Dirichlet is Natural

Danos, Vincent; Garnier, Ilias

doi:10.1016/j.entcs.2015.12.010

Cited by 8 publications

(3 citation statements)

References 9 publications

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“…As an aside, there is also a way to express Dirichlet via gamma distributions that is more familiar, see e.g. [30, 7.7.1] or [7,Prop. 4.1].…”

Section: Dirichlet Via Parallel Beta'smentioning

confidence: 99%

Stick Breaking, in Coalgebra and Probability

Jacobs

2022

Lecture Notes in Computer Science

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Stick breaking is an elementary operation that has been formulated and used within stochastic process theory. This paper extracts the essentials of stick breaking in terms of isomorphisms between discrete probability distributions (with full support) and sequences of numbers between zero and one. This works for both finite and infinite distributions. Stick breaking is a repetitive construction with a strong coalgebraic flavour. Indeed, it is shown that stick breaking turns discrete distributions with infinite full support on the natural numbers into a final coalgebra. Once isolated as a separate construction, the usefulness of stick breaking is illustrated in the description of various probability distributions, such as binomial & multinomial and beta & Dirichlet.

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“…As an aside, there is also a way to express Dirichlet via gamma distributions that is more familiar, see e.g. [30, 7.7.1] or [7,Prop. 4.1].…”

Section: Dirichlet Via Parallel Beta'smentioning

confidence: 99%

Stick Breaking, in Coalgebra and Probability

Jacobs

2022

Lecture Notes in Computer Science

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“…In this paper, we focus on step (ii), stochastic memoization: steps (i) and (iii) are studied extensively elsewhere (e.g. see [14] in the statistics literature, or [2,9,51] in the semantics literature, and references therein). This simple example of a memoized constant Bernoulli function is easy to implement using a memo-table, but already semantically complicated.…”

Section: Introductionmentioning

confidence: 99%

A model of stochastic memoization and name generation in probabilistic programming: categorical semantics via monads on presheaf categories

Kaddar,

Staton

2023

Electronic Notes in Theoretical Informatics and Computer Science

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Stochastic memoization is a higher-order construct of probabilistic programming languages that is key in Bayesian nonparametrics, a modular approach that allows us to extend models beyond their parametric limitations and compose them in an elegant and principled manner. Stochastic memoization is simple and useful in practice, but semantically elusive, particularly regarding dataflow transformations. As the naive implementation resorts to the state monad, which is not commutative, it is not clear if stochastic memoization preserves the dataflow property -- i.e., whether we can reorder the lines of a program without changing its semantics, provided the dataflow graph is preserved. In this paper, we give an operational and categorical semantics to stochastic memoization and name generation in the context of a minimal probabilistic programming language, for a restricted class of functions. Our contribution is a first model of stochastic memoization of constant Bernoulli functions with a non-enumerable type, which validates data flow transformations, bridging the gap between traditional probability theory and higher-order probability models. Our model uses a presheaf category and a novel probability monad on it.

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“…At one end of the spectrum, this line of work involves a foundational categorical analysis (e.g. [6,8,9,10,11,18,19,23,27]). At the other end of the spectrum is 'probabilistic programming', a popular method of statistical modelling using programs (e.g.…”

Section: Introductionmentioning

confidence: 99%

A Monad for Probabilistic Point Processes

Dash

Staton

2021

Electron. Proc. Theor. Comput. Sci.

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A point process on a space is a random bag of elements of that space. In this paper we explore programming with point processes in a monadic style. To this end we identify point processes on a space X with probability measures of bags of elements in X. We describe this view of point processes using the composition of the Giry and bag monads on the category of measurable spaces and functions and prove that this composition also forms a monad using a distributive law for monads. Finally, we define a morphism from a point process to its intensity measure, and show that this is a monad morphism. A special case of this monad morphism gives us Wald's Lemma, an identity used to calculate the expected value of the sum of a random number of random variables. Using our monad we define a range of point processes and point process operations and compositionally compute their corresponding intensity measures using the monad morphism.

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Dirichlet is Natural

Cited by 8 publications

References 9 publications

Stick Breaking, in Coalgebra and Probability

Stick Breaking, in Coalgebra and Probability

A model of stochastic memoization and name generation in probabilistic programming: categorical semantics via monads on presheaf categories

A Monad for Probabilistic Point Processes

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