Soft Execution

==ₛ
>ₛ
>=ₛ
<=ₛ
<ₛ

Omega.d(x, y)

Relaxation

In Omega you condition on predicates. A predicate is any function whose domain is Boolean. These are sometimes called indicator functions or characteristic functions. In particular, in Omega we condition on Bool valued random variables:

x = normal(0.0, 1.0)
y = x == 1.0
rand(y)

From this perspective, conditioning means to solve a constraint. It can be difficult to solve these constraints exactly, and so Omega supports softened constraints to make inference more tractable.

To soften predicates, use soft counterparts to primitive predicates. Suppose we construct a random variable of the form x == y. In the soft version we would write x ==ₛ y (or softeq(x, y)).

julia> x = normal(0.0, 1.0)
julia> y = x ==ₛ 1.0
julia> rand(y)
ϵ:-47439.72956833765

Softened predicates return values in unit interval [0, 1] as opposed to a simply true or false. Intuitively, 1 corresponds to true, and a high value (such as 0.999) corresonds to "nearly true". In practice, we encode this number in log scale [-Inf, 0] for numerical reasons. Mathematicallty, soft predicates they have the form:

\[k_\alpha(\rho(x, y))\]

If $\rho(x, y)$ denotes a notion of distance between $x$ and $y$. Distances are determined by the method Omega.d

Rather than output values of type Float64, soft predicate output values of type SoftBool.

Distances and Kernels

Omega has a number of built-in kernels:

By default, the squared exponential kernel kse is used with a default temperature parameter. The method withkernel can be used to choose which kernel is being applied within the context of a soft boolean operator.