# Bryan’s Maximum Entropy Method—Diagnosis of a Flawed Argument and Its Remedy

## Abstract

**:**

## 1. Introduction

## 2. Diagnosis of the Problem in Bryan’s MEM

#### 2.1. Tikhonov Regularization

#### 2.2. Maximum Entropy Method

#### 2.3. Numerical Evidence for the Inadequacy of the SVD Subspace

## 3. Remedy of the Problem

#### 3.1. Staying within MEM

#### 3.2. Beyond MEM

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BFGS | Broyden–Fletcher–Goldfarb–Shanno |

BR | Bayesian Reconstruction |

MEM | Maximum Entropy Method |

SJ | Shannon–Jaynes |

SVD | singular value decomposition |

TK | Tikhonov |

## Appendix A. Explicit SVD Example of (Non-) Projection to the Null-Space

## Appendix B. Explicit MEM Example with a Solution Outside of the SVD Subspace

## References

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**Figure 1.**The mock function $\rho $ (

**left**) and corresponding mock data (

**right**) deployed in the explicit example discussed in the main text, which shows that the extremum of the posterior is not necessarily located within the singular value decomposition (SVD) subspace.

**Figure 2.**The SVD basis functions ${U}_{1}\left(\omega \right)$, ${U}_{2}\left(\omega \right)$, and ${U}_{3}\left(\omega \right)$, which Bryan’s argument suggests should capture the extremum of the posterior for all three mock functions ${\rho}_{i}$ after exponentiation. Note that the SVD basis functions flatten out about $\omega =10$. Their domain extends to ${\omega}_{\mathrm{max}}=1000$ but their value stays close to zero.

**Figure 3.**Comparison of the behavior of the integrand of the Tikhonov (TK), MEM, and Bayesian Reconstruction (BR) regulators for the choice $\alpha =0.1$ and $m=1$ using (

**left**) a log-lin scale and (

**right**) a log-log scale. Note that the flattening off of the Shannon–Jaynes entropy towards vanishing $\rho $.

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Rothkopf, A.
Bryan’s Maximum Entropy Method—Diagnosis of a Flawed Argument and Its Remedy. *Data* **2020**, *5*, 85.
https://doi.org/10.3390/data5030085

**AMA Style**

Rothkopf A.
Bryan’s Maximum Entropy Method—Diagnosis of a Flawed Argument and Its Remedy. *Data*. 2020; 5(3):85.
https://doi.org/10.3390/data5030085

**Chicago/Turabian Style**

Rothkopf, Alexander.
2020. "Bryan’s Maximum Entropy Method—Diagnosis of a Flawed Argument and Its Remedy" *Data* 5, no. 3: 85.
https://doi.org/10.3390/data5030085