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Abstract
Higherorder probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measuretheoretic formalization of probability theory. Programs may use both higherorder functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higherorder functions well: the category of measurable spaces is not cartesian closed.
Here we introduce quasiBorel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higherorder functions; form a wellpointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasiBorel spaces for higherorder functions and probability by: showing that a wellknown construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasiBorel spaces.
Here we introduce quasiBorel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higherorder functions; form a wellpointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasiBorel spaces for higherorder functions and probability by: showing that a wellknown construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasiBorel spaces.
Original language  English 

Title of host publication  2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 
Publisher  Institute of Electrical and Electronics Engineers (IEEE) 
Pages  112 
Number of pages  12 
ISBN (Electronic)  9781509030187 
ISBN (Print)  9781509030194 
DOIs  
Publication status  Published  18 Aug 2017 
Event  2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science  Reykjavik, Reykjavik, Iceland Duration: 20 Jun 2017 → 23 Jun 2017 http://lics.siglog.org/lics17/ 
Publication series
Name  IEEE Symposium on Logic in Computer Science 

Publisher  IEEE 
ISSN (Electronic)  10436871 
Conference
Conference  2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science 

Abbreviated title  LICS 2017 
Country  Iceland 
City  Reykjavik 
Period  20/06/17 → 23/06/17 
Internet address 
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Chris Heunen
 School of Informatics  Reader
 Laboratory for Foundations of Computer Science
 Foundations of Computation
Person: Academic: Research Active