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3466Capacity of a Ternary Test With Uneven Follow-UpsA first test has three outcomes. If outcome A occurs, you may ask two binary follow-up questions; if outcome B occurs, you may ask one ternary follow-up; if outcome C occurs, the process must stop. What is the maximum number of equally likely states you can distinguish?数学简单derivation未尝试面试订阅3467Capacity of a Binary Gate With Different Menus on Each SideA first yes/no question is asked. On the yes branch, you may run one four-way diagnostic. On the no branch, you may run two further yes/no diagnostics. What is the maximum number of states that can be distinguished?数学简单derivation未尝试面试订阅3468Capacity of a Four-Way Triage With Partial Follow-UpA first test has four outcomes. On the first two branches you may ask one binary follow-up each; on the last two branches you must stop immediately. What is the maximum number of states you can distinguish?数学简单derivation未尝试面试订阅3469Capacity of a Ternary Gate With a Five-Way EscalationA first test has outcomes left, middle, right. If left occurs, you may run one five-way escalation test. If middle or right occurs, you may run one binary follow-up on that branch. How many states can be distinguished in total?数学简单derivation未尝试面试订阅3470Capacity of a Binary Gate With Deep and Shallow BranchesA first yes/no question is asked. If yes, you may then ask one ternary question followed by one binary question. If no, you may ask only one ternary question. What is the maximum number of distinguishable states?数学简单derivation未尝试面试订阅3471One Ternary Plus Three Binary Assays for One Fault or NoneYou may run one ternary assay and three binary assays non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3472Two Ternary and One Binary Assays for One Fault or NoneYou may run two ternary assays and one binary assay non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3473One Four-Way and Two Binary Assays for One Fault or NoneYou may run one four-way assay and two binary assays non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3474Binary Assays When Each Faulty Unit Can Fail in Two ModesYou have three yes/no assays run non-adaptively. Either no component is faulty, or exactly one component is faulty and it can be in one of two distinct failure modes. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3475One Ternary and Two Binary Assays With Three Bad Modes Per ComponentYou may run one ternary assay and two binary assays non-adaptively. Either nothing is wrong, or exactly one component is wrong and it can be in one of three bad modes. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3476Entropy Lower Bound for a 1/2, 1/4, 1/8, 1/8 PriorA hidden state has prior probabilities [0.5, 0.25, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3477Entropy Lower Bound for a 1/2, 1/4, 1/4 PriorA hidden state has prior probabilities [0.5, 0.25, 0.25]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3478Entropy Lower Bound for a 1/4, 1/4, 1/4, 1/8, 1/8 PriorA hidden state has prior probabilities [0.25, 0.25, 0.25, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3480Entropy Lower Bound for a Six-State Dyadic PriorA hidden state has prior probabilities [0.25, 0.25, 0.125, 0.125, 0.125, 0.125]. What is the Shannon lower bound, in yes/no questions on average, for any adaptive identification strategy?数学中等derivation未尝试面试订阅3481Expected Questions in a Hot-State-First Binary TreeA binary decision tree asks first whether the state is S1. If not, it asks whether it is S2. If not, it asks whether it is S3, and otherwise concludes S4. If prior probabilities are [0.5, 0.25, 0.15, 0.10], what is the expected number of questions?数学中等derivation未尝试面试订阅3482Expected Questions for a Balanced Three-State TriageA binary tree first asks whether the state is in S1, S2 . If yes, one more question separates S1 from S2. If no, the answer is S3. If prior probabilities are [0.5, 0.25, 0.25], what is the expected number of questions?数学中等derivation未尝试面试订阅3483Expected Questions With a Rare-State TailA binary tree uses code lengths [2, 2, 2, 3, 3] for states with priors [1/4, 1/4, 1/4, 1/8, 1/8]. What is the expected number of questions?数学中等derivation未尝试面试订阅3484Expected Questions Under a Heavy First StateA binary tree uses code lengths [1, 3, 3, 3, 3] for priors [1/2, 1/8, 1/8, 1/8, 1/8]. What is the expected number of questions?数学中等derivation未尝试面试订阅3485Expected Questions in a Ternary-Then-Binary StrategyA strategy first asks a ternary question that isolates state A on one branch, state B on another branch, and groups states C and D on the last branch. If the prior over [A, B, C, D] is [0.4, 0.3, 0.2, 0.1], what is the expected number of questions if the grouped branch uses one extra yes/no question?数学中等derivation未尝试面试订阅3486Why Adaptive Testing Beats Static MenusWhy can an adaptive questioning strategy distinguish more states than a non-adaptive test battery with the same primitive tests?数学中等essay未尝试面试订阅