As a high-level language and inference tool, we believe that probabilistic programming has much to offer to the robotics community. As robots become increasingly capable, they also pose greater risks. A wrong state estimate can result in serious damage and injury when robots are equipped with powerful motors. Eliminating the possibility of coding errors in the inference engine by compiling them from high-level specifications would be beneficial both from a safety and design-time perspective. Conversely, robotics can be a rich source of challenging inference problems related to robust long-term autonomy that probabilistic programming might help answer. We present a specific example problem of a common estimation task in autonomous robot and show how PP can address a (vexing) practical calibration issues that practitioners face.

]]>NetKAT is one such framework. In NetKAT, verification questions such as waypointing, reachability, and isolation can be naturally phrased as questions about program equivalence, and then be answered by a symbolic (worst-case PSPACE) decision procedure that scales well in practise.

This work studies the equivalence problem for Probabilistic NetKAT, i.e. NetKAT extended with a probabilistic choice operator. We show that the problem is decidable for the history-free fragment of the language and use the system to study resilient routing algorithms.

It remains open whether the equivalence problem for the full language is decidable. The problem is technically challenging because ProbNetKATs semantics is defined over the uncountable space of packet history sets and the calculus is powerful enough to encode continuous (i.e., atomless) distributions.

[ poster | extended-abstract | technical-report ]

]]>To demonstrate the usability of our characterization, we use it to prove that reordering the draws in a probabilistic program preserves contextual equivalence (ie, that sampling in our language is a commutative effect). This allows us to show, for example, that

**let x = e1 in let y = e2 in e3** and **let y = e2 in x = e1 in e3**

are contextual equivalent (provided **x** is not among the free variables of **e2**, and **y** is not among the free variables of **e1**), despite the fact that **e1** and **e2** may have the effect of drawing from the source of entropy.

We therefore propose a natural and intuitive way of comparing such probabilistic systems with respect to time by way of a faster-than relation. This relation is trace-based and requires that for any word and time bound, the faster process has a higher probability of producing a trace that begins with that word within the time bound. This means that time is accumulated along the trace, and it allows for the faster process to be slower than the slower process in some states along the trace, as long as the overall trace is faster. Our investigation of the faster-than relation is carried out in the context of semi-Markov processes, where the time behaviour can be governed by probability distributions other than the exponential distribution, since many systems encountered in practice do not have exponential time behaviour.

We investigate the computational complexity of the proposed relation, and show that is a difficult problem. Through a connection to probabilistic automata, we show that the relation in general is undeciable, and that if we restrict to only a single output label, the problem remains as hard as the Positivity problem for linear recurrence sequences, which has remained an open problem in number theory for many decades. Exploiting the connection to probabilistic automata further, we also show that the relation can not even be approximated.

Although the problem is a difficult one, we also obtain some decidability results. If we restrict ourselves to unambiguous processes, where every output label leads to a unique successor state, then the problem becomes decidable in coNP. If we instead restrict the time so that we only want the faster process to be faster up to a given time bound, then we are able to approximate the relation under the assumption that all timing distributions are slow, meaning that they must take some amount of time to fire a transition.

In this work we have investigated the computational complexity of the faster-than relation, but there are many other interesting aspects of the relation that we hope to investigate in future work, such as logical and compositional aspects.

Authors: Mathias Ruggaard Pedersen, Nathanaël Fijalkow, Giorgio Bacci, Kim Guldstrand Larsen, and Radu Mardare.

This extended abstract is based on a paper that has been accepted for LATA 2018, a preprint of which can be found on arXiv here.

]]>Staton has demonstrated that this class of kernels has various desirable properties like closure under composition and a Fubini-Tonelli result on swapping order of integration. Still, s-finite measures and kernels remain poorly understood and their basic theory still needs to be established.

We characterise their precise relationship to σ-finite and probability measures and kernels. This lets us establish basic results in the s-finite setting like Randomisation, Radon-Nikodým and Disintegration Theorems. These results explicate, respectively, how to understand s-finite probabilistic computation as deterministic computation with access to a Lebesgue measure oracle, when we can apply importance sampling in this setting and that s-finite measures have a well-defined theory of conditional probability.

Further, we sketch how s-finite kernels work together well with traditional game semantics techniques to give fully abstract models of higher-order probabilistic programming languages. The key insight here is that the intensional level of description of game semantics restricts higher-order functions to be tame enough such that classical measure theory (standard Borel spaces) suffices to describe function types, thus circumventing Aumann’s celebrated no-go theorem. Perhaps this explains why this no-go theorem is not encountered as an obstruction in implementing higher-order probabilistic languages?

Authors: Luke Ong, Matthijs Vákár.

Extended abstract: [pdf]

Poster: [pdf]

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Extended abstract: Support Method

]]>Extended abstract: Reinforcement Learning for Inference

]]>A quasi-Borel space is a set *X* together with a set of functions *M *⊆ (ℜ → *X*). The idea is that the reals ℜ are the source of randomness and *M* is a set of allowed random elements. Thus quasi-Borel spaces put the focus on random elements rather than σ-algebras, just as probabilists do.

Authors: Chris Heunen, Ohad Kammar, Sean Moss, Adam Ścibior, Sam Staton, Matthijs Vákár, Hongseok Yang.

Extended abstract: [pdf]

See also our LICS 2017 paper, A Convenient Category for Higher-Order Probability Theory, and our POPL 2018 paper, Denotational validation of higher-order Bayesian inference.

]]>We propose locale theory, and particularly non-spatial sublocales, as a constructive semantic framework for probabilistic programming. Whereas in measure theory, measurable spaces may not have a smallest probability-1 subspace (for a given probability distribution), some locales have smallest probability-1 sublocales, called *random* sublocales. Partial functions that almost surely terminate and discontinuous functions that are almost everywhere continuous become terminating and continuous, respectively, when restricted to random sublocales. We present a definition of disintegration and provide an example distribution where in locale theory, a unique continuous disintegration exists using random sublocales, whereas classically the disintegration is discontinuous and is only unique up to null sets.

Authors: Benjamin Sherman, Jared Tramontano, Michael Carbin

Extended Abstract: Constructive probabilistic semantics with non-spatial locales

]]>`Distributions`

provide fast, numerically stable methods for generating samples and computing statistics, e.g., log density. `Bijectors`

provide composable volume-tracking transformations with automatic caching. Together these enable modular construction of high dimensional distributions and transformations not possible with previous libraries (e.g., pixelCNNs, autoregressive flows, and reversible residual networks). They are the workhorse behind deep probabilistic programming systems like Edward and empower fast black-box inference in probabilistic models built on deep-network components. TensorFlow Distributions has proven an important part of the TensorFlow toolkit within Google and in the broader deep learning community.
Authors: Joshua V. Dillon, Ian Langmore, Eugene Brevdo, Srinivas Vasudevan,Brian Patton, Matt Hoffman, Dave Moore, Dustin Tran, Rif A. Saurous

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