Large-Scale Signal Detection: A Unified Perspective

Abstract

There is an overwhelmingly large literature and algorithms already available on "large-scale inference problems" based on different modeling techniques and cultures. Our primary goal in this article is not to add one more new methodology to the existing toolbox but instead (i) to clarify the mystery how these different simultaneous inference methods are connected, (ii) to provide an alternative more intuitive derivation of the formulas that leads to simpler expressions in order (iii) to develop a unified algorithm for practitioners. A detailed discussion on representation, estimation, inference, and model selection is given. Applications to a variety of real and simulated datasets show promise. We end with several future research directions.

Detecting new signals under background mismodeling

“…At the higher end of the spectrum, the postulated background (red dashed line) underestimates the true background distribution (green solid line). As a result, using (12) as background model increases the chance of false discoveries in this region. Conversely, at the lower end of the spectrum, g b (x) underestimates f b (x), reducing the sensitivity of the analysis.…”

“…are the true and the postulated background distributions, with pdfs as in (11) and (12), respectively. In our example, choosing M = 2 (see Section IV C), we obtain…”

“…The quantity ∑ M j=1 LP 2 j is known as deviance [12] and quantifies the departure of d(u; G, F) from one. If the deviance is equal to zero, we may expect that g is approximately equivalent to f ; hence, we test…”

“…First, a value M = 2 has been selected as in (27), and the respective deviance test (adequately adjusted via Bonferroni) indicates that the deviation of f b from g b is significant at a 6.430σ significance level (adjusted p-value of 6.399 · 10 −11 ). Additionally, the estimated comparison density in (15) lies outside the 2σ confidence bands in the region [0, 50] where the signal specified in (13) is expected to occur; hence, using (14) instead of (12) is recommended in order to improve the sensibility of the analysis in the signal region.…”

“…This approach allows the unification of many of the standard results of classical statistics by expressing them in terms of quantiles and comparison distributions and provides a simple and powerful framework for statistical learning and data analysis. The interested reader is directed to [9][10][11][12][13] and references therein, for recent advancements in mode detection, nonparametric time series, goodness-of-fit on prior distributions, and large-scale inference using an LP approach.…”