Random variables defined by their density/energy?

[zenna]

One thing that might be useful is to be able to define random variables by defining a density or energy (unnormalized density).

Sometimes this is the information we have or it's the most natural way to express a random variable. For instance, in #190, it might be best to define an action relative to its expected utility.

There seem to be some hairy implications:

Improper priors

If we give people free reign to define their own densities, there's nothing to stop them from constructing improper priors, which don't integrate to 1 or even to a finite value.

This seems undesirable, and makes the semantics more messy.

One approach is to only allow a special class of functions. That is, construct some DSL which preserves densities. I believe there is work in this area.

Another stance is that if a user constructs a model with an improper prior, then it's a modeling error. It's already possible to construct invalid models, e.g. by having a program error (e.g. divide by zero) or by conditioning on mutually inconsistent things (e.g. x<0 and x>0). Omega doesn't, and couldn't, prevent you from doing these, so maybe we should treat improper priors the same.

Non-generative

Densities don't specify how to actually generate values. It is true that in the normal case when we have a normal random variable, the generative model it defines once conditioned becomes in some sense wrong. But in another sense it remains generative, in the sense that there's a procedure we can use to construct values. In contrast if I had something like

logpdf(x::Graph, p) = length(nodes(x))

which defines an energy function where more nodes are more probable (oops, here's an example of how easy it is to construct improper priors, but let's ignore that for the moment). But it doesn't provide us with any means to actually create graphs.

One solution then is to say that we can add densities to parametric variables. Parametric variables are still generative, in the sense there is an algorithmic process to construct values, but hey have no probability interpretation. We only allow parametric variables in combination with a density to construct density defines random variables. For example:

nnodes = Choice(1:10)
Graph(\omega) = add_edges(Graph(1:nnodes(\omega)), \omega)
function add_edges(g, \omega)
  for v1 in nodes(g)
    for v2 in nodes(g)
      @~ Choice(true, false)(\omega) && add_edge(v1, v2, g)
    end
  end
  g
end
       ```

So Graph Is now a parametric random variable.  Then we might have something like:

```julia
logpdf(x::typeof(Graph), p) = length(nodes(x))

[zenna]

One observation is that you can think of factor as being this kind of DSL for manipulating densities. By only adding to the logpdf, it avoids problems of improper priors by construction.

[zenna]

Two issues:

1. What about conditional densities
2. Might it be preferable to be able to swap out different generative stories for the same random variable.