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Lecture# Differential Privacy: Hypothesis Testing and Composition

Description

This lecture covers the concept of differential privacy, focusing on partial secrecy and hypothesis testing. It explains the conditions for a mechanism to be (ε,δ)-differentially private and how to test hypotheses for differential privacy. The lecture also delves into the composition of differentially private mechanisms, discussing homogeneous and heterogeneous compositions. The speaker explores the privacy region for differential privacy and the optimal privacy region under composition. The lecture concludes by addressing the question of how privacy degrades after repeated accesses and the tightest bound on privacy under composition.

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Composition of relations

In the mathematics of binary relations, the composition of relations is the forming of a new binary relation R; S from two given binary relations R and S. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions. The word uncle indicates a compound relation: for a person to be an uncle, he must be the brother of a parent.

Pea

The pea is most commonly the small spherical seed or the seed-pod of the flowering plant species Pisum sativum. Each pod contains several peas, which can be green or yellow. Botanically, pea pods are fruit, since they contain seeds and develop from the ovary of a (pea) flower. The name is also used to describe other edible seeds from the Fabaceae such as the pigeon pea (Cajanus cajan), the cowpea (Vigna unguiculata), and the seeds from several species of Lathyrus. Peas are annual plants, with a life cycle of one year.

Homogeneous relation

In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product X × X. This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people. Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations.

Pigeon pea

The pigeon pea (Cajanus cajan) is a perennial legume from the family Fabaceae native to the Old World. The pigeon pea is widely cultivated in tropical and semitropical regions around the world, being commonly consumed in South Asia, Southeast Asia, Africa, Latin America and the Caribbean. The scientific name for the genus Cajanus and the species cajan derive from the Malay word katjang meaning legume in reference to the bean of the plant.

Test statistic

A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing. A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test. In general, a test statistic is selected or defined in such a way as to quantify, within observed data, behaviours that would distinguish the null from the alternative hypothesis, where such an alternative is prescribed, or that would characterize the null hypothesis if there is no explicitly stated alternative hypothesis.