Imprecise Probability

Transcription

1 Imprecise Probability Alexander Karlsson University of Skövde School of Humanities and Informatics 6th October D W 0 L 0

2 Introduction The term imprecise probability refers to a collection of theories which describe probabilities in an imprecise way. The following theories ordered in increasing generality belong to the concept of imprecise probability [8].. Possibility theory 2. Belief and plausibility functions 3. Choquet capacities of order 2 4. Coherent upper and lower probabilities 5. Coherent lower previsions 6. Sets of probability measures 7. Sets of desirable gambles 8. Partial preference orderings 9. Partial comparative probability orderings Hence all kinds of uncertainty one can express by belief and plausibility functions can also be expressed by coherent upper and lower probabilities, but not vice versa. This report focus on the theory of coherent lower previsions since it has many interesting properties and also constitutes a basis for 7-9. The theories in 6-9 solve some problems which can be found in coherent lower previsions (for further detail see [8]). Section 2 depicts the foundation of coherent lower previsions as in [5, 6, 7, 8] and how it relates to decision theory and information. Section 3 explains how one can apply it on a simple example [6] and section 4 compares the theory with other imprecise uncertainty methods. 2 Coherent lower previsions Consider an uncertain reward X, i.e. you do not know the value of X, and define your lower prevision P (X) as your supremum buying price for X. This means that before you know the true value of X, you are willing to pay any amount µ P (X) in order to receive X. Since an uncertain reward could be interpreted as a gamble we will hereafter use that term. Formally a gamble is defined as a function X : Ω R where Ω = {ω,..., ω n } is a possibility space. Also define P (X) as your infimum selling price for the gamble X. These definitions are epistemic and behavioural (subjective) since they describe your behaviour in terms of buying and selling prices under your current state of knowledge about gambles [6]. The interpretation can also be logical (objective), i.e., the prices are uniquely defined by available evidence [6]. Buying a negative gamble for a 2

3 Buy X Undetermined Sell X 0 Figure : Buying or selling a gamble? negative amount of money is equal to selling the gamble for the corresponding amount, thus P ( X) = P (X) () This means that the theory can be completely defined in terms of lower previsions (or upper previsions). When P (X) = P (X) = P (X) one calls P (X) a precise prevision. Since we have defined a supremum buying price and an infimum selling price there exist an interval of prices ( P (X), P (X) ) where selling or buying gambles are undetermined (see figure ). This interval depicts the degree of imprecision and reflects the amount of information missing in order to define a precise price for X. As we will see later (in section 2.4), this is the property that enables for indecision when used as a decision theory. Upper and lower probabilities of events are special types of previsions where all gambles are indicator functions, i.e., the gamble I A will return a reward of one unit if A occurs and zero otherwise. 2. Rationality principles The theory of lower previsions relies on three rationality principles: avoiding sure loss, coherence and natural extension, which all can be given an interpretation of rational behaviour. The principles are defined in terms of marginally desirable transactions (or gambles). Let X µ be a transaction, i.e, you pay µ in order to receive X. We call a transaction desirable when µ < P (X) and marginally desirable when µ = P (X). Example Consider the gamble X where you have assessed the lower and upper prevision P (X) and P (X), then the gambles X P (X) and P (X) X are marginally desirable transactions. 2.. Avoiding sure loss Avoiding sure loss states that rational behaviour excludes the possibility of buying gambles for prices that imply a sure loss. Formally [7, 2] [ n ] sup λ i [X i (ω) P (X i )] 0 (3) i= for every finite set of gambles X,..., X n, where λ i 0. (2) 3

4 Example 2 ([2]) Assume that the following previsions have been assessed about the outcome: (W)in, (D)raw and (L)oss, of a football game P (I W ) = 0.65 P (I D ) = 0.25 P (I L ) = 0.4 P (I W ) = 0.6 P (I D ) = 0.2 P (I L ) = 0.35 These assessments imply that the gamble Z = I W + I D + I L ( ) is marginally desirable. But I W, I D and I L are mutually exclusive and thus Z Coherence Lower previsions are coherent when it does not exist a combination of transactions which impose a higher lower prevision (price) for some gamble X 0 than an assessed prevision (price) P (X 0 ) [7]. When this principle fails to hold, your behaviour is irrational since implications of your previsions are in contradiction with your assessed previsions. Formally one can express this as [7, 2] [ n ] sup λ i [X i (ω) P (X i )] λ 0 [X 0 (ω) P (X 0 )] 0 (4) i= since if this does not hold n λ i [X i (ω) P (X i )] < λ 0 [X 0 (ω) P (X 0 )] (5) i= ω, but then there exist an ɛ such that [2] n λ i [X i (ω) P (X i )] + ɛ < λ 0 [X 0 (ω) (P (X 0 ) + ɛ)] (6) i= The left hand side is desirable so the gamble on the right hand side is even more desirable and this is equal to a supremum buying price P (X 0 ) + ɛ, which is higher than the specified prevision P (X 0 ). Example 3 ([2]) Consider the football game again, where we have the following previsions P (I W ) = 0.52 P (I D ) = 0.6 P (I L ) = 0.3 P (I W ) = 0.27 P (I D ) = 0.27 P (I L ) = 0.2 which means that the gamble Z = I W I L 0.2 is marginally desirable. But I W + I L = I D so Z = I D = 0.52 I D, which means that P (I D ) = Since this is less than the assessed prevision P (I D ) we have incoherence (inconsistency). 4

5 When the specified gambles constitute a linear space an alternative representation of coherence exists [6] P (X) inf X(ω) (7) P (λx) = λp (X) (8) P (X + Y ) P (X) + P (Y ) (9), which can be given the following behavioural interpretation [6]:. You should always be willing to pay at least as much as the smallest possible reward 2. The set of acceptable gambles should not depend on the unit of utility 3. The price of buying two gambles X and Y in one transaction cannot be less than buying them separately 2..3 Natural extension In order to draw conclusions, making inference for previsions of new gambles one uses the principle of natural extension. The idea of natural extension is that your assessed previsions provide information about your behaviour for new gambles for which you have not yet specified previsions (X 0 ). Formally natural extension is defined by [7, 2] (provided that the principle of avoiding sure loss holds) { } n E(X 0 ) = sup µ : X 0 (ω) µ λ i [X i (ω) P (X i )], ω, λ i 0 (0) i= One can interpret natural extension in the following way: since the right hand side is a desirable gamble implied by the assessed previsions, so is the gamble on the left hand side, and your inferred prevision is the largest of these prices [7]. Loosely, natural extension is the largest buying price that you can be forced to pay based on implications of your assessed previsions. 2.2 Generalised Bayes Rule When new information is obtained through observations as an event I B, one wants to update the initial previsions to previsions conditional on I B. This can be done with the Generalised Bayes Rule (GBR) which is a consequence of coherence between P (X) and P (X I B ) (see [5] for further detail) P (I B (X P (X I B ))) = 0 () In the case when X = I A is an event and P = P i.e. a precise prevision, the GBR is reduced to Bayes rule [5] P (I A I B ) = P (I AI B ) P (I B ) (2) 5

6 2.3 Comparative probability judgements A nice feature of coherent lower previsions is that they can express comparative probability judgements from natural language [6]. Such judgements are transformed to constraints on precise previsions which then can be found by linear programming (LP) techniques. The constraints constitute a convex set in linear space and can thus be characterised by its convex hull [3]. As an example [8] of a comparative probability judgement consider A is at least c times as probable as B, which can be modelled by P (I A ci B ) 0 (see section 3 for a complete example). 2.4 A theory of decision Since coherent lower previsions describe behaviour (decisions) in terms of buying and selling gambles; it can easily be mapped to a theory of decision. Consider a set of acts {a,..., a n } with corresponding gambles X i. Then, in order to decide if and act a i is preferred to a j one calculates the difference of lower and upper previsions for the gambles [7]. P (X i X j ) P (X i X j ) (3) If P (X i X j ) > 0, a i is preferred to a j and if P (X i X j ) < 0, a j is preferred to a i. If neither of these conditions hold, one cannot calculate a preference due to lack of information. This can be seen as a drawback, however according to Walley [7] Some critics have seen this as a weakness of imprecise probabilities, but I see it as a virtue. It simply reflects the fact that, when information is scarce and probability judgments are imprecise, there may be more than one reasonable course of action. 2.5 Previsions and observations Consider a binary experiment i.e. there are two possible outcomes, where one makes observations X,..., X n. Assume that the observations are generated from some process with parameter θ [0, ] and that the observations are conditional independent given this parameter. We want to model how our belief change as we make more observations i.e. when more information becomes available. Also assume that we do not have any prior information about the experiment. In the Bayesian setting the solution would have been to choose a non-informative prior and update with Bayes rule as more observations become available. A problem with this approach [5, Section 5..5] is that one cannot adequately model the amount of current information, i.e., θ = 0.5 can be the best estimate after 000 observation but also reflect you prior ignorance. However, by using previsions one can model this by specifying P (θ) and P (θ), which in the case of complete ignorance becomes the vacuous prevision P (θ) = 0, P (θ) =, and as the number of observations increases = P (θ) P (θ) decreases. Thus 6

7 adequately reflects the amount of information since when more information becomes available one should be able to specify a more precise prevision (price). The reasoning above induces that imprecision is tightly coupled to the amount of available information [5, Section 5..5]. 2.6 Previsions and information aggregation Cooman et al. [] have further extended the theory of lower previsions in order to show that it can be used to solve a problem set in what is called the Sandia challenge problems defined by Oberkampf et al. [4]. The overall problem in this set is to calculate the total amount of uncertainty for a system response function y = (a + b) a (4) given assessments, closed intervals of the parameters a and b. Problem 2 and 3 involves assessments from several, possible conflicting but equally reliable sources, i.e., the intersections of intervals can be empty. Let { } n ( n kp k )(Z) = inf P (Z) : P M(P k ) (5) ( n kp k )(Z) = inf { P (Z) : P k= } n M(P k ) k= (6) where M(P k ) is the set of previsions that pointwise dominates P k. Cooman et al. propose the following protocol for aggregating information from multiple sources []:. If sources are mutually consistent i.e. the intersection of the closed intervals is non-empty, then n k= P k is the prevision that every source accepts 2. If sources are mutually inconsistent, discard those sources which cannot be trusted and use the remaining sources as in 3. If the sources are still mutually inconsistent then use n k= P k Rule 3 can be highly imprecise but as Cooman et al. point out There is no unique solution in case of inconsistency: everything depends on how much information is available about the reliability of the given information. 3 Walley s football example This section demonstrates how one can apply the theory of coherent lower previsions to a simple example. 7

8 Example 4 ([8]) We return to the football example as stated in example 2 where a football expert makes the following judgements about the outcome. Not win is at least as probable as win 2. Win is at least as probable as draw 3. Draw is at least as probable as loss The goal is to find the set of previsions M that satisfy the judgements -3. The first step is to translate these judgments into expressions of previsions. Judgement can be expressed as P (I D + I L I W ) 0, which implies that you are willing to exchange I W against I D + I L since you expect this to generate a non-negative reward. Judgements 2 and 3 can be expressed in a similar way by P (I W I D ) 0 and P (I D I L ) 0. Comparative probability judgements are now expressed as constraints on a precise prevision in the following way :. P (I D + I L I W ) 0 2. P (I W I D ) 0 3. P (I D I L ) 0 The next step is to verify that these statements avoid sure loss. If this is not the case there is a combination of gambles that always produce a net loss. Verifying this property is equal to finding at least one solution to the constraints -3 [5], which can be done by LP-techniques. In this case the uniform distribution (P (I W ), P (I D ), P (I L )) = ( 3, 3, 3 ) satisfies the constraints so avoiding sure loss is fulfilled. If this had not been the case, we are forced to go back and reassess the judgements until they satisfy this property. In order to develop a model for making inference, we must now find the set of previsions M that satisfy these judgments. We know from LP theory that M is convex and can therefore be characterised by its convex hull (extreme points). The convex set M in this example has the following extreme points (found by half-space intersection calculations) ext M = { ( 3, 3, 3 ), ( 2, 2, 0), ( 2, 4, 4 )}. We also get from LP theory [3] that a linear function with linear constraints has its maximum or minimum at an extreme point, which implies that it suffices to do calculations at these points. The natural extension E of a gamble X can therefore be calculated as [5] E = min{p (X) : P ext M} (7) E = max{p (X) : P ext M} (8) This procedure will always generate coherent lower previsions [5, Section 2.6.3]. As an example of inference for prevision of new gambles consider I D +I L, which thus becomes equivalent to minimise and maximise among extreme points E(I D + I L ) = min {( ), 8 ( ), ( )} = 2 (9)

9 0 D W 0 L 0 Figure 2: Graphical representation of the model in example 4 {( E(I D + I L ) = max 3 + ) ( ) (, , 4 + )} = (20) If new information via an event I B is obtained about the football game, one can calculate new conditional previsions P (X I B ) by applying the GBR to each point in ext M [5, Section 6.4.2]. The model can be visualised by drawing M in a probability simplex as shown in figure 2. The simplex constitutes the triangular plane in three dimensional space with corner points (, 0, 0), (0,, 0) and (0, 0, ) (the LP-problem loses one degree of freedom due to the probability constraint). 4 Other theories of imprecision 4. Previsions and upper-lower probabilities A problem with upper and lower probabilities is that they cannot model all types of uncertainty, in particular, they cannot model comparative probability 9

10 0 D W 0 L 0 Figure 3: Blue - Upper and lower probabilities, Red - Coherent lower previsions judgements such as A is more probable than B [8]. Consider the three judgments in example 3. The second statement cannot be expressed in terms of upper and lower probabilities, for example P (I W ) P (I D ) is too strong [8]. Assume that one wants to express the model in example 3 through upper and lower probabilities of M. This can be done by maximising and minimising for each event I W, I D and I L over the extreme points [8] 3 P (I W ) 2 4 P (I D) 2 0 P (I L ) 3 (2) If one calculates the set of probabilities that satisfies these upper and lower probability constraints, two additional extreme points: ( 3, 2, ) ( 6 and 5 2, 4, ) 3 are added [8]. This means that one can lose information when summarising with upper and lower probabilities; the new model is not equally specific as M. The problem with upper and lower probabilities is that they can only describe M by drawing lines parallel to each side of the probability simplex (see figure 3, blue region). This drawback is also a property of Dempster-Shafer theory since it is a special case of upper and lower probabilities [6]. Furthermore, in Dempster-Shafer theory there are no methods for verifying consistency between 0

11 model and conclusion, and the interpretation of belief functions is unsettled [6]. Since rules such as Dempster s rule can result in unintuitive conclusions, the importance of clear interpretation and methods for verifying consistency cannot be underestimated. Another drawback with upper and lower probabilities is that the assessments can be unintuitive, i.e. instead of specifying upper and lower bounds (previsions) for a quantity X, one needs to specify an upper and lower bound for a probability of the event that X is in a certain interval [6]. 4.2 Previsions and robust Bayes In the framework of robust Bayes, one uses a set of probability distributions where each distribution is interpreted as a hypotheses for the true distribution [8]. Walley argues [8, 6] that in many cases it is misleading to regard a set of probability distributions as hypothesis for a correct distribution since one cannot properly define the meaning of a correct distribution. He also argues that a set of probability distributions is not as clearly related to decision making and in order to use it as such one needs to calculate upper and lower previsions [5, Section 5.9.4]. Although the two theories are similar they differ in interpretation; robust Bayes emphasise assessments of plausible precise probability distributions for a correct distribution, while the theory of coherent lower previsions emphasises a set of previsions that satisfies certain constraints [5, Section 5.9.4]. 5 Conclusions Imprecise probability is a collection of theories that can be ordered by their ability to model uncertainty. Coherent lower previsions can model most types of uncertainty, including comparative probability judgements, and it contains belief functions, plausibility functions and upper-lower probabilities as special cases. Coherent lower previsions have a clear interpretation that makes it easy to use as a decision theory. There also exist methods for consistency (coherence) and inference with natural extension follows from rationality principles.

12 Figure 4: Visualising previsions about a football game 6 Appendix - Demonstration tool The purpose of this tool is to visualise and calculate implications of comparative probability judgements for the football example. The tool also visualises the consequences of describing a model with upper and lower probabilities instead of coherent lower previsions. It finds all extreme points by calculating half-space intersections with LP-techniques. 2

13 References [] de cooman, G., and Troffaes, M. C. M. Coherent lower previsions in systems modeling: products and aggregation rules. Reliability Engineering & System Safety 85 (2004), [2] Miranda, E. An introduction to the theory of coherent lower previsions. 2nd SIPTA School on Imprecise Probabilities, (2006). [3] Nash, S. G., and Sofer, A. Linear and Nonlinear Programming. McGRaw-Hill, 996. [4] Oberkampf, W. L., Helton, J. C., Joslyn, C. A., Wojtkiewicz, S. F., and Ferson, S. Challenge problems: uncertainty in systems response given uncertain parameters. Reliability Engineering & System Safety 85 (2004), 9. [5] Walley, P. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 99. [6] Walley, P. Measures of uncertainty in expert systems. Artificial Intelligence 83 (996), 58. [7] Walley, P. Coherent upper and lower previsions. Imprecise Probabilities Project, lower prev/culp.pdf (997). [8] Walley, P. Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24 (2000),