# Deep Gaussian Processes

Andreas Damianou, University of Sheffield
Neil D. Lawrence, University of Sheffield

in Proceedings of the Sixteenth International Workshop on Artificial Intelligence and Statistics 31, pp 207-215

#### Abstract

In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP latent variable model (GP-LVM). We perform inference in the model by approximate variational marginalization. This results in a strict lower bound on the marginal likelihood of the model which we use for model selection (number of layers and nodes per layer). Deep belief networks are typically applied to relatively large data sets using stochastic gradient descent for optimization. Our fully Bayesian treatment allows for the application of deep models even when data is scarce. Model selection by our variational bound shows that a five layer hierarchy is justified even when modelling a digit data set containing only 150 examples.

  @InProceedings{damianou-deepgp13, title = {Deep Gaussian Processes}, author = {Andreas Damianou and Neil D. Lawrence}, booktitle = {Proceedings of the Sixteenth International Workshop on Artificial Intelligence and Statistics}, pages = {207}, year = {2013}, editor = {Carlos Carvalho and Pradeep Ravikumar}, volume = {31}, address = {AZ, USA}, month = {00}, publisher = {JMLR W\&CP 31}, edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/2013-04-29-damianou-deepgp13.md}, url = {http://inverseprobability.com/publications/damianou-deepgp13.html}, abstract = {In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP latent variable model (GP-LVM). We perform inference in the model by approximate variational marginalization. This results in a strict lower bound on the marginal likelihood of the model which we use for model selection (number of layers and nodes per layer). Deep belief networks are typically applied to relatively large data sets using stochastic gradient descent for optimization. Our fully Bayesian treatment allows for the application of deep models even when data is scarce. Model selection by our variational bound shows that a five layer hierarchy is justified even when modelling a digit data set containing only 150 examples.}, crossref = {Carvalho:aistats13}, key = {Damianou:deepgp13}, linkpdf = {ftp://ftp.dcs.shef.ac.uk/home/neil/deepGPsAISTATS.pdf}, linksoftware = {https://github.com/SheffieldML/deepGP}, OPTgroup = {} }
 %T Deep Gaussian Processes %A Andreas Damianou and Neil D. Lawrence %B %C Proceedings of the Sixteenth International Workshop on Artificial Intelligence and Statistics %D %E Carlos Carvalho and Pradeep Ravikumar %F damianou-deepgp13 %I JMLR W\&CP 31 %P 207--215 %R %U http://inverseprobability.com/publications/damianou-deepgp13.html %V 31 %X In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP latent variable model (GP-LVM). We perform inference in the model by approximate variational marginalization. This results in a strict lower bound on the marginal likelihood of the model which we use for model selection (number of layers and nodes per layer). Deep belief networks are typically applied to relatively large data sets using stochastic gradient descent for optimization. Our fully Bayesian treatment allows for the application of deep models even when data is scarce. Model selection by our variational bound shows that a five layer hierarchy is justified even when modelling a digit data set containing only 150 examples. 
 TY - CPAPER TI - Deep Gaussian Processes AU - Andreas Damianou AU - Neil D. Lawrence BT - Proceedings of the Sixteenth International Workshop on Artificial Intelligence and Statistics PY - 2013/04/29 DA - 2013/04/29 ED - Carlos Carvalho ED - Pradeep Ravikumar ID - damianou-deepgp13 PB - JMLR W\&CP 31 SP - 207 EP - 215 L1 - ftp://ftp.dcs.shef.ac.uk/home/neil/deepGPsAISTATS.pdf UR - http://inverseprobability.com/publications/damianou-deepgp13.html AB - In this paper we introduce deep Gaussian process (GP) models. Deep GPs are a deep belief network based on Gaussian process mappings. The data is modeled as the output of a multivariate GP. The inputs to that Gaussian process are then governed by another GP. A single layer model is equivalent to a standard GP or the GP latent variable model (GP-LVM). We perform inference in the model by approximate variational marginalization. This results in a strict lower bound on the marginal likelihood of the model which we use for model selection (number of layers and nodes per layer). Deep belief networks are typically applied to relatively large data sets using stochastic gradient descent for optimization. Our fully Bayesian treatment allows for the application of deep models even when data is scarce. Model selection by our variational bound shows that a five layer hierarchy is justified even when modelling a digit data set containing only 150 examples. ER - 
 Damianou, A. & Lawrence, N.D.. (2013). Deep Gaussian Processes. Proceedings of the Sixteenth International Workshop on Artificial Intelligence and Statistics 31:207-215