Handling loops in variational message passing algorithm.

[albertpod]

I am trying to build the inference algorithm in the coupled random walk model:

We assume the following factorization: q = PosteriorFactorization(x_0, x, γ_x, w_0, w, γ_w)
However, ForneyLab throws an error when calling variationalAlgorithm(q, free_energy=false) due to the loop (green edges):
ArgumentError: The input graph contains a loop around Interface 2 (m) of GaussianMeanPrecision gaussianmeanprecision_3

In my opinion, if the loop occurs, ForneyLab should still be able to build the algorithm, but the user would have to provide a schedule.

Here is the code snippet:

using ForneyLab

n_samples = 2

fg = FactorGraph()

# Hidden process model
@RV x_0 ~ GaussianMeanVariance(placeholder(:m_x_0),
                               placeholder(:v_x_0))

@RV γ_x ~ ForneyLab.Gamma(placeholder(:a_x), placeholder(:b_x))

# Noise model
@RV w_0 ~ GaussianMeanVariance(placeholder(:m_w_0),
                               placeholder(:v_w_0))

@RV γ_w ~ ForneyLab.Gamma(placeholder(:a_w), placeholder(:b_w))

# Transition and observation models
x = Vector{Variable}(undef, n_samples)
w = Vector{Variable}(undef, n_samples)
y = Vector{Variable}(undef, n_samples)

x_i_min = x_0
w_i_min = w_0
for i in 1:n_samples
    global x_i_min, w_i_min
    @RV x[i] ~ GaussianMeanPrecision(x_i_min, γ_x)
    @RV w[i] ~ GaussianMeanPrecision(w_i_min, γ_w)
    @RV y[i] = x[i] + w[i]

    # Data placeholder
    placeholder(y[i], :y, index=i)

    # Reset state for next step
    x_i_min = x[i]
    w_i_min = w[i]
end

q = PosteriorFactorization(x_0, x, γ_x, w_0, w, γ_w, ids=[:X0 :X :ΓX :W0 :W :ΓW])
algo = variationalAlgorithm(q, free_energy=false)