scholarly journals Sequential Mode Estimation with Oracle Queries

2020 ◽  
Vol 34 (04) ◽  
pp. 5644-5651
Author(s):  
Dhruti Shah ◽  
Tuhinangshu Choudhury ◽  
Nikhil Karamchandani ◽  
Aditya Gopalan
Keyword(s):  
Probability Distribution ◽  
Lower Bounds ◽  
Pac Learning ◽  
Query Complexity ◽  
Underlying Distribution ◽  
Mode Estimation ◽  
Estimation Algorithms ◽  
Sequential Mode ◽  
Two Indices

We consider the problem of adaptively PAC-learning a probability distribution 𝒫's mode by querying an oracle for information about a sequence of i.i.d. samples X1, X2, … generated from 𝒫. We consider two different query models: (a) each query is an index i for which the oracle reveals the value of the sample Xi, (b) each query is comprised of two indices i and j for which the oracle reveals if the samples Xi and Xj are the same or not. For these query models, we give sequential mode-estimation algorithms which, at each time t, either make a query to the corresponding oracle based on past observations, or decide to stop and output an estimate for the distribution's mode, required to be correct with a specified confidence. We analyze the query complexity of these algorithms for any underlying distribution 𝒫, and derive corresponding lower bounds on the optimal query complexity under the two querying models.

scholarly journals Quantum conditional query complexity

2017 ◽  
Vol 17 (7&8) ◽  
pp. 541-567
Author(s):  
Imdad S.B. Sardharwalla ◽  
Sergii Strelchuk ◽  
Richard Jozsa
Keyword(s):  
Boolean Functions ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Query Complexity ◽  
Equivalence Testing ◽  
Conditional Probabilities ◽  
Underlying Distribution ◽  
New Type ◽  
Testing Equivalence ◽  
Oracle Access

We define and study a new type of quantum oracle, the quantum conditional oracle, which provides oracle access to the conditional probabilities associated with an underlying distribution. Amongst other properties, we (a) obtain highly efficient quantum algorithms for identity testing, equivalence testing and uniformity testing of probability distributions; (b) study the power of these oracles for testing properties of boolean functions, and obtain an algorithm for checking whether an n-input m-output boolean function is balanced or e-far from balanced; and (c) give an algorithm, requiring O˜(n/e) queries, for testing whether an n-dimensional quantum state is maximally mixed or not.

Choice Functions Generated by Mallows and Plackett–Luce Relations

2019 ◽  
Vol 15 (02) ◽  
pp. 191-213 ◽  
Author(s):  
Vidal Kamdem Tagne ◽  
Siméon Fotso ◽  
Louis Aimé Fono ◽  
Eyke Hüllermeier
Keyword(s):  
Probability Distribution ◽  
Structural Properties ◽  
Probability Distributions ◽  
Preference Relation ◽  
Choice Functions ◽  
Strong Completeness ◽  
Underlying Distribution ◽  
Preference Relations ◽  
Fuzzy Preference Relations ◽  
Consistency Properties

The rationality and consistency of preference-based choice functions is often studied for (fuzzy) preference relations having specific properties, such as strong completeness, transitivity, or certain properties on triplets. In this paper, we turn our attention to another type of preference relation, namely relations that are induced as pairwise marginal of an underlying probability distribution on complete rankings (permutations) of all given alternatives. Such relations are necessarily reciprocal and, depending on the underlying distribution, obey additional structural properties. More specifically, we study relations induced by two important probability distributions, the Mallows and the Plackett–Luce distribution, which are well-known in the literature on statistics of rank data. Assuming preference relations of this kind, we study seven different choice functions and their four rationality and consistency properties.

Some Properties of Divisor Methods for Legislative Apportionment and Proportional Representation

1982 ◽  
Vol 76 (3) ◽  
pp. 575-584 ◽  
Author(s):  
Cyril Carter
Keyword(s):  
Probability Distribution ◽  
Lower Bounds ◽  
Proportional Representation ◽  
Upper And Lower Bounds ◽  
Rigorous Analysis ◽  
Political Choice ◽  
Divisor Methods ◽  
Divisor Method

Some rather elegant properties of linear divisor methods are derived and used to establish upper and lower bounds on the possible variation between apportionment and exact quota entitlement. A probability distribution is derived for this variation, and it is shown that the probability of the variation exceeding one seat is very small with the major fractions linear divisor method.A less rigorous analysis of the nonlinear equal proportions method shows that in practice it is very similar to the major fractions method, but with a very slight bias in favor of small parties (or states). It is concluded that there is no “best” apportionment method, but a knowledge of the properties of the various methods enables a political choice of the most appropriate method.

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