Notes on Category Theory with examples from basic mathematics

Perrone, Paolo

doi:10.48550/arxiv.1912.10642

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…The composition of these two functors then defines a comonad on PROCTHEORY which, in particular, takes FUNC to PREFUNC. This is closely related to [86,Ex. 4…”

Section: B Subsuming the Framework Of Classical Causal Modelingmentioning

confidence: 84%

Unscrambling the omelette of causation and inference: The framework of causal-inferential theories

Schmid,

Selby,

Spekkens

2020

Preprint

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Using a process-theoretic formalism, we introduce the notion of a causal-inferential theory: a triple consisting of a theory of causal influences, a theory of inferences (of both the Boolean and Bayesian varieties), and a specification of how these interact. Recasting the notions of operational and realist theories in this mold clarifies what a realist account of an experiment offers beyond an operational account. It also yields a novel characterization of the assumptions and implications of standard no-go theorems for realist representations of operational quantum theory, namely, those based on Bell's notion of locality and those based on generalized noncontextuality. Moreover, our process-theoretic characterization of generalised noncontextuality is shown to be implied by an even more natural principle which we term Leibnizianity. Most strikingly, our framework offers a way forward in a research program that seeks to circumvent these no-go results. Specifically, we argue that if one can identify axioms for a realist causal-inferential theory such that the notions of causation and inference can differ from their conventional (classical) interpretations, then one has the means of defining an intrinsically quantum notion of realism, and thereby a realist representation of operational quantum theory that salvages the spirit of locality and of noncontextuality. CONTENTS

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“…The composition of these two functors then defines a comonad on PROCTHEORY which, in particular, takes FUNC to PREFUNC. This is closely related to [86,Ex. 4…”

Section: B Subsuming the Framework Of Classical Causal Modelingmentioning

confidence: 84%

Unscrambling the omelette of causation and inference: The framework of causal-inferential theories

Schmid,

Selby,

Spekkens

2020

Preprint

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“…It is then straightforward to verify that C W is indeed also a Markov category. We can alternatively think of C W as the co-Kleisli category of the reader comonad 5 W ⊗ on C (see for example [25,Section 5.3]). Note that, while the reader comonad is usually defined on cartesian monoidal categories, the only property of cartesian monoidal categories that is actually used in the definition is that the object W has a comonoid structure, and thus this co-Kleisli category still makes sense in our context.…”

Section: Parametric Markov Categoriesmentioning

confidence: 99%

Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability

Fritz¹,

Gonda²,

Perrone³

et al. 2020

Preprint

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Markov categories are a recent categorical approach to the mathematical foundations of probability and statistics. Here, this approach is advanced by stating and proving equivalent conditions for second-order stochastic dominance, a widely used way of comparing probability distributions by their spread. Furthermore, we lay the foundation for the theory of comparing statistical experiments within Markov categories by stating and proving the classical Blackwell-Sherman-Stein Theorem. Our version not only offers new insight into the proof, but its abstract nature also makes the result more general, automatically specializing to the standard Blackwell-Sherman-Stein Theorem in measure-theoretic probability as well as a Bayesian version that involves prior-dependent garbling. Along the way, we define and characterize representable Markov categories, within which one can talk about Markov kernels to or from spaces of distributions. We do so by exploring the relation between Markov categories and Kleisli categories of probability monads.

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“…I will implement that extension by equipping each joint density kernel with an extra likelihood factor via a writer monad, referring to the resulting factors as likelihoood weights. Perrone [Perrone, 2019] described the writer monad for arbitrary monoids (A, , 1) in his Example 5.1.7, so Proposition 6.12 will only quickly review its construction. Proposition 6.12 (Weights form a writer monad).…”

Section: Unnormalized Densities Via a Writer Monad Of Weightsmentioning

confidence: 99%

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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Notes on Category Theory with examples from basic mathematics

Cited by 3 publications

References 7 publications

Unscrambling the omelette of causation and inference: The framework of causal-inferential theories

Unscrambling the omelette of causation and inference: The framework of causal-inferential theories

Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability

Towards compositional probabilistic programming

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