Thank you, that is very helpful! So if I wanted to model a Gaussian Markov random field, say for example this simple one with four latent nodes (a-d) and two observations (x and y):
x
|
a---b
| |
c---d---y
what would be the correct way to specify it in the @model function?
Hi @ElOceanografo!
To give a tiny bit of background, RxInfer.jl uses a particular type of factor graphs, which establishes the connection between the random variables. I will not go into details about the computational graph behind the package. Still, it technically means that if I want to implement your model, I need to add additional nodes (functions) in your graph representation.
For example, you can specify the graph as follows:
x
|
(f_x)
|
a---(f_a)---b
| |
(f_a) |
| |
c--------(f_cb)--d--(f_d)--y
The difference with your specification is that I explicitly show the functional dependence between your variables. The graph I showed is still incomplete, though I hope it will help you to understand what’s missing from the RxInfer.jl perspective when implementing your graph.
Now, depending on the functional form of fs (let’s say Gaussians, i.e., f_i(i, x) = N(i|x, 1.0) i in {x, a, d} and f_cb=N(d|c+b, 1.0)) you can write the model and inference in RxIner.jl as:
using RxInfer
@model function loopy_model()
y = datavar(Float64)
x_0 = datavar(Float64)
# variances are chosen arbitrary
x ~ Normal(mean = x_0, var = 1.0)
a ~ Normal(mean = x, var = 1.0) # f_x
b ~ Normal(mean = a, var = 1.0) # f_a
c ~ Normal(mean = a, var = 1.0) # f_a
d ~ Normal(mean = c+b, var = 1.0) # f_cb
y ~ Normal(mean = d, var = 1.0)
end
result = inference(
model = loopy_model(),
data = (y = 1.0, x_0=1.0),
initmessages = (
a = vague(NormalMeanPrecision),
),
iterations = 50,
free_energy = true
)
@model macro lets you specify the model in a probabilistic way, RxInfer.jl handles the graphical equivalent of the model under the hood for fast computations.
So if I’m understanding the modeling language correctly, the left-hand side of each ~ is a node, and each node can only appear in one ~ statement. The right-hand side of each ~ corresponds to a factor. If a node has more than one edge coming into it, it needs to be explicitly written as a function of all the connected nodes. So a graph like this
x---a b---y
\ /
d
|
c
|
z
could be specified like this:
x = datavar(Float64)
y = datavar(Float64)
z = datavar(Float64)
# v is arbitrary
a ~ Normal(x, v)
b ~ Normal(y, v)
c ~ Normal(z, v)
d ~ Normal((a + b + c) / 3, v)
Is that right?
RxInfer.jl uses Forney-style Factor Graphs (FFGs) as the graphical representation of the generative model. We use edges to represent variables and nodes for factors/functions. Indeed, you can think of the left-hand side of ~ as a variable defined by an edge (or a node in your description/ it really doesn’t matter for now). The right-hand side of ~ is indeed a factor.
If a node has more than one edge coming into it (in the case of a multi-argument function), you would need to pass all the required variables inside the function.
For example, if you have a gaussian factor node N(y|x, z), it has three edges for out y, mean x, and variance z. In your model definition, you would have to write something like this:
...
y ~ Normal(mean=x, variance=z)
...
As for the model you have specified, yes, that could be an option.