Relatively Complete Verification of Probabilistic Programs: An Expressive Language for Expectation-Based Reasoning

We study a syntax for specifying quantitative “assertions” - functions mapping program states to numbers - for probabilistic program verification. We prove that our syntax is expressive in the following sense: Given any probabilistic program C, if a function f is expressible in our syntax, then the function mapping each initial state σ to the expected value of f evaluated in the final states reached after termination C on σ (also called the weakest preexpectation wp[C](f)) is also expressible in our syntax. As a consequence, we obtain a relatively complete verification system for verifying expected values and probabilities in the sense of Cook: Apart from a single reasoning step about the inequality of two functions given as syntactic expressions in our language, given f, g, and C, we can check whether g ≤ wp[C](f)

A Pre-expectation Calculus for Probabilistic Sensitivity

Sensitivity properties describe how changes to the input of a program affect the output, typically by upper bounding the distance between the outputs of two runs by a monotone function of the distance between the corresponding inputs. When programs are probabilistic, the distance between outputs is a distance between distributions. The Kantorovich lifting provides a general way of defining a distance between distributions by lifting the distance of the underlying sample space; by choosing an appropriate distance on the base space, one can recover other usual probabilistic distances, such as the Total Variation distance. We develop a relational pre-expectation calculus to upper bound the Kantorovich distance between two executions of a probabilistic program. We illustrate our methods by proving algorithmic stability of a machine learning algorithm, convergence of a reinforcement learning algorithm, and fast mixing for card shuffling algorithms. We also consider some extensions: using our calculus to show convergence of Markov chains to the uniform distribution over states and an asynchronous extension to reason about pairs of program executions with different control flow

'MRI-negative PET-positive' temporal lobe epilepsy (TLE) and mesial TLE differ with quantitative MRI and PET: a case control study

Background: \u27MRI negative PET positive temporal lobe epilepsy\u27 represents a substantial minority of temporal lobe epilepsy (TLE). Clinicopathological and qualitative imaging differences from mesial temporal lobe epilepsy are reported. We aimed to compare TLE with hippocampal sclerosis (HS+ve) and non lesional TLE without HS (HS-ve) on MRI, with respect to quantitative FDG-PET and MRI measures.Methods: 30 consecutive HS-ve patients with well-lateralised EEG were compared with 30 age- and sex-matched HS+ve patients with well-lateralised EEG. Cerebral, cortical lobar and hippocampal volumetric and co-registered FDG-PET metabolic analyses were performed.Results: There was no difference in whole brain, cerebral or cerebral cortical volumes. Both groups showed marginally smaller cerebral volumes ipsilateral to epileptogenic side (HS-ve 0.99, p = 0.02, HS+ve 0.98, p &lt; 0.001). In HS+ve, the ratio of epileptogenic cerebrum to whole brain volume was less (p = 0.02); the ratio of epileptogenic cerebral cortex to whole brain in the HS+ve group approached significance (p = 0.06). Relative volume deficits were seen in HS+ve in insular and temporal lobes. Both groups showed marked ipsilateral hypometabolism (p &lt; 0.001), most marked in temporal cortex. Mean hypointensity was more marked in epileptogenic-to-contralateral hippocampus in HS+ve (ratio: 0.86 vs 0.95, p &lt; 0.001). The mean FDG-PET ratio of ipsilateral to contralateral cerebral cortex however was low in both groups (ratio: HS-ve 0.97, p &lt; 0.0001; HS+ve 0.98, p = 0.003), and more marked in HS-ve across all lobes except insula.Conclusion: Overall, HS+ve patients showed more hippocampal, but also marginally more ipsilateral cerebral and cerebrocortical atrophy, greater ipsilateral hippocampal hypometabolism but similar ipsilateral cerebral cortical hypometabolism, confirming structural and functional differences between these groups.<br /