# Conditional Deep Gaussian Processes: Empirical Bayes Hyperdata Learning

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Background

#### 3.1. Gaussian Process

#### 3.2. Deep Gaussian Process

#### 3.3. Marginal Prior, Covariance and Marginal Likelihood

## 4. Model

#### 4.1. Conditional Deep Gaussian Process

**Lemma**

**1.**

#### 4.2. When Conditional DGP Is Almost a GP

**Lemma**

**2.**

**Proof.**

**Remark**

**1.**

#### 4.3. Non-Gaussian Aspect

## 5. Results

#### 5.1. Mauna Loa Data

#### 5.2. Airline Data

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GP | Gaussian Process |

DGP | Deep Gaussian Process |

DKL | Deep Kernel Learning |

SE | Squared Exponential |

## Appendix A

**Lemma**

**A1.**

**Proof.**

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**Figure 1.**Extrapolation of standardized CO

_{2}time series data (yellow dots for training and red dots for test) using GP with three kernels. The dark solid line represents the predictive mean, and the shaded area is the the model’s confidence. Panel (

**a**) displays the result using a single GP with an SE kernel. Panel (

**b**) was obtained following the kernel composition in [3]. Panel (

**c**) came from using the effective kernel of 2-layer zero-mean DGP with SE used in both layers [23]. (

**a**) SE kernel; (

**b**) SE+periodic SE+RQ kernel; (

**c**) SE[SE] kernel.

**Figure 2.**Extrapolation of standardized CO

_{2}using DKL and variational inference [6] for the DGP implemented in GPFlux [57]. Panel (

**a**) was obtained using the DKL with three-layer RELU network. Panel (

**b**) shows the results from the two-layer zero-mean DGP model. Panel (

**c**) shows the results of the three-layer zero-mean DGP. (

**a**) DKL; (

**b**) Two-layer DGP; (

**c**) Three-layer DGP.

**Figure 3.**Extrapolation of the standardized CO

_{2}using conditional DGP. Panel (

**a**) is for the two-layer model, and (b) for the three-layer model. Top and middle panels shows the mean and confidence in the posterior over the latent functions. See text for details. (

**a**) Two-layer conditional DGP; (

**b**) Three-layer conditional DGP.

**Figure 4.**Extrapolation of the standardized airline data with three different GPs. (

**a**) SE kernel; (

**b**) SE+periodic SE+RQ kernel; (

**c**) SE[SE] kernel.

**Figure 5.**Extrapolation of the standardized airline data using DKL (

**a**), 2-layer DGP (

**b**) and 3-layer DGP (c). (

**a**) DKL; (

**b**) Two-layer DGP; (c) Three-layer DGP.

**Figure 6.**Extrapolation of airline data using conditional DGP. The upper panel shows the learned latent function and uncertainty from hyperdata learning, and the bottom panel shows the extrapolation from the past data. (

**a**) The first model had 23 hyperdata supporting the latent GP. (

**b**) The other model had 13. (

**a**) 2-layer cDGP with 23 hyperdata; (

**b**) 2-layer cDGP with 13 hyperdata.

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Lu, C.-K.; Shafto, P.
Conditional Deep Gaussian Processes: Empirical Bayes Hyperdata Learning. *Entropy* **2021**, *23*, 1387.
https://doi.org/10.3390/e23111387

**AMA Style**

Lu C-K, Shafto P.
Conditional Deep Gaussian Processes: Empirical Bayes Hyperdata Learning. *Entropy*. 2021; 23(11):1387.
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**Chicago/Turabian Style**

Lu, Chi-Ken, and Patrick Shafto.
2021. "Conditional Deep Gaussian Processes: Empirical Bayes Hyperdata Learning" *Entropy* 23, no. 11: 1387.
https://doi.org/10.3390/e23111387