Conditioning in Probabilistic Programming

Abstract

This paper investigates the semantic intricacies of conditioning, a main feature in probabilistic programming. Our study is based on an extension of the imperative probabilistic guarded command language pGCL with conditioning. We provide a weakest precondition (wp) semantics and an operational semantics. To deal with possibly diverging program behaviour we consider liberal preconditions. We show that diverging program behaviour plays a key role when defining conditioning. We establish that weakest preconditions coincide with conditional expected rewards in Markov chains-the operational semantics-and that the wp-semantics conservatively extends the existing semantics of pGCL (without conditioning). An extension of these results with non-determinism turns out to be problematic: although an operational semantics using Markov decision processes is rather straightforward, we show that providing an inductive wp-semantics in this setting is impossible. Finally, we present two program transformations which eliminate conditioning from any program. The first transformation hoists conditioning while updating the probabilistic choices in the program, while the second transformation replaces conditioning-in the same vein as rejection sampling-by a program with loops. In addition, we present a last program transformation that replaces an independent identically distributed loop with conditioning.

Fine-Grained Semantics for Probabilistic Programs

“…Listing 6 shows an example program using √ x. Usually, semantics do not explicitly address partial functions [23,24,28,33] or use is filled in [34] because its semantics can handle non-termination. However, the intersection between and is not filled because [34] cannot distinguish non-termination from observation failure.…”

“…Other languages handle both non-termination and observation failure by subprobability distributions, which makes it impossible to conclude that the missing weight is due to observation failure (and not due to non-termination) [4,24,34]. The semantics in [28] cannot directly express that the missing weight is due to observation failure (rather, the semantics are undefined due to a division by zero). However, the semantics enables a careful reader to determine that the missing weight is due to observation failure (by investigating the conditional weakest precondition and the conditional weakest liberal precondition).…”

“…This is not surprising, given that this is also the semantics of exceptions in most deterministic languages. Most existing semantics either cannot express Listing 8 ( [23,34] have no assertions, [35] has no iteration) or cannot distinguish failing an assertion from non-termination [24,28,33]. The consequence of the latter is that removing the first two assertions from Listing 8 does not affect the semantics.…”

“…For later work [24], Kozen also switches to sub-probability kernels, which has the advantage of avoiding redundancies. A different approach uses weakest precondition to define the semantics, as in [28]. Staton et al [35] use a different concept of measurable functions A → P (R ≥0 × B) (where P (S) denotes the set of all probability measures on S).…”

“…Typing. Some probabilistic languages are untyped [4,28], while others are limited to just a single type: R n [23,24] [33]. Some languages provide more interesting types including sum types, distribution types and tuples [34,35].…”