This paper identifies several key properties of a monad that allow us to formulate the basics of conditional probability theory, using states for distributions/measures and predicates for events/probability density functions (pdf’s). The distribution monad for discrete probability and the Giry monad for continuous probability are leading examples. Our categorical description handles discrete an...

We study conditional probability from the perspective of complexity theory of functions and operators in analysis, building on work by Ko (1983), Friedman (1984), and Kawamura and Cook (2010). For some random variable X in {0, 1}N whose distribution is continuous and polynomial-time computable, and some polynomial-time computable function f : {0, 1}N → [0, 1] for which the random variable f(X) ...

Clark & Thornton’s type-1/-2 distinction is not well-defined. The classes of type-1 and type-2 problems are too broad: many nocomputable functions are type-1 and type-2 learnable. They are also too narrow: trivial functions, such as identity, are neither type-1 nor type-2 learnable. Moreover, the scope of type-1 and type-2 problems appears to be equivalent. Overall, this distinction does not ap...

This paper deals with the fundamental problem of estimating the distribution function (df) of the duration of the longest path in the stochastic activity network such as PERT network. First a technique is introduced to reduce variance in Conditional Monte Carlo Sampling (CMCS). Second, based on this technique a new procedure is developed for CMCS. Third, a combined approach of simulation and ap...

The entropy of an ergodic finite-alphabet process can be computed from a single typical sample path x1 using the entropy of the k-block empirical probability and letting k grow with n roughly like logn. We further assume that the distribution of the process is a g-measure with respect to a continuous and regular g-function. We prove large deviation principles for conditional, non-conditional an...

We study the problem of computing conditional probabilities, a fundamental operation in statistics and machine learning. In the elementary discrete setting, a ratio of probabilities defines conditional probability. In the abstract setting, conditional probability is defined axiomatically and the search for more constructive definitions is the subject of a rich literature in probability theory a...