[Task]: MvNormalWishart node to enable joint μ–Λ priors for NormalMixture

[albertpod]

It'd be nice to have MvNormalWishart factor/node (Normal–Wishart) to express a joint prior over component mean and precision:

Why

NormalMixture currently enforces separate mean-field over means and precisions, which can degrade inference (it really does). A joint μ–Λ node (prior) will let us keep mean-field at the mixture level while coupling mean and precision as a single interface, not the nicest, but a pragmatic workaround until mixture nodes support true joint interfaces.

Minimal proposal

• New node: (μ, Λ) ~ MvNormalWishart(m0, κ, ν, W)

Intended usage with NormalMixture:

# per component k
μΛ[k] ~ MvNormalWishart(m0, κ, ν, W)
y[i] ~ NormalMixture(switch = z[i], mp = μΛ)

Note

MvNormalWishart exists in ExponentialFamily.jl (distribution side), which is helpful, this request is specifically for a ReactiveMP node/factor with conjugate message passing.

[wouterwln]

Just dumping my thoughts to help @esther-van-pelt get started on this issue. Correct me if I'm wrong @bvdmitri : AFAIK we don't have a straightforward way of redefining the number of interfaces we can accept in a node. There is one very straightforward way, which is building a composite model. I'd advise Esther to just implement the MvNormalWishart distribution and its associated rules, and then afterwards we can see if it might be worthwhile to reimplement NormalMixture as a composite model with multiple implementations based on the number of interfaces provided. I'd say, implement the MvNormalWishart first.

[bvdmitri]

I agree with you, we can start simple and implement MvNormalWishart node simply outputing the distribution on the :out edge and start experimenting with extending the mixture node itself either with composite models or standalone nodes

[abpolym]

Wait isn't this just what we did in our paper (@wmkouw and me)? We have derived update rules for a joint matrix-normal wishart (MvNormalWishart) https://arxiv.org/pdf/2506.02710. I could help make this node since we spent a long time on the update rules for this joint. Also for our IWAI 2025 submission we then created this repo that includes a MatrixNormalWishart node for message passing.

[bvdmitri]

If we have work done by @abpolym and @wmkouw we should simply reuse it. @esther-van-pelt seems like a good challenge for you

[wmkouw]

The repo Tim mentions only has the matrix normal, matrix normal Wishart and mv location scale T dists. It does not have the joint parameter MvNormal distributions.

But those dists have been on my todo list for months. I also want to use them for 5SSD0, to, for instance, explain the difference between exact Bayesian inference, p(\mu,\Lambda | data), and mean-field variational, q(\mu)q(\Lambda).

I would be happy to pick this up with someone, but Tim needs to finish a paper first.

[wouterwln]

Hi! We are moving to a cleaner Epic -> Feature -> Task issue hierarchy to better organize our backlog.
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[wouterwln]

@esther-van-pelt let's look at this together next dev session. Node should be easy to make. Integration with NormalMixture node (or any other Normal node) won't work out of the box and is a significant time investment (Feature level). We can leave the node in a decent shape though and it will be good exercise creating nodes.