solv-ing asymmetric valuation networks,known as the fusion algorithm,works simultaneously on all three sets of valuations de?ned in the model:utility valuations V,probability valuationsΓand indicator valuationsΥ.Similarly to standard evaluation algorithms for in?uence diagrams,the fusion algo-rithm marginalizes out the variables in reverse temporal order by combining all relevant valuations.However,the fusion algorithm only works on the ef-fective state spaces of the variables,i.e.,the legitimate state con?gurations as de?ned by the indicator valuations.For instance,the combination of two probability valuationsρ1andρ2is de?ned as:

(ρ1?ρ2)(ˉx)=(ρ2?ρ1)(ˉx)=ρ1 ˉx↓dom(ρ1) ρ2 ˉx↓dom(ρ2) , whereˉx is in the e?ective state space of the variables in the domain of both inρ1andρ2.

To specify the fusion algorithm w.r.t.a variable X we denote by V X,ΓX andΥX the utility,probability and indicator valuations including X in their domain.The fusion operation Fus X(V∪Γ∪Υ)w.r.t.the valuations V∪Γ∪Υis then de?ned di?erently depending on the type of variable X.If X is a chance variable,then:6

Fus X(V∪Γ∪Υ)= v? ρ

6Note that the projection notation,i.e.↓,refers to the standard marginalization of a variable except that we only work with the e?ective state space.

Advances in Decision Graphs17 If X is a decision variable,then:

Fus X(V∪Γ∪Υ)= v↓(dom(V X)\{X}) ∪ (ρ?ζX)↓(dom(V X∪ΓX∪ΥX)\{X})

∪{V\V X}∪{Υ\ΥX}∪{Γ\ΓX},

(6) whereζX is the indicator valuation representing the optimal policy for X.

Note that Shenoy(2000)also derives cases where the fusion algorithm can be optimized based on the speci?c valuations involved.

Alternatively,in the frameworks where asymmetry is represented explic-itly in the graphical structure,the solution algorithms use this information to construct a secondary structure for solving the model.The secondary structure is used to decompose(or unfold)the original asymmetric deci-sion problem into a collection of symmetric subproblems that can be solved independently;the actual decomposition is performed by traversing in the temporal order and iteratively instantiating the variables which“produce”asymmetry.For instance,in the asymmetric in?uence diagram we instantiate the variables which appear in the domain of either a guard or a restrictive function.The resulting subproblems can then be organized in a tree structure (see Fig.12)having the property that a solution to the original decision prob-lem can be found be combining the solutions from the“smaller”symmetric decision

problems.

R=g,e

Fig.12.The?gure illustrates the decomposition tree for the reactor problem de-picted in Figure11.Note that decomposing w.r.t.R produces subproblems which are structurally identical but di?er in the state space of the decision variable B.

18Thomas D.Nielsen and Finn V.Jensen

Obviously,both types of frameworks have their advantages and disad-vantages.For example,the asymmetric valuation network relies on arti?cial states to represent asymmetry.From a computational perspective this has the undesirable e?ect that we in general work with larger valuations during the solution phase,as compared to e.g.the sequential valuation networks or asymmetric in?uence diagram.On the other hand,when e.g.evaluating an asymmetric in?uence diagram,the calculations are performed according to its decomposition tree.This implies that the same calculations may be required in di?erent subproblems,i.e.,we may perform redundant calculations.Obvi-ously,this problem does not occur when evaluating an asymmetric valuation network using the fusion algorithm.One way around this problem could be to maintain a cache of previous calculations e.g.a hash table indexed with the calculated potentials,see also(Cano et al.2000).

5Model Analysis

5.1Sensitivity analysis in in?uence diagrams

When solving an in?uence diagram the aim is to determine an optimal policy for each decision involved.These policies(as well as the maximum expected utility of the in?uence diagram)are sensitive to variations in both the utilities and the probabilities.Unfortunately,it is usually the case that these values are di?cult to elicit and subject to second-order uncertainty.7This makes it desirable to be able to determine how robust the solution is to variations in both the utilities and the probabilities,see also(Poh and Horvitz1993).Based on such an analysis,the modeler may focus his/her attention on parameter values for which the solution is particularly sensitive(a situation known as one-way sensitivity analysis).Another important type of analysis concerns the joint variation of a set of parameters(n-way sensitivity analysis).This type of analysis yields little information about the individual parameters,but provides general insight into the overall robustness of the model.

In what follows we make a distinction between value sensitivity and deci-sion sensitivity when referring to sensitivity analysis.Value sensitivity con-cerns variations in the maximum expected utility when changing a set of parameters,and decision sensitivity refers to changes in the optimal strat-egy.

Analysis of decision sensitivity can furthermore be characterized as either threshold proximity or probabilistic sensitivity analysis.Threshold proximity uses a distance measure to determine the values a parameter(or a set of pa-rameters)can be assigned without changing the optimal strategy found w.r.t. the initial values of the parameters.Probabilistic sensitivity analysis assigns

Advances in Decision Graphs19 a probability distribution to each parameter under investigation.Hence,the analysis is based on the probability of obtaining a di?erent optimal strategy rather than determining the admissible domain in which the parameters can be varied,see e.g.(Doubilet et al.1985).

A method based on value sensitivity has been proposed in(Felli and Hazen1998).This method uses the expected value of perfect information as a sensitivity indicator,and similar to probabilistic sensitivity analysis it requires that a probability distribution is assigned to each parameter under investigation.More precisely,letˉt be the uncertain parameters,and let?0 be the optimal strategy found w.r.t.the initial values of the parameters. Ifρ?andρ?0denote the expected utilities of the in?uence diagram under the strategies?and?0,respectively,then the expected value of perfect information(denoted by EV P I(ˉt))is given by:

[ρ?(ˉt)]?ρ?0(ˉt)].

EV P I(ˉt)=Eˉt[max

?

This expectation can be very di?cult to evaluate.Hence simulations are usu-ally applied.A detailed description and comparison of the methods outlined above,can be found in(Felli and Hazen1998)and(Felli and Hazen1999).

A drawback of employing sensitivity analysis based on probability distri-butions over the uncertain parameters is the elicitation of these distributions. Given that the decision maker has a(partial)preference ordering over the outcomes,the uncertain utility parameters are not independent.I.e.,we work with joint probability distributions over the utility parameters.8This makes it di?cult to elicit these distributions,in particular,when the utilities have no straightforward interpretation.Parameter dependence also has an impact on the computational complexity.Even when the parameters are assumed to be independent,the calculations can be rather cumbersome,and Monte Carlo methods are usually applied to sample values for the parameters.For instance, when applying the method based on the expected value of perfect informa-tion we need to compute di?i=[max?[ρ?(ˉt i)]?ρ?0(ˉt i)],for each generated sampleˉt i;the expectation is then approximated by EV P I(ˉt)≈1

8Analogously to the standard assumptions within the learning community,see e.g.(Cooper and Herskovits1991),it is usually assumed that the probability parameters are independent.

20Thomas D.Nielsen and Finn V.Jensen

the decision variables in reverse temporal order.For each decision variable, a set of linear constraints is computed which in turn forms the basis for the analysis;as all constraints are linear,n-way sensitivity analysis comes free of charge after one-way sensitivity analysis has been performed.More-over,when working with uncertain utility parameters the time complexity of the algorithms is roughly linear in the number of uncertain parameters (the basic unit of computation is one propagation);unfortunately,in case of n-way sensitivity analysis w.r.t.probability parameters the time complexity is exponential in the number of uncertain parameters.

Finally,it should be noted that decision sensitivity has recently been sub-ject to criticism on the ground that it tends to overestimate the sensitivity of a model(di?erent strategies may yield almost the same expected utility),see e.g.(Felli and Hazen1999).On the other hand,one may argue that if a pa-rameter induces signi?cant changes in the expected utility without in?uencing the optimal strategy,then the model is also insensitive to the parameter.To overcome the problem of overestimating model sensitivity,one may perform the decision analysis iteratively.That is,after decision sensitivity has been analyzed we can calculate the expected utility of the optimal strategy,? ,in-duced by changing the parameters.If the expected utility is not signi?cantly di?erent from the expected utility of the initial strategy,then the analysis is repeated and the strategy is allowed to change accordingly(i.e.,? is disre-garded).Thus,by analyzing decision sensitivity iteratively,we may avoid the problem of overestimating the sensitivity of the model.

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数至少为2，而V2中的每个结点度数至多为2，从而它满足t条件t=1，因此存在从V1到V2的匹配，故可分配。 5.此平面图具有五个面，如下图所示。 a b c d e f g r1r2 r3 r4 r5 ?r1，边界为abca，D(r1)=3; ?r2，边界为acga，D(r2)=3; ?r3，边界为cegc，D(r3)=3; ?r4，边界为cdec，D(r4)=3; ?r5，边界为abcdefega，D(r5)=8;无限面 6.设该连通简单平面图的面数为r，由欧拉公式可得，6?12+r=2，所以 r=8，其8个面分别设为r1,r2,r3,r4,r5,r6,r7,r8。因是简单图，故每个面至少由3条边围成。只要有一个面是由多于3条边所围成的，那就有所有面的次数之和 8∑ i=1 D(r i)>3×8=24。但是，已知所有面的次数之和等于边数的两倍，即2×12=24。因此每个面只能由3条边围成。 2

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软件研发部岗位职责-标准化文件发布号：（9456-EUATWK-MWUB-WUNN-INNUL-DDQTY-KII

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