# 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. arXiv:2005.12998 [Math], January.
Matthews, Alexander Graeme de Garis. 2017. 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. Acta Numerica 19: 451–559.

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