# 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

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 $$M$$.

## 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).
@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.
@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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