Omega Internals
Omega is a library for causal and probabilistic inference in Julia. It shares many characteristics with other probabilistic languages, such as Pyro, Gen, Stan, and Turing, but has a few important differences. At a high level, the salient properties of Omega are that:
<!– The conceptual model of Omega could be summarized as follows: –>
Random Variables
A very important type of object in Omega is a random variable. The simplest kind of random variable is a primitive random variable. There is an infintie set of primitive variables, that are all mutually independent. Random variables in this set can be constructed using ~. For example:
ID = 1
X = ID ~ StdUniform{Float64}()StdUniform{Flaot64} is a singleton type – it has a single element StdUniform{Float64}().
A random variable in Omega is a function of the form
\[X : \Omega \to T\]
In Julia, a random variable is any function X such that X(\omega::AbstractΩ) is well-defined, that is, the method exists, and X(\omega::AbstractΩ) for all \omega::AbstractΩ is well-defined.
For those familiar with other probabilistic programming languages, the conceptual differences with Omega are that:
- Omega random variables are pure functions
Patterns
μ = ~ Normal(0, 1)
X1 = Normal(μ, 1)
X2 = Normal(μ, 1)
X3 = Normal(μ, 1)Sampling
Unlike other PPLs, random variables in Omega are not smaplers themselves, per-se. They can be sampled from, using randsample.
X = ID ~ StdUniform{Float64}()
randsample(X)Distribution Families
Probabilistic Models
In Omega, there is no explicit notion of a probabilistic model, there are only random variables. Conceptually, it can be useful to think of a probabilsitic model as a collection of random variables.
Independence and Conditional Independence
When constructing a probabilistic model, it's common to want to:
- Construct multiple random variables that are
Conditioning
In Omega, conditioning is a process that transforms a random variable into a new one.
Conditioning is performed by a function cnd, which naturally has the type:
\[cnd : (\Omega \to T) \times (\Omega \to Bool) \to (\Omega \to T)\]