Bandit Algorithms

2.6 Conditional expectation 32 Proposition2.8.IfX≥ 0 is a nonnegative random variable, then E[X] = ∫∞ 0 P(X > x)dx. The integ ...

2.6 Conditional expectation 33 Eq. (2.8) does not generalize to continuous random variables becauseP(Y=y) in the denominator mig ...

2.6 Conditional expectation 34 conditional expectation ofXgivenHis denoted byE[X|H] and defined to be anyH-measurable random var ...

2.6 Conditional expectation 35 andE[X|H] is uniquely defined only up to events ofP-measure zero, none of this should be of a sig ...

2.7 Notes 36 The above list of abstract properties will be used over and over again. We encourage the reader to study the list c ...

2.7 Notes 37 5 Can you think of a set that is not Borel measurable? Such sets exist, but do not arise naturally in applications. ...

2.7 Notes 38 normal distribution and its density p(x) = 1 √ 2 π exp(−x^2 /2), (2.10) which can be integrated over intervals to o ...

2.7 Notes 39 is thatQis forcingP(to be nil when it is nil). One can also remember the direction ofby thinking (symbolically) o ...

2.8 Bibliographic remarks 40 15 Thesupportof a measureμon (X,B(X)) is Supp(μ) ={x∈X:μ(U)>0 for all neighborhoodsUofx}. WhenXi ...

2.9 Exercises 41 proofs and the book is comprehensive. The factorization lemma (Lemma 2.5) is stated in the book by Kallenberg [ ...

2.9 Exercises 42 Hint As suggested after the lemma, this can be arranged for by makingGcoarse- grained. Hence, choose Ω =Y=X=R,X ...

2.9 Exercises 43 (j)Is it true or not thatA,B,C are mutually independent if and only if P(A∩B∩C) =P(A)P(B)P(C)? Prove your claim ...

2.9 Exercises 44 2.17 Demonstrate using an example that in general, for dependent random variables,E[XY] =E[X]E[Y] does not hold ...

This material will be published by Cambridge University Press as Bandit Algorithms by Tor Lattimore and Csaba Szepesvari. This p ...

3.1 Stochastic processes 46 infinite sequence such that x= ∑∞ t=1 Ft(x)2−t. We can viewF 1 ,F 2 ,...as (binary valued) random va ...

3.2 Markov chains 47 3.2 Markov chains A Markov chain is an infinite sequence of random elementsX 1 ,X 2 ,...where the condition ...

3.3 Martingales and stopping times 48 The word ‘homogeneous’ refers to the fact that the probability kernel does not change with ...

3.3 Martingales and stopping times 49 the gambler at the end of roundtcan decide to stop (δt= 1) or continue (δt= 0) based on th ...

3.4 Notes 50 live a very long life! One application of Doob’s optional stopping theorem is a useful and apriori surprising gener ...

3.4 Notes 51 where Θ is the parameter space andPθis a measure on some measurable space (Ω,F). This notation is often more conven ...