(Probability kernels: conditioning and random measures)

Also as: MMF500 Graduate course within Probability Theory.

This page contains all the necessary information about the course. Some of the information, such as learning outcomes, teachers, literature and examination, can also be found in a separate course PM.

Short course description

The starting point of this course is the notion of a (probability) kernel, which may be thought of as family of (probability) measures indexed by a parameter.

More than anything, kernels offer a nice and intuitive way of working with conditional expectations and probabilities: in the elementary definition, the conditional distribution of a random variable X given an observation y of Y is described by a distribution which depends on y, and here this is formalized by letting y be the parameter.

Any statistical model is a family of distributions indexed by a parameter, which we want to fit to data, so one again encounters a kernel. For example, the Poisson distribution family Poi(lambda), lambda>0, constitutes a probability kernel parametrized by the mean parameter. Moreover, in hierarchical statistical models one creates a joint model by hierarchical composition of probability kernels.

Kernels also have a central role in the theory of Markov processes, as they formalize the notion of transition probabilities.

By letting the parameter in the kernel be drawn randomly, we obtain a random measure. Random measures are ubiquitous in probability theory, in particular in stochastic geometry and the study of point processes (random counting measures). Heuristically, a point process is a generalization of a random sample, where we allow the sample size to be random and the sample points to be dependent. This viewpoint is the basis of much of temporal and spatial statistics.

Literature

The main book of the course is:

• Baccelli, F., Błaszczyszyn, B., & Karray, M. (2024). Random measures, point processes, and stochastic geometry. Inria (can be downloaded here).

In different places we (might) also make use of:

• Daley, D. J., & Vere-Jones, D. (2003). An introduction to the theory of point processes: volume I: elementary theory and methods. Springer (found here).
• Daley, D. J., & Vere-Jones, D. (2008). An introduction to the theory of point processes: volume II: general theory and structure. Springer (found here).
• Brémaud, P. (2020). Point process calculus in time and space. Springer (found here).
• Kechris, A. (2005).  Classical Descriptive Set Theory. Springer (found here).
• Bauer, H. (2001). Measure and Integration Theory (https://doi.org/10.1515/9783110866209).

If the field of random measures, point processes and stochastic geometry interests you, here is an additional list of books which cover the field from either a probabilistic or a statistical point of view: 

• Baccelli, F., & Błaszczyszyn, B. (2009). Stochastic geometry and wireless networks: Volume I & II. Inria.
• Baddeley, A., Rubak, E., & Turner, R. (2016). Spatial point patterns: methodology and applications with R . CRC.
• Benes, V., & Rataj, J. (2007). Stochastic geometry: selected topics. Springer.
• Chiu, S. N., Stoyan, D., Kendall, W. S., & Mecke, J. (2013). Stochastic geometry and its applications. Wiley.
• Coupier, D. (Ed.). (2019). Stochastic geometry: modern research frontiers  (Vol. 2237). Springer.
• Diggle, P. J. (2013). Statistical analysis of spatial and spatio-temporal point patterns. CRC press.
• Gelfand, A. E., & Schliep, E. M. (2018). Bayesian inference and computing for spatial point patterns. In: NSF-CBMS regional conference series in probability and statistics (Vol. 10, pp. i-125). Institute of Mathematical Statistics and the American Statistical Association.
• Illian, J., Penttinen, A., Stoyan, H., & Stoyan, D. (2008). Statistical analysis and modelling of spatial point patterns. Wiley.
• Jansen S. (2018). Gibbsian point processes. LMU München.
• Kallenberg, O. (2017). Random measures, theory and applications. Springer.
• Last, G., & Penrose, M. (2018). Lectures on the Poisson process. Cambridge University Press.
• Molchanov, I. (2005). Theory of random sets. Springer.
• Møller, J., & Waagepetersen, R. P. (2003). Statistical inference and simulation for spatial point processes. CRC.
• Schneider, R., & Weil, W. (2008). Stochastic and integral geometry. Springer.
• van Lieshout, M. N. M. (2000). Markov point processes and their applications. World Scientific.
• van Lieshout, M. N. M. (2019). Theory of spatial statistics: A concise introduction. Chapman & Hall/CRC.

Schedule, lecture room & program

From week 13 (starting with 24 March) until week 22 (starting with 26 May), 2025, the lectures will be given as follows:

• Wednesdays: 8:15 - 9:45 in MVH12
• Thursdays: 13:15-15:00 in MVH12

Lectures

Link to Moritz' lecture notes: LectureKernels1-4-1.pdf

Link to the overleaf project for Ottmar's lecture notes where you can freely make changes/additions: https://www.overleaf.com/2117846663jprtxskncbbp#6ee5e5

Weekday	Date	Time	Content	Book chapter	Lecturer	Lecture notes
Wednesday	2025-03-26	08:15-09:45	Introduction		Moritz
Thursday	2025-03-26	13:15-15:00	Measures and kernels		Moritz
Wednesday	2025-04-02	08:15-09:45	Disintegration		Moritz
Thursday	2025-04-03	13:15-15:00	Composition		Moritz	LectureKernels1-4-1.pdf
Wednesday	2025-04-09	08:15-09:45	Polish and LCHS spaces + locally finite measures + basics of construction of random measures.	Parts of Kechris chapter 1.3,  Kallenberg chapter 1.1. and Baccelli chapter 1.1.	Ottmar	RMs_L4.pdf
Thursday	2025-04-10	13:15-15:00	The space of locally finite measures, measure decomposition, definition of random measures and point processes, intensity measures/functions, random sets, set processes, coverage models (germ-grain and Boolean models), Voronoi tessellations.	Parts of Kallenberg chapter 1, Daley & Vere-Jones chapter A2 & 9 and Baccelli et al chapter 1.1, 1.2, 9.1, 9.2 & 10.1.	Ottmar	RMs_L5.pdf
Wednesday	2025-04-16	08:15-09:45		No lecture
Thursday	2025-04-17	13:15-15:00		No lecture
Wednesday	2025-04-23	08:15-09:45		No lecture
Thursday	2025-04-24	13:15-15:00		No lecture
Wednesday	2025-04-30	08:15-09:45	Laplace functionals, generating functionals, void probabilities, higher order (factorial) moments, product densities.	Baccelli et al chapter 1.3, 1.4, 1.6, 2.1.2 & 2.3.4.	Ottmar	For now: Go to the overleaf project to get the slides.
Thursday	2025-05-01	13:15-15:00		No lecture
Wednesday	2025-05-07	08:15-09:45	Poisson processes and superpositions.	Baccelli et al chapter 2.1 & 2.2.1.	Ottmar	For now: Go to the overleaf project to get the slides.
Thursday	2025-05-08	13:15-15:00	Thinnings, images, displacements, markings and mixtures of RMs/PPs.	Baccelli et al chapter 2.2	Ottmar	For now: Go to the overleaf project to get the slides.
Wednesday	2025-05-14	08:15-09:45	Mixtures of RMs/PPs, stationarity, Cox processes, cluster PPs	Baccelli et al chapter 2.2.7, 2.3.1 & 2.3.3 (note that 2.3.4 has already been covered; 2.3.2 which is about Gibbs processes will be done in an alternative way by Mathis)	Ottmar	For now: Go to the overleaf project to get the slides.
Thursday	2025-05-15	13:15-15:00	Palm theory	3.1-3.3 (minus 3.1.2, 3.1.3 & 3.2.3)	Ottmar	For now: Go to the overleaf project to get the slides.
Wednesday	2025-05-21	08:15-09:45	Infinitely divisible and stable PPs + limit theorems		Sergey	2025-05-20 Limit theorems for PPs.pdf
Thursday	2025-05-22	13:15-15:00	Gibbs processes		Mathis	Gibbs.pdf
Wednesday	2025-05-28	08:15-09:45	Optimal transport		Axel	Lecture-14_OT.pdf Lecture-14_OT_Handout.pdf
Wednesday	2025-05-28	13:15-15:00	PhD defence,  Mattias Byhlén: Hyperuniformity and hyperfluctuations for random measures on Euclidean and non-Euclidean spaces

The exact schedule might appear on TimeEdit at some point.

Excercises

The sections we have covered in Baccelli et. al. have exercises in the following sections of the book:

Section 1: Exercises in Section 1.7

Section 2: Exercises in Section 2.7.1-2.7.3

Section 3: Exercises in Section 3.5

If time and your interest permit, there are exercises on random sets in Section 9.3

Examination

• Oral exam.

Note that the main book contains solutions to all of its exercises, and we strongly encourage you to work through as many of the exercises as you can (the oral exam might involve questions about specific exercises in the book).

People at MV or in Sweden who (in)directly are/have been working with random measures, point processes and stochastic geometry

The field of random measures and point processes has a long history in Sweden, particularly at our department. To put the things you learn in the course, and related topics, into a local context, we here present a list of researchers at the department who are/have been working with aspects of the field (which may be particularly interesting for masters students who are looking for a masters project):

• Berman, Robert
• Björklund, Michael
• Broman, Erik
• Cronie, Ottmar
• Häggström, Olle
• Jagers, Peter
• Jonasson, Johan
• Lindvall, Torgny 
• Palö Forsström, Malin
• Rootzén, Holger
• Rudemo, Mats
• Schauer, Moritz
• Steif, Jeff
• Särkkä, Aila
• Södergren, Anders
• Tykesson, Johan
• Zuyev, Sergey

We should also mention the following individuals who have been actively working in the field, in Sweden (at some point):

• Kallenberg, Olav: by many considered one of the main figures in the field of random measures. He used to work at MV, before leaving for the US; in his book from 2017, Random Measures, Theory and Applications, he thanks our colleague Peter Jagers for introducing him to the field.
• Matérn, Bertil: by many considered one of the founders of spatial statistics, was active at the Forestry Research Institute of Sweden.
• Palm, Conny: introduced the notion of Palm distributions/measures (no, Palm does not relate to a tree or a part of the hand here... ;)). He was active at KTH and Ericsson.
• Hermann Thorisson: once a PhD student of Torgny Lindvall at MV, he is a key figure in the field of stochastic geometry.

Please inform us if we have missed anyone who should be mentioned!

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