A framework for generalized group testing with inhibitors and its potential application in neuroscience

The main goal of group testing with inhibitors (GTI) is to efficiently identify a small number of defective items and inhibitor items in a large set of items. A test on a subset of items is positive if the subset satisfies some specific properties. Inhibitor items cancel the effects of defective items, which often make the outcome of a test containing defective items negative. Different GTI models can be formulated by considering how specific properties have different cancellation effects. This work introduces generalized GTI (GGTI) in which a new type of items is added, i.e., hybrid items. A hybrid item plays the roles of both defectives items and inhibitor items. Since the number of instances of GGTI is large (more than 7 million), we introduce a framework for classifying all types of items non-adaptively, i.e., all tests are designed in advance. We then explain how GGTI can be used to classify neurons in neuroscience. Finally, we show how to realize our proposed scheme in practice

Error-Tolerant Non-Adaptive Learning of a Hidden Hypergraph

We consider the problem of learning the hypergraph using edge-detecting queries. In this model, the learner is allowed to query whether a set of vertices includes an edge from a hidden hypergraph. Except a few, all previous algorithms assume that a query\u27s result is always correct. In this paper we study the problem of learning a hypergraph where alpha -fraction of the queries are incorrect. The main contribution of this paper is generalizing the well-known structure CFF (Cover Free Family) to be Dense (we will call it DCFF - Dense Cover Free Family) while presenting three different constructions for DCFF. Later, we use these constructions wisely to give a polynomial time non-adaptive learning algorithm for a hypergraph problem with at most alpha-fracion incorrect queries. The hypergraph problem is also known as both monotone DNF learning problem, and complexes group testing problem

Concomitant Group Testing

In this paper, we introduce a variation of the group testing problem capturing the idea that a positive test requires a combination of multiple ``types'' of item. Specifically, we assume that there are multiple disjoint \emph{semi-defective sets}, and a test is positive if and only if it contains at least one item from each of these sets. The goal is to reliably identify all of the semi-defective sets using as few tests as possible, and we refer to this problem as \textit{Concomitant Group Testing} (ConcGT). We derive a variety of algorithms for this task, focusing primarily on the case that there are two semi-defective sets. Our algorithms are distinguished by (i) whether they are deterministic (zero-error) or randomized (small-error), and (ii) whether they are non-adaptive, fully adaptive, or have limited adaptivity (e.g., 2 or 3 stages). Both our deterministic adaptive algorithm and our randomized algorithms (non-adaptive or limited adaptivity) are order-optimal in broad scaling regimes of interest, and improve significantly over baseline results that are based on solving a more general problem as an intermediate step (e.g., hypergraph learning).Comment: 15 pages, 3 figures, 1 tabl

Conservative statistical post-election audits

There are many sources of error in counting votes: the apparent winner might not be the rightful winner. Hand tallies of the votes in a random sample of precincts can be used to test the hypothesis that a full manual recount would find a different outcome. This paper develops a conservative sequential test based on the vote-counting errors found in a hand tally of a simple or stratified random sample of precincts. The procedure includes a natural escalation: If the hypothesis that the apparent outcome is incorrect is not rejected at stage $s$, more precincts are audited. Eventually, either the hypothesis is rejected--and the apparent outcome is confirmed--or all precincts have been audited and the true outcome is known. The test uses a priori bounds on the overstatement of the margin that could result from error in each precinct. Such bounds can be derived from the reported counts in each precinct and upper bounds on the number of votes cast in each precinct. The test allows errors in different precincts to be treated differently to reflect voting technology or precinct sizes. It is not optimal, but it is conservative: the chance of erroneously confirming the outcome of a contest if a full manual recount would show a different outcome is no larger than the nominal significance level. The approach also gives a conservative $P$-value for the hypothesis that a full manual recount would find a different outcome, given the errors found in a fixed size sample. This is illustrated with two contests from November, 2006: the U.S. Senate race in Minnesota and a school board race for the Sausalito Marin City School District in California, a small contest in which voters could vote for up to three candidates.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS161 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org