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1 The Pacic Institute for the Mathematical Sciences Surprise Maximization D. Borwein Department of Mathematics University of Western Ontario London, Ontario, Canada N6A 5B7 J. M. Borwein CECM Department of Mathematics and Statistics Simon Fraser University Burnaby, B.C., Canada V5A S6 P. Marechal CECM Department of Mathematics and Statistics Simon Fraser University Burnaby, B.C., Canada V5A S6 Preprint number: PIMS-98-9 Received on November 26, 998.

3 Surprise Maximization D. Borwein, J.M. Borwein y and P. Marechal z y Department of Mathematics University of Western Ontario London, Ontario, Canada N6A 5B7 dborwein@@uwo.ca Research supported by NSERC CECM { Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A S6 borwein@@cecm.sfu.ca Research partially supported by NSERC and the Shrum Endowment. z Same address marechal@@cecm.sfu.ca Pacic Institute for the Mathematical Sciences Postdoctoral Fellow. January 8, 999 Abstract The optimization problems arising from an information theoretic formulation of the Surprise Examination (or Unexpected Hanging) Paradox are examined and solved. They provide a nice application of both the Kuhn-Tucker Theorem and the Jensen inequality. Keywords Surprise examination paradox, entropy optimization, Kuhn-Tucker conditions, Jensen inequality.

4 Introduction In this paper, we study the optimization problems arising from an entropic approach to the so-called Surprise Examination (or Unexpected Hanging) Paradox. The idea of such an approach was proposed by Karl Narveson and presented in a recent article by Timothy Y. Chow []. (We shall not discuss here the dierent approaches to the resolution of the paradox itself; the reader interested in these aspects is invited to consult [].) An event (such as a test given by a teacher or a surprise tax audit) occurs once every m days, with probability p i on day i, i ; : : : ; m. We wish to nd a probability distribution that maximizes the average surprise caused by the event when it occurs. We consider a measure of surprise analogous to the one used in the denition of the Shannon entropy (see [2], for example). The surprise on day i will be the negative of the logarithm of the probability that the event occurs on day i given that it has not occurred so far. The event `test occurs on day i' will be simply denoted by i, and its probability will be denoted by P (i) or p i. The event `test does not occur on day i' will be denoted by i. The quantity to be maximized can therefore be written as?? P (i) ln P i ; : : : ; (i? ) : () i Using Bayes' formula for conditional probabilities, we obtain P? i ; : : : ; (i? ) P? ; : : : ; (i? ) i P (i) P? ; : : : ; (i? ) P (i)?? P () + + P (i? ) P (i)? P (i) + + P (m) : Consequently, we are led to consider the following optimization problem: n (P m ) inf S m (p) : u; p o in which S m is the extended real-valued function dened on R m by 8 S m (p) h p ; x >< ln x? x if x and y > ; p i with h(x; y) 4 y if x and y ; m i >: + otherwise, 2

5 and u is the vector whose entries are all equal to. (For all p satisfying the constraint in (P ), S m (p) clearly coincide with the negative of the quantity in (). Note that S m (p) can be regarded as Kullback-Leibler information measure of p relative to its tail q: q (q ; : : : ; q m ) with q i 4 m p ; i ; : : : ; m: Also of interest is the continuous time formulation of the above problem. Suppose that the event occurs at some point in the time interval [; T ]. By analogy with the discrete case, it is reasonable to consider the following optimization problem: n (P) inf S(p) ; p 2 L? [; T ] ; u; p o? in which S is the functional dened on L [; T ] by S(p) h p(t); T t p(t ) dt and u denotes the function identically equal to unity on [; T ]. dt; 2 A preliminary result We rst establish the convexity of (the negative of) our measures of surprise. Lemma The function h dened above is closed and convex. Proof: Immediate from the fact that h is the convex conugate of the indicator function? (; ) C ; where C 4 n (; ) 2 R 2? exp o : 2 From Lemma, we deduce that S m and S are convex. Indeed, we have S m (p) i h(p i ; [Jp] i ) and S(p) h? p(t); [J p](t) dt; in which J is the (m m)-matrix whose entries are m? in the upper triangle (including the diagonal) and elsewhere, and J is the linear mapping dened by J : L? [; T ]? C? [; T ] p 7? [J p](t) T 3 t p(t ) dt :

6 3 Discrete time analysis By the Kuhn-Tucker Theorem (cf. [3], Section 28), for x to be be a solution for (P m ) it is necessary and sucient that the Kuhn-Tucker Conditions are satised : (a)? hu; pi; (b) there exists 2 R such that m (p) hu; i (p). As a matter of fact, it is obvious that the optimal value in (P m ) is not? and one can easily check that (P m ) has a feasible solution in the relative interior of the domain of S m (see [3], Corollary ). Recall that such a, when it exists, is said to be a Lagrange multiplier for (P m ). Clearly, S m is dierentiable in the interior of its domain, and we k (p) ln m k? ik Consequently, Condition (b) becomes ln m k? ik i ; where k 4 p k k p : i? ; k ; : : : ; m: (2) P Now, by denition, m, so that the last of Equations (2) gives ln m? i, from which we obtain the recursion m ; k exp? ; k m? ; : : : ; : (3) Note that, since k? exp? k the above recursion can be rewritten as k+ exp(? k ) exp? k+ m ; k? k exp (? k ) ; k m; : : : ; 2: (4) The values of the k 's can be obtain as shown in Figure below. Figure 2 shows examples of optimal probability distributions, for m 7 and m 5. 4 ;

7 Figure : Recursion for the k 's Figure 2: Optimal distributions for m 7 (left) and m 5 (right). 5

8 Finally, from Condition (a) above and the values of the k 's, we see that the components of p must obey the following recursion : k? p ; p k k? p ; k 2; : : : ; m: (5) The vector p dened above satises Conditions (a) and (b), and therefore solves Problem (P ). Moreover, its components form an increasing sequence. As a matter of fact, we have p k k (p k + : : : + p m ) and p k? k? (p k? + : : : + p m ); from which we deduce, using (4), that p k k (? k? ) p k? k?? exp k? k exp(? k ) (6) exp k? k > : Remark: As pointed out in [], the (optimal) probability that the event occurs on the ith-to-the-last day, given that it has not occurred so far does not depend on m. This is immediate from Recursion (4) and from the equality P (m? i ; : : : ; (m? i? )) p m?i m?i p? m?i : Furthermore, the fact that the k 's are dened via a backward recursion implies that p m?i p m?i? does not depend on m either (cf. Eq. (6)). 2 4 Transitional comments Note rst that we could have obtained the solution by considering the optimization problem n o (Pm) inf Sm(p; q) q Jp ; ; p ; where S m(p; q) 4 h(p ; q ). The corresponding Kuhn-Tucker Conditions read 6

9 (a')? hu; pi and q? Jp; (b') there exist 2 R and ( ; : : : ; m ) 2 R m such that m(p; q) q) (p; q) + : : : + m (p; q) in which we have dened f and f (f ; : : : ; f m ) by f(p; q) 4? hu; pi and f(p; q) 4 q? Jp: It is then easy to check that the 's derived from (a') and (b') coincide with the 's of the previous paragraph multiplied by m. Some immutable characteristics of the optimal probability distribution were pointed out in Remark. One may also be interested in the asymptotic properties of Problem (P m ) when m tends to innity. We shall simply mention here three facts. Firstly, the ratio between the last and the rst components of p (m) tends to a nite value. Indeed, from Eq. (6), we get p m (m) lim m p (m) lim my m 2 (exp (m)? (m) ) 2:32988 : : : : (The numerical computation is ustied here by the inequality exp (m) + ( (m) ) 2 for (m) 2 [; ].) Secondly, mp (m)? (m) tends to as m tends to. As a matter of fact, let us denote t m p (m). Then we see that the sequence ft m g is dened by the recurence t ; t k+ t k exp(?t k ); k ; 2; : : : : We deduce that t?? k+ t? k t? k (exp(t k)? ), which tends to exp () as k tends to innity (since t k tends to ). Consequently, mt m m m? k exp(t k )? t k + mt also tends to. Thirdly, the optimal value V (P m ) of (P m ) tends to as m tends to innity. To prove this claim, we show that lim inf V (P m ) lim sup V (P m ). The rst equality is easily obtained from V (P m ) S m m ; : : : ; m ln m ln 2m ln m?? m 2m : 7

10 (We made use of the Stirling formula in the above approximation.) The second equality results from the following three facts : (i) V (P m ) m + p (m) m ln m? p (m) m ; (ii) m? m tends to as m tends to innity; (iii) m?p (m) m ln m. Here, we have dened m m? i p (m) i ln p(m) i q (m) i+? p (m) i and m m? i p (m) i ln p(m) i q (m) i? p (m) i Fact (i) is an immediate consequence of the denitions. As for (ii), we have m? m? m? i m? p (m) m?i ln(? t i+ )?p (m) m i ln(? t i+ ) ; since t i and mp m (m) O(). Finally the proof of (iii) is deferred to the end of Section 5 (cf. Corollary ) since it relies on Theorem below. 5 Continuous time analysis Theorem For all p 2 L ([; T ]), we have S(p), that is p(t) ln T p(t) R T p(t ) dt? p(t) with equality if and only if p is constant on [; T ]. dt ; Proof: We can assume that p is (almost everywhere) nonnegative, for otherwise S(p). Observe that S(p) p(t) ln p(t) q(t)? p(t) dt? p(t) ln p(t)? p(t) dt + T p(t) ln p(t) dt? T q() ln q(); 8 q (t) ln q(t) dt :

11 in which we have put q(t) 4 [J p](t). The theorem will therefore be proved if we can show that p(t) ln p(t) dt T q() ln q(); (7) with equality if and only if p is constant. Now, applying Jensen's inequality to the strictly convex function g : x 7 x ln x? x yields Z T p(t) p(t) ln T q() q()? p(t) dt?; q() from which (7) follows immediately. 2 Theorem shows that the (unique) solution of Problem (P) is the uniform probability density on [; T ]. Another immediate consequence of Theorem, which brings us back to the considerations of Section 4 is the following result: Corollary With the notation of section 4, we have m?p (m) m ln m: Proof: Apply Theorem with T and p(t) p (m) i if t 2 i? m ; i m (i ; : : : ; m): 2 This completes the proof that the optimal value of (P m ) tends to (which is also the optimal value of (P)), as claimed in Section 4. 6 Conclusion The entropic formulation of the Surprise Examination problem provides a beautiful example of the application of concepts from the theory of convex constrained optimization. Its originality lies mainly in the recursivity of the (discrete time) solution which follows from the Kuhn-Tucker Conditions. 9

12 References [] T. Y. Chow, The surprise examination or unexpected hanging paradox, The American Mathematical Monthly, 5 (998), pp. 4{5. [2] A. Renyi, Calcul des probabilites. Avec un appendice sur la theorie de l'information, Dunod, Paris, 966. Traduit de l'allemand par C. Bloch. Collection Universitaire de Mathematiques, No. 2. [3] R. T. Rockafellar, Convex analysis, Princeton University Press, Princeton, N.J., 97. Princeton Mathematical Series, No. 28.

Examples of Convex Functions and Classifications of Normed Spaces

Journal of Convex Analysis Volume 1 (1994), No.1, 61 73 Examples of Convex Functions and Classifications of Normed Spaces Jon Borwein 1 Department of Mathematics and Statistics, Simon Fraser University