Bayesian nonparametric statistics

Updating more dimensions than datapoints



It is hard to explain what happens to the posterior in this case

Dirichlet priors, other measure priors, Gaussian Process regression, reparameterisations etc. 🏗

For now, this is just a bookmark to the general measure theoretic notation that unifies, in principle, the various Bayesian nonparametric methods.

A textbook on general theory is Schervish (2012).

Particular applications are outlined in Matthews (2017) (GP regression) and Stuart (2010) (inverse problems). A brief introduction the kind of measure-theoretic notation we need in the infinite-dimensional Hilbert space settings is in Alexanderian (2021), giving Bayes’ formula as \[ \frac{d \mu_{\text {post }}^{y}}{d \mu_{\text {pr }}} \propto \pi_{\text {like }}(\boldsymbol{y} \mid m), \] where the left hand side is the Radon-Nikodym derivative of \(\mu_{\text {post }}^{y}\) with respect to \(\mu_{\text {pr }}\).

They observe

Note that in the finite-dimensional setting the abstract form of the Bayes’ formula above can be reduced to the familiar form of Bayes’ formula in terms of PDFs. Specifically, working in finite-dimensions, with \(\mu_{\mathrm{pr}}\) and \(\mu_{\mathrm{post}}^{y}\) that are absolutely continuous with respect to the Lebesgue measure \(\lambda\), the prior and posterior measures admit Lebesgue densities \(\pi_{\mathrm{pr}}\) and \(\pi_{\text {post }}\), respectively. Then, we note \[ \pi_{\mathrm{post}}(m \mid \boldsymbol{y})=\frac{d \mu_{\mathrm{post}}^{y}}{d \lambda}(m)=\frac{d \mu_{\mathrm{post}}^{y}}{d \mu_{\mathrm{pr}}}(m) \frac{d \mu_{\mathrm{pr}}}{d \lambda}(m) \propto \pi_{\mathrm{like}}(\boldsymbol{y} \mid m) \pi_{\mathrm{pr}}(m) \]

References

Alexanderian, Alen. 2021. Optimal Experimental Design for Infinite-Dimensional Bayesian Inverse Problems Governed by PDEs: A Review.” arXiv:2005.12998 [Math], January.
Matthews, Alexander Graeme de Garis. 2017. Scalable Gaussian Process Inference Using Variational Methods.” Thesis, University of Cambridge.
Schervish, Mark J. 2012. Theory of Statistics. Springer Series in Statistics. New York, NY: Springer Science & Business Media.
Stuart, A. M. 2010. Inverse Problems: A Bayesian Perspective.” Acta Numerica 19: 451–559.

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