The EfProb Library for Probabilistic Calculations

EfProb is an abbreviation of Effectus Probability. It is the name of a library for probability calculations in Python. EfProb offers a uniform language for discrete, continuous and quantum probability. For each of these three cases, the basic ingredients of the language are states, predicates, and channels. Probabilities are typically calculated as validities of predicates in states. States can be updated (conditioned) with predicates. Channels can be used for state transformation and for predicate transformation. This short paper gives an overview of the use of EfProb

Lower and Upper Conditioning in Quantum Bayesian Theory

Updating a probability distribution in the light of new evidence is a very basic operation in Bayesian probability theory. It is also known as state revision or simply as conditioning. This paper recalls how locally updating a joint state can equivalently be described via inference using the channel extracted from the state (via disintegration). This paper also investigates the quantum analogues of conditioning, and in particular the analogues of this equivalence between updating a joint state and inference. The main finding is that in order to obtain a similar equivalence, we have to distinguish two forms of quantum conditioning, which we call lower and upper conditioning. They are known from the literature, but the common framework in which we describe them and the equivalence result are new.Comment: In Proceedings QPL 2018, arXiv:1901.0947

Disintegration and Bayesian Inversion via String Diagrams

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.Comment: Accepted for publication in Mathematical Structures in Computer Scienc

Affine monads and lazy structures for Bayesian programming

We show that streams and lazy data structures are a natural idiom for programming with infinite-dimensional Bayesian methods such as Poisson processes, Gaussian processes, jump processes, Dirichlet processes, and Beta processes. The crucial semantic idea, inspired by developments in synthetic probability theory, is to work with two separate monads: an affine monad of probability, which supports laziness, and a commutative, non-affine monad of measures, which does not. (Affine means that T(1)≅ 1.) We show that the separation is important from a decidability perspective, and that the recent model of quasi-Borel spaces supports these two monads. To perform Bayesian inference with these examples, we introduce new inference methods that are specially adapted to laziness; they are proven correct by reference to the Metropolis-Hastings-Green method. Our theoretical development is implemented as a Haskell library, LazyPPL

Control-Data Separation and Logical Condition Propagation for Efficient Inference on Probabilistic Programs

We introduce a novel sampling algorithm for Bayesian inference on imperative probabilistic programs. It features a hierarchical architecture that separates control flows from data: the top-level samples a control flow, and the bottom level samples data values along the control flow picked by the top level. This separation allows us to plug various language-based analysis techniques in probabilistic program sampling; specifically, we use logical backward propagation of observations for sampling efficiency. We implemented our algorithm on top of Anglican. The experimental results demonstrate our algorithm's efficiency, especially for programs with while loops and rare observations.Comment: 11 pages with appendice

Logical Aspects of Probability and Quantum Computation

Most of the work presented in this document can be read as a sequel to previous work of the author and collaborators, which has been published and appears in [DSZ16, DSZ17, ABdSZ17]. In [ABdSZ17], the mathematical description of quantum homomorphisms of graphs and more generally of relational structures, using the language of category theory is given. In particular, we introduced the concept of ‘quantum’ monad. In this thesis we show that the quantum monad fits nicely into the categorical framework of effectus theory, developed by Jacobs et al. [Jac15, CJWW15]. Effectus theory is an emergent field in categorical logic aiming to describe logic and probability, from the point of view of classical and quantum computation. The main contribution in the first part of this document prove that the Kleisli category of the quantum monad on relational structures is an effectus. The second part is rather different. There, distinct facets of the equivalence relation on graphs called cospectrality are described: algebraic, combinatorial and logical relations are presented as sufficient conditions on graphs for having the same spectrum (i.e. being ‘cospectral’). Other equivalence of graphs (called fractional isomorphism) is also related using some ‘game’ comonads from Abramsky et al. [ADW17, Sha17, AS18]. We also describe a sufficient condition for a pair of graphs to be cospectral using the quantum monad: two Kleisli morphisms (going in opposite directions) between them satisfying certain compatibility requirement

Dagger and Dilation in the Category of Von Neumann algebras

This doctoral thesis is a mathematical study of quantum computing, concentrating on two related, but independent topics. First up are dilations, covered in chapter 2. In chapter 3 "diamond, andthen, dagger" we turn to the second topic: effectus theory. Both chapters, or rather parts, can be read separately and feature a comprehensive introduction of their own

The Category of Von Neumann Algebras

In this dissertation we study the category of completely positive normal contractive maps between von Neumann algebras. It includes an extensive introduction to the basic theory of $C^*$-algebras and von Neumann algebras.Comment: Ph.D. thesi