automatic statistician (wip)

27 February 2017

notes on (Duvenaud et al., n.d.) and (Lloyd et al., 2014). these papers are part of the automatic statistician project.

summary

understanding: 5/10
code: https://github.com/jamesrobertlloyd/gp-structure-search, https://github.com/jamesrobertlloyd/gpss-research

the problem: find good kernels for GPs.

kernels are closed under addition and multiplication. we form a generative grammar for kernel construction: \begin{align} \mathcal S &\to \mathcal S + \mathcal B \\
\mathcal S &\to \mathcal S \times \mathcal B \\
\mathcal B &\to \mathcal B’ \end{align} where \(\mathcal S\) is a subexpression (e.g. \(\mathrm{LIN} \times \mathrm{LIN}\)) and \(\mathcal B\) is a base kernel; one of \(\mathrm{SE}\) (exponentiated square difference), \(\mathrm{PER}\) (periodic), \(\mathrm{LIN}\) (linear), \(\mathrm{RQ}\) (rational quadratic).

how do we search over expressions? we use greedy search: at each stage, we choose the highest scoring kernel and expand it by applying all possible operators.

how do we score kernels? using the bayesian information criterion: \begin{align} \mathrm{BIC}(M) = -2 \log p(D \given M) + |M| \log n. \end{align} here, \(M\) is kernel, \(D\) is data, \(n\) is number of data points \(|M|\) is number of kernel parameters, \(p(D \given M)\) is the evidence.

? this is some sort of approximation to \(p(D) = \int p(D \given M) p(M) \,\mathrm dM\) if we had a prior on \(M\).

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references

1. Duvenaud, D., Lloyd, J. R., Grosse, R., Tenenbaum, J. B., & Ghahramani, Z. Structure Discovery in Nonparametric Regression through Compositional Kernel Search. 30th International Conference on Machine Learning (June 2013).
   • Bib

   @inproceedings{duvenaud2013structure,
     title = {Structure Discovery in Nonparametric Regression through Compositional Kernel Search},
     author = {Duvenaud, David and Lloyd, James Robert and Grosse, Roger and Tenenbaum, Joshua B and Ghahramani, Zoubin},
     booktitle = {30th International Conference on Machine Learning (June 2013)},
     organization = {International Machine Learning Society}
   }
   

2. Lloyd, J. R., Duvenaud, D., Grosse, R., Tenenbaum, J., & Ghahramani, Z. (2014). Automatic Construction and Natural-Language Description of Nonparametric Regression Models. Twenty-Eighth AAAI Conference on Artificial Intelligence.
   • Bib

   @inproceedings{lloyd2014automatic,
     title = {Automatic Construction and Natural-Language Description of Nonparametric Regression Models},
     author = {Lloyd, James Robert and Duvenaud, David and Grosse, Roger and Tenenbaum, Joshua and Ghahramani, Zoubin},
     booktitle = {Twenty-Eighth AAAI Conference on Artificial Intelligence},
     year = {2014}
   }
   

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