Deep Gaussian Processes (DGPs) are introduced as a Bayesian model that extends Gaussian Process (GP) models to deep architectures. DGPs use multiple layers of GPs, where each layer's inputs are governed by another GP. This allows for hierarchical modeling of data, with each layer acting as both an input and output for adjacent layers. A single layer of a DGP is equivalent to a standard GP or GP-LVM. The model uses variational inference to approximate the marginal likelihood, enabling model selection based on the number of layers and nodes. This approach allows for deep learning even with scarce data, as demonstrated by the successful modeling of a 150-example digit dataset with a five-layer hierarchy. The paper discusses the challenges of traditional deep learning methods, which often require large datasets and are intractable for smaller data. DGPs offer a Bayesian approach that can handle smaller datasets and provide a variational lower bound for model selection. The model is trained using variational inference, which allows for efficient computation and automatic structure discovery in the hierarchy. DGPs are shown to be effective in both toy and real-world data sets. In a regression task, DGPs outperformed traditional methods in predicting unseen data. In a motion capture experiment, DGPs successfully modeled human motion with a hierarchical structure, discovering common subspaces and automatically determining the relevant dimensions. In a digit recognition task, DGPs demonstrated the ability to learn increasingly abstract features, with the model's performance improving as the depth of the hierarchy increased. The paper also discusses the computational efficiency of DGPs, noting that the complexity of each GP mapping is reduced from O(N^3) to O(NM^2) through the use of variational inference. This makes DGPs suitable for deeper architectures and larger datasets. The authors conclude that DGPs provide a powerful framework for deep learning, capable of modeling complex data with abstract representations, even when data is scarce. Future work includes extending DGPs to larger datasets and exploring their application in multitask learning and nonstationary data modeling.Deep Gaussian Processes (DGPs) are introduced as a Bayesian model that extends Gaussian Process (GP) models to deep architectures. DGPs use multiple layers of GPs, where each layer's inputs are governed by another GP. This allows for hierarchical modeling of data, with each layer acting as both an input and output for adjacent layers. A single layer of a DGP is equivalent to a standard GP or GP-LVM. The model uses variational inference to approximate the marginal likelihood, enabling model selection based on the number of layers and nodes. This approach allows for deep learning even with scarce data, as demonstrated by the successful modeling of a 150-example digit dataset with a five-layer hierarchy. The paper discusses the challenges of traditional deep learning methods, which often require large datasets and are intractable for smaller data. DGPs offer a Bayesian approach that can handle smaller datasets and provide a variational lower bound for model selection. The model is trained using variational inference, which allows for efficient computation and automatic structure discovery in the hierarchy. DGPs are shown to be effective in both toy and real-world data sets. In a regression task, DGPs outperformed traditional methods in predicting unseen data. In a motion capture experiment, DGPs successfully modeled human motion with a hierarchical structure, discovering common subspaces and automatically determining the relevant dimensions. In a digit recognition task, DGPs demonstrated the ability to learn increasingly abstract features, with the model's performance improving as the depth of the hierarchy increased. The paper also discusses the computational efficiency of DGPs, noting that the complexity of each GP mapping is reduced from O(N^3) to O(NM^2) through the use of variational inference. This makes DGPs suitable for deeper architectures and larger datasets. The authors conclude that DGPs provide a powerful framework for deep learning, capable of modeling complex data with abstract representations, even when data is scarce. Future work includes extending DGPs to larger datasets and exploring their application in multitask learning and nonstationary data modeling.