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**1 - 4**of**4**### Submitted to POPL 2016 Symbolic Bayesian Inference by Lazy Partial Evaluation

"... Bayesian inference, of posterior knowledge based on prior knowl-edge and observed evidence, is typically implemented by applying Bayes’s theorem, solving an equation in which the posterior mul-tiplied by the probability of an observation equals a joint proba-bility. But when we observe a value of a ..."

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Bayesian inference, of posterior knowledge based on prior knowl-edge and observed evidence, is typically implemented by applying Bayes’s theorem, solving an equation in which the posterior mul-tiplied by the probability of an observation equals a joint proba-bility. But when we observe a value of a continuous variable, the observation usually has probability zero, and Bayes’s theorem says only that zero times the unknown is zero. To infer a posterior dis-tribution from a zero-probability observation, we turn to the sta-tistical technique of disintegration. The classic formulation of dis-integration tells us only what constitutes a posterior distribution, not how to compute it. But by representing all distributions and observations as terms of our probabilistic language, core Hakaru, we have developed the first constructive method of computing dis-integrations, solving the problem of drawing inferences from zero-probability observations. Our method uses a lazy partial evaluator to transform terms of core Hakaru, and we argue its correctness by a semantics of core Hakaru in which monadic terms denote mea-sures. The method, which has been implemented in a larger system, is useful not only on its own but also in composition with sampling and other inference methods commonly used in machine learning. 1.

### ARNAUD SPIWACK

"... Abstract. This article describes a one-sided variant of system l whose typing corresponds to linear sequent calculus and its application. A polarised version of the system is introduced to control the reduction strategy. The polarised type system is then extended to dependent linear types. The syste ..."

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Abstract. This article describes a one-sided variant of system l whose typing corresponds to linear sequent calculus and its application. A polarised version of the system is introduced to control the reduction strategy. The polarised type system is then extended to dependent linear types. The system with dependent type supports dependent elimination of positive connectives.

### DEPENDENT TYPES IN HASKELL: THEORY AND PRACTICE

"... 2.1 Type classes and dictionaries....................... 4 ..."

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### Sound and Complete Bidirectional Typechecking for Higher-Rank Polymorphism and Indexed Types

"... Bidirectional typechecking, in which terms either synthesize a type or are checked against a known type, has become popular for its scalability, its error reporting, and its ease of implementation. Fol-lowing principles from proof theory, bidirectional typing can be ap-plied to many type constructs. ..."

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Bidirectional typechecking, in which terms either synthesize a type or are checked against a known type, has become popular for its scalability, its error reporting, and its ease of implementation. Fol-lowing principles from proof theory, bidirectional typing can be ap-plied to many type constructs. The principles underlying a bidirec-tional approach to indexed types (generalized algebraic datatypes) are less clear. Building on proof-theoretic treatments of equality, we give a declarative specification of typing based on focaliza-tion. This approach permits declarative rules for coverage of pat-tern matching, as well as support for first-class existential types using a focalized subtyping judgment. We use refinement types to avoid explicitly passing equality proofs in our term syntax, making our calculus close to languages such as Haskell and OCaml. An explicit rule deduces when a type is principal, leading to reliable substitution principles for a rich type system with significant type inference. We also give a set of algorithmic typing rules, and prove that it is sound and complete with respect to the declarative system. The proof requires a number of technical innovations, including proving soundness and completeness in a mutually-recursive fashion. 1.