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In Search of a Global Belief Model for Discrete-Time Uncertain Processes

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In Search of a Global Belief Model for Discrete-Time Uncertain Processes View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Ghent University Academic Bibliography ISIPTA 2019 In Search of a Global Belief Model for Discrete-Time Uncertain Processes Natan T’Joens NATAN.TJOENS@UGENT.BE Jasper De Bock JASPER.DEBOCK@UGENT.BE Gert de Cooman GERT.DECOOMAN@UGENT.BE ELIS – FLip, Ghent University, Belgium Abstract partial information about a ‘precise’ probability measure To model discrete-time uncertain processes, we argue or charge. The game-theoretic framework by Shafer and for the use of a global belief model in the form of an Vovk has no need for such assumptions. However, since upper expectation that satisfies a number of simple it defines global upper expectations in a constructive way and intuitive axioms. We motivate these axioms on using the concept of a ‘supermartingale’, it lacks a concrete the basis of two possible interpretations for this upper identification in terms of mathematical properties. expectation: a behavioural interpretation similar to that Our aim here is to establish a global belief model, in the of Walley’s, and an interpretation in terms of upper en- form of an upper expectation, that extends the information velopes of linear expectations. Subsequently, we show gathered in the local models by using a number of mathe- that the most conservative upper expectation satisfy- ing our axioms coincides with a particular version of matical properties. Notably, this model will not be bound to the game-theoretic upper expectation introduced by a single interpretation. Instead, its characterising properties Shafer and Vovk. This has two important implications. can be justified starting from a number of different inter- On the one hand, it guarantees that there is a unique pretations. We here consider and discuss two of the most most conservative global belief model satisfying our significant; see Section4. We will then define the desired axioms. On the other hand, it shows that Shafer and global model as the most conservative—the unique model Vovk’s model can be given an axiomatic characterisa- that does not include any additional information apart from tion, thereby providing an alternative motivation for what is given—under this particular set of properties. In adopting this model, even outside their framework. Section5, we give a complete axiomatisation for this most Keywords: Game-theoretic probability, Upper expec- conservative model, serving as an alternative definition. tations, Uncertain processes, Coherence Finally and most importantly, we show that the obtained model is equal to a version of the global upper expectation 1. Introduction defined by Shafer and Vovk [3, 4]. On the one hand, this serves as an additional motivation for the use of our model. There are various ways in which discrete-time uncertain On the other hand, it gives a concrete axiomatisation for processes, such as Markov processes, can be described this game-theoretic upper expectation. mathematically. For many, measure theory has been the preferred framework to describe the uncertain dynamics of these processes. Others may use martingales or a game- 2. Upper Expectations theoretic approach to do so. The common starting point for all these approaches are the local belief models. They We denote the set of all natural numbers, without 0, by N, describe the dynamics of the process from one time instant and let N0 := N [ f0g. The set of extended real numbers to the next. In a measure-theoretic context, they are given is denoted by R := R [ f+¥;−¥g and is endowed with in the form of (sets of) probability charges or (sets of) mea- the usual order topology (corresponding to the two-point sures on the local state space; in a game-theoretic context, compactification of R). The set of positive real numbers is sets of allowable bets are used. When local state-spaces denoted by R> and the set of non-negative real numbers by are assumed finite, as in our case, these descriptions are R≥. We also adopt the conventions that +¥ − ¥ = −¥ + all mathematically equivalent. However, it is how these ¥ = +¥ and 0 · (+¥) = 0 · (−¥) = 0. local models are extended to a global level that differs Informally, we will consider a subject that is uncertain greatly from one theory to another. Measure theory uses the about the value that some variable Y assumes in a non- concept of sigma-additivity to do this, leading to a math- empty set Y . More formally, we call any map on Y a ematically elegant, but rather abstract framework. Apart variable; our informal Y is a special case: it corresponds to from being bounded to the constraint of measurability, it the identity map on Y . A subject’s uncertainty about the moreover relies on the questionable assumption of ‘preci- unknown value of Y can then be represented by an upper sion’, in the sense that imprecision is always regarded as expectation E: an extended real-valued map on some subset © N. T’Joens, J. De Bock & G. de Cooman. IN SEARCH OF A GLOBAL BELIEF MODEL FOR DISCRETE-TIME UNCERTAIN PROCESSES D of the set L (Y ) of all extended real-valued variables on expectations on L (Y ) that are self-conjugate, meaning Y . An element f of L (Y ) is simply called an extended that E( f ) = −E(− f ) for all f 2 L (Y ). According to [1, real variable. We say that f is bounded below if there is a Theorem 8.15], linear expectations on L (Y ) are on their real c such that f (y) ≥ c for all y 2 Y , and we say that f is turn in a one-to-one relation with probability charges on bounded above if − f is bounded below. An important role the powerset P(Y ) of Y , being maps m : P(Y ) ! R≥ that will be reserved for elements f of L (Y ) that are bounded, are finitely additive and where m(Y ) = 1 and m(/0) = 0. meaning that they are bounded above and below. These These probability charges are more general than the conven- bounded real-valued variables on Y are called gambles, tional notion of a probability measure, which additionally and we use L (Y ) to denote the set of all of them. The requires s-additivity. If Y is finite, however, this distinc- set of all bounded below elements of L (Y ) is denoted by tion disappears. L b(Y ). It follows from the discussion above, that coherent up- Consider now the special case that E is at least defined on per expectations can be interpreted in two possible ways: the set of all bounded real-valued variables; so L (Y ) ⊆ D. in a direct behavioural way in terms of selling prices for Then we call E coherent [9] if it satisfies the following three gambles, or as a supremum over—an upper envelope of—a coherence axioms: set of linear expectations. In this paper, we will not bound ourselves to any of these two interpretations. Instead, we C1. E( f ) ≤ sup f for all f 2 ( ); L Y will motivate the defining properties of our proposed global C2. E( f + g) ≤ E( f ) + E(g) for all f ;g 2 L (Y ); belief model in terms of both of these interpretations. We conclude this section by introducing a method for C3. E(l f ) = lE( f ) for all l 2 R≥ and f 2 L (Y ). extending the domain of a coherent upper expectation on If we let E be the conjugate lower expectation associated L (Y ) to the set L b(Y ) of all extended real variables that with E, meaning that E(− f ) := −E( f ) for all f 2 D, the are bounded below. This will prove to be particularly useful following additional properties follow from C1–C3: when we introduce game-theoretic upper expectations in Section6. We limit ourselves to the special case where C4. f ≤ g ) E( f ) ≤ E(g) for all f ;g 2 L (Y ); Y is finite. To obtain the desired upper expectation E0 on L b(Y ), we will impose the following continuity property: C5. inf f ≤ E( f ) ≤ E( f ) ≤ sup f for all f 2 L (Y ); C8. For any non-decreasing sequence f fng in L (Y ): C6. E( f +m) = E( f )+m for all m 2 R and all f 2 L (Y ); n2N0 0 0 1 C7. for any sequence f fngn2N0 in L (Y ): lim E ( fn) = E ( lim fn): n!+¥ n!+¥ lim supj f − fnj = 0 ) lim E( fn) = E( f ): n!+¥ n!+¥ If limn!+¥ fn is a gamble, then if Y is finite, C8 is implied by C7, and therefore a consequence of coherence (C1–C3). Proof of C4–C7 We only prove C5. This clearly implies Property C8 can therefore be regarded as a generalisation that E is real-valued on L (Y ), and the remaining proper- of C7 to extended real variables that are bounded below. ties then follow from [9, Section 2.6.1.]. Moreover, any coherent upper expectation E on L (Y ) can ( ) = First, note that E 0 0 because of C3 and our con- be uniquely extended to L b(Y ) if we impose C8. vention that 0 · (+¥) = 0 · (−¥) = 0. Therefore, for all f 2 ( ) ≤ ( f ) + (− f ) L Y , it follows from C2 that 0 E E , or Proposition 1 Consider any finite set Y and a coherent + − = − + equivalently, due to our convention that ¥ ¥ ¥ upper expectation E on L (Y ). Then there exists a unique ¥ = +¥, that −E(− f ) ≤ E( f ). Applying C1 to both sides, 0 coherent upper expectation E on L b(Y ) that satisfies C8 f = − (− f ) ≤ − (− f ) ≤ ( f ) ≤ f we find that inf sup E E sup .
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