This is the Moment for Probabilistic Loops

Abstract

We present a novel static analysis technique to derive higher moments for program variables for a large class of probabilistic loops with potentially uncountable state spaces. Our approach is fully automatic, meaning it does not rely on externally provided invariants or templates. We employ algebraic techniques based on linear recurrences and introduce program transformations to simplify probabilistic programs while preserving their statistical properties. We develop power reduction techniques to further simplify the polynomial arithmetic of probabilistic programs and define the theory of moment-computable probabilistic loops for which higher moments can precisely be computed. Our work has applications towards recovering probability distributions of random variables and computing tail probabilities. The empirical evaluation of our results demonstrates the applicability of our work on many challenging examples.CCS Concepts: • Mathematics of computing → Markov processes; • Computing methodologies → Symbolic and algebraic algorithms.

Distribution Estimation for Probabilistic Loops

“…This section reviews relevant terminology from statistics and probabilistic programs; for further details we refer to [12,31]. Throughout the paper, N denotes the set of natural numbers.…”

“…[19,40]). Alternatively, for restricted classes of probabilistic loops with polynomial updates, symbolic methods from algorithmic combinatorics can be used to compute the exact (higherorder) moments of random program variables x by expressing these moments as closed-form expressions over loop iterations and some initial values [2,31]. That is, [2,31] derive the expected value of the kth moment of variable x at loop iteration n, denoted as E(x k (n)), in closed form, where x k (n) specifies the value of x k at loop iteration n, and k, n ∈ N.…”

“…[19], they cannot be applied to infinite-state PPs with potentially unbounded loops. More recently, static program analysis was combined with statistical techniques to infer higher-order statistical moments of random program variables in PPs with restricted loops and polynomial updates [2,31]. Our work complements these techniques with an algorithmic approach to compute the distributions of random variables in PPs with unbounded loops for which (some) higher-order moments are known (see Algorithm 1).…”

“…1 has polynomial loop updates over random variables r, w drawn from a normal distribution with zero mean and unit variance. This PP satisfies the programming model of [2,31], and as such higher-order moments of this PP can be computed using [2,31]. Based on the first-and second-order moments of r in the Vasicek model PP, we estimate the distribution of the random variable r using Maximum Entropy and Gram-Charlier expansion (see Section 3.1).…”