# Subordinator priors and completely random measures

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A subordinator is an a.s. non-decreasing LĂ©vy process. These can be used a random measures, which means they can be used as priors, which means they are handy for Bayesian nonparametrics. These priors include as a special case Dirichlet process priors in a way I will make precise maybe one day.

A rule of thumb is that while Gaussian process regression is good for giving you curves and more generally, fields, a subordinator prior gives you measures. They are always kinda lumpy measures. It is easy to conserve mass with these things.

Barndorff-Nielsen, O. E., and J. Schmiegel. 2004. â€śLĂ©vy-Based Spatial-Temporal Modelling, with Applications to Turbulence.â€ť Russian Mathematical Surveys 59 (1): 65. https://doi.org/10.1070/RM2004v059n01ABEH000701.

Higdon, Dave. 2002. â€śSpace and Space-Time Modeling Using Process Convolutions.â€ť In Quantitative Methods for Current Environmental Issues, edited by Clive W. Anderson, Vic Barnett, Philip C. Chatwin, and Abdel H. El-Shaarawi, 37â€“56. London: Springer. https://doi.org/10.1007/978-1-4471-0657-9_2.

Hjort, Nils Lid, Chris Holmes, Peter MĂĽller, and Stephen G. Walker, eds. 2010. â€śModels Beyond the Dirichlet Process.â€ť In Bayesian Nonparametrics. Cambridge University Press. https://doi.org/10.2139/ssrn.1526505.

James, Lancelot F. 2005. â€śBayesian Poisson Process Partition Calculus with an Application to Bayesian LĂ©vy Moving Averages.â€ť Annals of Statistics 33 (4): 1771â€“99. https://doi.org/10.1214/009053605000000336.

Nieto-Barajas, Luis E., Igor PrĂĽnster, and Stephen G. Walker. 2004. â€śNormalized Random Measures Driven by Increasing Additive Processes.â€ť Annals of Statistics 32 (6): 2343â€“60. https://doi.org/10.1214/009053604000000625.

Walker, Stephen G., Paul Damien, PuruShottam W. Laud, and Adrian F. M. Smith. 1999. â€śBayesian Nonparametric Inference for Random Distributions and Related Functions.â€ť Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61 (3): 485â€“527. https://doi.org/10.1111/1467-9868.00190.

Wolpert, Robert L., and Katja Ickstadt. 1998. â€śSimulation of LĂ©vy Random Fields.â€ť In Practical Nonparametric and Semiparametric Bayesian Statistics, edited by Dipak Dey, Peter MĂĽller, and Debajyoti Sinha, 227â€“42. Lecture Notes in Statistics. New York, NY: Springer. https://doi.org/10.1007/978-1-4612-1732-9_12.