By Thomas Ondra
Two-armed response-adaptive medical trials are modelled as Markov determination difficulties to pursue overriding pursuits: to begin with, to spot some of the best remedy on the finish of the trial and, secondly, to maintain the variety of sufferers receiving the inferior therapy small. Such medical trial designs are vitally important, specifically for infrequent ailments. Thomas Ondra provides the most resolution thoughts for Markov determination difficulties and gives a close description how one can receive optimum allocation sequences.
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Extra info for Optimized Response-Adaptive Clinical Trials: Sequential Treatment Allocation Based on Markov Decision Problems
3) = P π (Y1 = a|X1 = s) = P π (X1 = s, Y1 = a|X1 = s). 2) is true for t = 1, . . , n − 1 then P π (Xn = σ|X1 = s) P π (Xn−1 = k, Yn−1 = a|X1 = s)p(σ|k, a) = k∈S a∈Ak P π˜ (Xn−1 = k, Yn−1 = a|X1 = s)p(σ|k, a) = = k∈S a∈Ak P π˜ (Xn = σ|x1 = s). Finally, we calculate P π˜ (Xn = σ, Yn = a|X1 = s) = P π˜ (Yn = a|Xn = σ)P π˜ (Xn = σ|X1 = s) = P π (Yn = a|Xn = σ, X1 = s)P π (Xn = σ|X1 = s) = P π (Xn = σ, Yn = a|X1 = s). Now we are introducing some vector based notation, which we use in the rest of the chapter.
This would ensure the existence of a solution of the Bellman equations. To do so we need to show that B : V → V is a contraction mapping. 12. The Operator B : V → V is a contraction mapping. Proof. Let v ∈ V be arbitrary. Bv = supd∈DDM rd + λPd v = supd∈DDM rd + λPd v ≤ supd∈DDM rd + λ Pd v ≤ c + λ v ≤ c˜, hence B : V → V . If u = v the inequality Bv − Bu ≤ K u − v is fulﬁlled trivially for every K ∈ R, so let u = v. Now choose ε > 0 so that √ ε< v−u . √ Observe that this yields ε < ε v − u . We will use this fact later.
D RH If S and A are ﬁnite and the rewards are bounded then the supremum exists and can be replaced by the maximum of the right hand ∗ (s) = v π ∗ (s), side of the equation above. Clearly, in this case we have vN N so the expected total reward equals the value of a Markov decision problem if ∗an optimal policy π ∗ is used. If we use a ε–optimal policy π ∗ (s). 1 Policy Evaluation Now we want to ﬁnd a method which allows us to calculate the π (s) for a given policy π. We want to do this expected total reward vN in a backward inductive way.