# Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Unfolding

#### 2.1. Simulated Spectra

#### 2.2. Defining Equations of Unfolding

#### 2.3. Likelihood Function $P\left(\mathbf{D}\right|\mathbf{T})$

#### 2.4. Motivation for the Regularization

#### 2.5. Regularization Functions $S\left(\mathbf{T}\right)$

- Entropy regularization$${S}_{1}\left(\mathbf{T}\right)=-\left[-\sum _{t=1}^{N}\frac{{T}_{t}}{\sum {T}_{{t}^{\prime}}}log\left(\frac{{T}_{t}}{\sum {T}_{{t}^{\prime}}}\right)\right].$$
- Curvature regularization$${S}_{2}\left(\mathbf{T}\right)=\sum _{t=2}^{N-1}{({\mathsf{\Delta}}_{t+1,t}-{\mathsf{\Delta}}_{t,t-1})}^{2},$$$${\mathsf{\Delta}}_{{t}_{1},{t}_{2}}={T}_{{t}_{1}}-{T}_{{t}_{2}}.$$
- First derivative regularization$${S}_{3}\left(\mathbf{T}\right)=\sum _{t=2}^{N-1}\frac{|{\delta}_{t+1,t}-{\delta}_{t,t-1}|}{|{\delta}_{t+1,t}+{\delta}_{t,t-1}|},$$$${\delta}_{{t}_{1},{t}_{2}}=\frac{\frac{{T}_{{t}_{1}}}{{W}_{{t}_{1}}}-\frac{{T}_{{t}_{2}}}{{W}_{{t}_{2}}}}{{C}_{{t}_{1}}-{C}_{{t}_{2}}}$$

#### 2.6. Sampling the Likelihood Function

## 3. Regularization

#### 3.1. Evolution of Posteriors with $\tau $

#### 3.2. Evolution of Curvature, Entropy, and Derivatives with $\tau $

#### 3.3. Evolution of Bin Cross-Correlations with $\tau $

#### 3.4. Evolution of ${\chi}^{2}/\mathrm{ndf}$ and Bin Uncertainties with $\tau $

## 4. Combined Sampling as a Faster Algorithm

## 5. Accidental Minima of ${\mathbf{\chi}}^{\mathbf{2}}\left(\mathbf{\tau}\right)/\mathrm{ndf}$

## 6. Hidden Minima of ${\mathbf{\chi}}^{\mathbf{2}}\left(\mathbf{\tau}\right)/\mathrm{ndf}$

## 7. Real Minima of ${\mathbf{\chi}}^{\mathbf{2}}\left(\mathbf{\tau}\right)/\mathrm{ndf}$

## 8. Results

## 9. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Unfolding components of the spectrum ${\eta}^{t\overline{t}}$: (

**a**) particle spectra (green), pseudo data (blue), unfolding result $\widehat{\mathbf{T}}$ (red), (

**b**) efficiency $\u03f5$ and acceptance $\eta $ corrections of statistically independent sets A and B, and (

**c**) normalized migration matrix ${M}_{A}$.

**Figure 2.**(

**a**) View of a part of the 16-dimensional log-likelihood $logL\left(\mathbf{T}\right)$ function as function of the 6th and 9th bin (

**b**) normalized maximal values of $logL({T}_{6},{T}_{9})$, and (

**c**) normalized maximal values of $L({T}_{6},{T}_{9})$.

**Figure 3.**Marginalized 1D posteriors in the 6th (

**a**) and 9th (

**b**) bin of the ${\eta}^{t\overline{t}}$ spectrum without regularization applied.

**Figure 4.**Unfolding the double-peaked ${\eta}^{t\overline{t}}$ over-binned spectrum for different values of the regularization strength parameter $\tau $. The parameter $\tau $ is normalized, such that $\tau ={\tau}_{\mathrm{rel}}$, see Section 3.

**Figure 5.**Relative ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ as function of the regularization strength parameter $\tau $ and its minimum at ${\tau}_{\mathrm{opt}}=2089$. The parameter $\tau $ is normalized, such that $\tau ={\tau}_{\mathrm{rel}}$, see Section 3. The vertical line represents the minimum of ${\chi}^{2}/\mathrm{ndf}$.

**Figure 6.**The envelope of normalized regularization functions $S\left(\mathbf{T}\right)$ in the 6th and 9th bin. For sampling purposes, the gradient of $L\left(\mathbf{T}\right)-S\left(\mathbf{T}\right)$ was used.

**Figure 7.**Posterior shifting and narrowing with increasing the regularization strength parameter $\tau $ in a selected single bin: (

**a**) no regularization applied $\tau =0$ (

**b**) $\tau =2089$ (

**c**) $\tau \approx {10}^{4}$.

**Figure 8.**Mostly decreasing (

**a**) curvature, (

**b**) entropy, and (

**c**) derivatives of the unfolded spectrum ${\eta}^{t\overline{t}}$ with respect to $\tau $. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.

**Figure 9.**Cross-bin correlation matrix built from the correlation factor of likelihood, $L({T}_{i},{T}_{j})$ while using the curvature regularization for three different values of $\tau $.

**Figure 10.**The averaged cross bin correlations ${\overline{C}}_{\mathrm{abs}}$ (black) and $\overline{C}$ (pink) using (

**a**) curvature, (

**b**) entropy, and (

**c**) derivative regularization for the ${\eta}^{t\overline{t}}$ spectrum. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.

**Figure 11.**Variable ${\chi}_{\mathrm{num}}\left(\tau \right)$ of the spectrum ${\eta}^{t\overline{t}}$ using (

**a**) curvature (

**b**) entropy and (

**c**) derivative regularization illustrating the effect of spectra smoothing. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds. The red line indicates the minimal value of the ${\chi}_{\mathrm{num}}\left(\tau \right)$.

**Figure 12.**Variable ${\chi}_{\mathrm{denom}}\left(\tau \right)$ of the spectrum ${\eta}^{t\overline{t}}$ using (

**a**) curvature, (

**b**) entropy, and (

**c**) derivative regularization illustrating the effect of narrowing the posteriors and decreasing the uncertainty. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.

**Figure 13.**Variables ${\chi}^{2}/\mathrm{ndf}$ and ${\chi}_{\mathrm{num}}/{\chi}_{\mathrm{denom}}$ of the ${\eta}^{t\overline{t}}$ spectrum using (

**a**) curvature, (

**b**) entropy, and (

**c**) derivative regularization showing good correspondence. The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds.

**Figure 14.**Variable ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ of the spectrum ${\eta}^{t\overline{t}}$ using (

**a**) curvature, (

**b**) entropy, and (

**c**) derivative regularization comparing combined (faster) sampling (blue) and full sampling (red). The uncertainty band is evaluated as a standard deviation over 20 independent unfolding runs initiated with different random seeds. The vertical dotted lines indicate positions of ${\chi}^{2}/\mathrm{ndf}$ minima for each sampling case.

**Figure 15.**Variables (

**a**) ${\chi}_{\mathrm{num}}\left(\tau \right)$, (

**b**) ${\chi}_{\mathrm{denom}}\left(\tau \right)$ for the full sampling method; and (

**c**) ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ using full (red) and combined (blue) sampling of the ${m}^{t\overline{t}}$ spectrum with an accidental minimum at $\tau \approx 7000$ (curvature regularization). The vertical dotted lines indicate positions of ${\chi}^{2}/\mathrm{ndf}$ minima for each sampling case.

**Figure 16.**The result of unfolding (

**a**) without regularization (

**b**) with regularization and (

**c**) with regularization applied only at second half of the spectrum ${m}^{t\overline{t}}$ while using the curvature in the case of a accidental minimum in ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ for one representative random seed.

**Figure 17.**Variables (

**a**) ${\chi}_{\mathrm{num}}\left(\tau \right)$, (

**b**) ${\chi}_{\mathrm{denom}}\left(\tau \right)$ for the full sampling method; and, (

**c**) ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ using full (red) and combined (blue) sampling of the ${p}_{T}^{t,\mathrm{had}}$ spectrum with a vanishing minimum (derivative regularization). The vertical dotted lines indicate positions of ${\chi}^{2}/\mathrm{ndf}$ minima for each sampling case.

**Figure 18.**Result of unfolding (

**a**) without regularization and (

**b**) with regularization of the spectrum ${p}_{T}^{t,\mathrm{had}}$ using the derivatives in case of a hidden minimum in ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ for one representative random seed. Spectrum becomes smoother, but ${\chi}^{2}/\mathrm{ndf}$ does not improve due to the narrowing of posteriors.

**Figure 19.**Variables (

**a**) ${\chi}_{\mathrm{num}}\left(\tau \right)$, (

**b**) ${\chi}_{\mathrm{denom}}\left(\tau \right)$ for the full sampling method; and (

**c**) ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ using full (red) and combined (blue) sampling of the ${\eta}^{t\overline{t}}$ spectrum with the real minimum at $\tau \approx 900$ (curvature regularization). The vertical dotted lines indicate positions of ${\chi}^{2}/\mathrm{ndf}$ minima for each sampling case.

**Figure 20.**The result of unfolding (

**a**) without regularization and (

**b**) with regularization of the spectrum ${\eta}^{t\overline{t}}$ while using the curvature in case of a real minimum in ${\chi}^{2}/\mathrm{ndf}$ for one representative random seed.

**Figure 21.**Result of unfolding (

**a**) without regularization and (

**c**) with regularization of the spectrum ${\eta}^{t,\mathrm{had}}$ using the curvature regularization for one representative random seed. Variable ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ (

**b**) while using full (red) and combined (blue) sampling of the ${\eta}^{t,\mathrm{had}}$ spectrum with minimum at $\tau =1698$ (curvature regularization). In this case, the regularization is not needed. The vertical dotted lines indicate positions of ${\chi}^{2}/\mathrm{ndf}$ minima for each sampling case.

**Figure 22.**Result of unfolding (

**a**) without regularization and (

**c**) with regularization of the spectrum ${p}_{T}^{t\overline{t}}$ using entropy regularization for one representative random seed. Variable ${\chi}^{2}\left(\tau \right)/\mathrm{ndf}$ (

**b**) using full (red) and combined (blue) sampling of the ${p}_{T}^{t\overline{t}}$ spectrum with minimum at $\tau =2089$ (entropy regularization). The vertical dotted lines indicate the positions of ${\chi}^{2}/\mathrm{ndf}$ minima for each sampling case.

**Table 1.**Results of minima type and basic characteristics of the migration matrix M: averaged on-diagonal factor ${f}_{\mathrm{diag}}$ and correlation matrix $\rho $.

Spectrum | Type | Minimum | Method | ${\mathit{f}}_{\mathbf{diag}}$ | $\mathit{\rho}$ |
---|---|---|---|---|---|

${m}^{t\overline{t}}$ | falling | Accidental | Curvature | 0.51 | 0.92 |

${p}_{T}^{t,\mathrm{had}}$ | falling | Hidden | Derivative | 0.48 | 0.86 |

${\eta}^{t,\mathrm{had}}$ | double-peaked | Real | Curvature | 0.75 | 0.93 |

${\eta}^{t\overline{t}}$ | double-peaked | Real | Curvature | 0.49 | 0.94 |

${p}_{T}^{t\overline{t}}$ | falling | Real | Entropy | 0.29 | 0.86 |

**Table 2.**Relative curvature, entropy and derivatives of the unfolded spectra ($\tau =0$) with respect to the curvature, entropy, and derivatives of the particle level spectra.

Spectrum | Minimum | $\frac{{\mathit{C}}_{0}}{{\mathit{C}}_{\mathbf{ptcl}}}$ | $\frac{{\mathit{E}}_{0}}{{\mathit{E}}_{\mathbf{ptcl}}}$ | $\frac{{\mathit{D}}_{0}}{{\mathit{D}}_{\mathbf{ptcl}}}$ |
---|---|---|---|---|

${m}^{t\overline{t}}$ | Accidental | 1.1 | 1.0 | 2.7 |

${p}_{T}^{t,\mathrm{had}}$ | Hidden | 0.98 | 1.0 | 11 |

${\eta}^{t,\mathrm{had}}$ | Real | 1.1 | 1.0 | 0.96 |

${\eta}^{t\overline{t}}$ | Real | 3.7 | 1.0 | 3.5 |

${p}_{T}^{t\overline{t}}$ | Real | 14 | 0.97 | 11 |

**Table 3.**Time needed to produce ${\chi}^{2}/\mathrm{ndf}\left(\tau \right)$ curve using combined sampling ${T}_{\mathrm{comb}}$ and full sampling ${T}_{\mathrm{full}}$ for 68 points of $\tau $ in the region [0; ${10}^{5}$]. Time in seconds is rounded to hundreds.

Spectrum | Minimum | ${\mathit{T}}_{\mathbf{comb}}$[s] | ${\mathit{T}}_{\mathbf{full}}$[s] | $\frac{{\mathit{T}}_{\mathbf{comb}}}{{\mathit{T}}_{\mathbf{full}}}$ |
---|---|---|---|---|

${m}^{t\overline{t}}$ | Accidental | 900 | 30,700 | 0.029 |

${p}_{T}^{t,\mathrm{had}}$ | Hidden | 1200 | 41,300 | 0.030 |

${\eta}^{t,\mathrm{had}}$ | Real | 600 | 14,500 | 0.039 |

${\eta}^{t\overline{t}}$ | Real | 1000 | 33,700 | 0.029 |

${p}_{T}^{t\overline{t}}$ | Real | 1300 | 44,700 | 0.030 |

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Baroň, P.; Kvita, J.
Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method. *Symmetry* **2020**, *12*, 2100.
https://doi.org/10.3390/sym12122100

**AMA Style**

Baroň P, Kvita J.
Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method. *Symmetry*. 2020; 12(12):2100.
https://doi.org/10.3390/sym12122100

**Chicago/Turabian Style**

Baroň, Petr, and Jiří Kvita.
2020. "Extending the Fully Bayesian Unfolding with Regularization Using a Combined Sampling Method" *Symmetry* 12, no. 12: 2100.
https://doi.org/10.3390/sym12122100