Free gs-Monoidal Categories and Free Markov Categories

Fritz, Tobias; Liang, Wendong

doi:10.1007/s10485-023-09717-0

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…These include (co)parameterized categories of systems that may modify their parameters [Capucci et al, 2021] and hierarchical string diagrams for rewriting higher-order computations [Alvarez-picallo et al, 2022]. Recent work on Markov categories of probabilistic mappings has provided denotational semantics to probabilistic programs [Staton, 2017, abstract categorical descriptions of conditioning, disintegration, sufficient statistics, conditional independence [Cho andJacobs, 2019, Fritz, 2020], and generalized causal models [Fritz andKlingler, 2023, Fritz andLiang, 2023].…”

Section: Motivationmentioning

confidence: 99%

“…underlying hypergraph, which yields a functor I follow Fritz and Liang [2023] in denoting hyp(•) : MonCat ! Hyp.…”

Section: Free Copy/delete and Markov Categoriesmentioning

confidence: 99%

“…Definition 6.22 (Free copy/delete category [Fritz and Liang, 2023]). The free CD category FreeCD ⌃ for ⌃ 2 FinHyp is a subcategory FreeCD ⌃ ✓ cospan(FinHyp/⌃) where…”

Section: Free Copy/delete and Markov Categoriesmentioning

confidence: 99%

“…Generalized causal models [Fritz and Klingler, 2023] [Fritz and Liang, 2023]). Given a fixed monoidal signature ⌃ 2 FinHyp (see Section 6.2.4), a generalized causal model…”

Section: Unnormalized Densities Via a Writer Monad Of Weightsmentioning

confidence: 99%

“…Reasoning about directed graphical models or probabilistic programs compositionally requires providing a graphical syntax interpretable into @Joint(Stoch). Recent work [Fritz andKlingler, 2023, Fritz andLiang, 2023] treated a combinatorial syntax of string diagrams as generalized causal models. This section first reviews the definitions of a generalized causal model and its factorization of a Markov kernel, then applies that syntax to this chapter's novel constructions.…”

Section: Chapter 5 Inference Programming With Combinatorsmentioning

confidence: 99%

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Towards compositional probabilistic programming

Sennesh

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List of AcronymsCK Curse of Knowledge. The tendency to communicate as if others know what we know, because actually thinking about what they will understand takes effort beyond just thinking of what we want to communicate.DSL domain specific language. A programming language designed specifically for a narrow problem domain, often by embedding in a general-purpose language that provides infrastructure. ELBOEvidence Lower Bound. Lower bound on the log-evidence of a generative model. EUBO Evidence Upper Bound. Upper bound on the log-evidence of a generative model. fMRI functional Magnetic Resonance Imaging. Functional magnetic resonance imaging measures brain activity by detecting changes associated with blood flow. GMM Gaussian mixture model. A probabilistic generative model (PGM) of clustering in which cluster indices and parameters are hidden but learned by how they parameterize the observation of continuous data points with a Gaussian likelihood. HMM Hidden Markov Model. A state space model (SSM) in which the latent state has Markovian dynamics, i.e. the state at each time-step depends only on the immediately previous time-step. KL Kullback-Leibler divergence. The average log ratio of two probability densities q and p over the same sample space. MLE maximum likelihood estimation. Calculation of a point estimate of latent variables or model parameters to maximize the likelihood. PDF probability density function. Intuitively, the derivative of a probability measure around a single point in a sample space. vi PGM probabilistic generative model. A statistical model, written in terms of probability densities, that describes a process for generating possible observations.PPL probabilistic programming language. A programming language augmented with primitives for reasoning about uncertainty, usually in the forms of random sampling and observation.SMC symmetric monoidal category. A category with, in addition to the typical sequential composition, a parallel composition whose ordering can be arbitrarily associated or flipped around.SSM state space model. A generative model for time-series data in which each observation is generated based upon the time-evolution of a latent variable through a state space.SPW strictly properly weighted. A relationship between a probability measure and an unnormalized measure whereby drawing samples and weights from the probability measure will, in expectation, equal the value of the unnormalized measure or density. VAE variational autoencoder. A deep generative modeling framework in which not only do neural networks parameterize conditional densities of the generative model, the posterior density is approximated with a neural proposal and the networks are all trained jointly by variational methods.vii Besides my family, I would first and foremost like to express my gratitude to my advisor, Jan-Willem van de Meent, and my co-advisors, Lisa Feldman Barrett, and Karen Quigley. I sent them a cold email late one November, in between tech jobs, while trying to put together graduate applications....

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Section: Motivationmentioning

confidence: 99%

“…underlying hypergraph, which yields a functor I follow Fritz and Liang [2023] in denoting hyp(•) : MonCat ! Hyp.…”

Section: Free Copy/delete and Markov Categoriesmentioning

confidence: 99%

“…Definition 6.22 (Free copy/delete category [Fritz and Liang, 2023]). The free CD category FreeCD ⌃ for ⌃ 2 FinHyp is a subcategory FreeCD ⌃ ✓ cospan(FinHyp/⌃) where…”

Section: Free Copy/delete and Markov Categoriesmentioning

confidence: 99%

“…Generalized causal models [Fritz and Klingler, 2023] [Fritz and Liang, 2023]). Given a fixed monoidal signature ⌃ 2 FinHyp (see Section 6.2.4), a generalized causal model…”

Section: Unnormalized Densities Via a Writer Monad Of Weightsmentioning

confidence: 99%

Section: Chapter 5 Inference Programming With Combinatorsmentioning

confidence: 99%

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Towards compositional probabilistic programming

Sennesh

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Categorical Information Geometry

Perrone

2023

Lecture Notes in Computer Science

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A category-theoretic proof of the ergodic decomposition theorem

Moss

Perrone²

2023

Ergod. Th. Dynam. Sys.

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The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here, we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way, we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories), and the invariant $\sigma $ -algebra of a dynamical system can be seen as a colimit in the category of Markov kernels. In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much more intuitive than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.

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Free gs-Monoidal Categories and Free Markov Categories

Cited by 6 publications

References 18 publications

Towards compositional probabilistic programming

Towards compositional probabilistic programming

Categorical Information Geometry

A category-theoretic proof of the ergodic decomposition theorem

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