nalyzing Covariance Matrix for Independent Likelihoods in Frequency Bins

by Phenomenology TechnologyOctober 28th, 2024

Too Long; Didn't Read

This article discusses the covariance matrix in relation to total likelihoods across frequency bins, emphasizing a signal-only analysis. By computing the covariance, we find that the peaks are statistically independent, enabling the total likelihood to be expressed as the product of individual likelihoods.

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Authors:

(1) Dorian W. P. Amaral, Department of Physics and Astronomy, Rice University and These authors contributed approximately equally to this work;

(2) Mudit Jain, Department of Physics and Astronomy, Rice University, Theoretical Particle Physics and Cosmology, King’s College London and These authors contributed approximately equally to this work;

(3) Mustafa A. Amin, Department of Physics and Astronomy, Rice University;

(4) Christopher Tunnell, Department of Physics and Astronomy, Rice University.

Table of Links

Abstract and 1 Introduction

2 Calculating the Stochastic Wave Vector Dark Matter Signal

2.1 The Dark Photon Field

2.2 The Detector Signal

3 Statistical Analysis and 3.1 Signal Likelihood

3.2 Projected Exclusions

4 Application to Accelerometer Studies

4.1 Recasting Generalised Limits onto B − L Dark Matter

5 Future Directions

6 Conclusions, Acknowledgments, and References

A Equipartition between Longitudinal and Transverse Modes

B Derivation of Marginal Likelihood with Stochastic Field Amplitude

C Covariance Matrix

D The Case of the Gradient of a Scalar

C Covariance Matrix

To treat the total likelihood as the product of the individual likelihoods in each frequency bin, we must check that the covariance matrix is diagonal.

We will consider a signal-only analysis, discarding the noise, since the noise merely adds to the power and is uncorrelated between different frequency bins. We may write the values of the three peaks as

We wish to compute the quantity

We can do this using the expression for the raw moments,

where, for us, σ = 1/ √ 2. Aside from this, we need to know that

We then get that

Crucially, we get that the covariance between peaks is 0, allowing us to treat them as statistically independent and hence permitting us to express the total likelihood as the product of the individual likelihoods.