feat(ar): ConjugateAR node with structured MvNormalGamma (θ, γ) posterior

[RetrospectiveRotations]

Summary

Adds ConjugateAR, an autoregressive node that carries the AR coefficients θ and the
transition precision γ jointly on a single edge w as an MvNormalGamma, instead of the
mean-field q(θ)q(γ) split of the existing AR node. Because (θ, γ) are jointly conjugate
for the AR likelihood (a Bayesian linear regression with unknown noise precision), the
structured posterior q(θ, γ) is exactly MvNormalGamma — more accurate than mean-field and
still closed-form.

Also registers MvNormalGamma as a prior factor node so models can write
w ~ MvNormalGamma(μ, Λ, α, β).

⚠️ Dependency

This branch depends on ReactiveBayes/ExponentialFamily.jl#287 (the MvNormalGamma
distribution + prod rule). That PR must land first; this one should not be merged until the
ExponentialFamily release containing MvNormalGamma is available.

What's included

• ConjugateAR node [y, x, w] reusing ARMeta.
• ConjugateAR(:w) marginal rule — the structured BLR Normal-Gamma posterior, with
  expected sufficient statistics from q(y, x): C = E[xxᵀ], b = E[x·y₁], a = E[y₁²].
• ConjugateAR(:w) message rule — the AR likelihood as a Normal-Gamma factor; its product
  with the prior at the equality node reproduces the :w marginal (natural parameters add).
• ConjugateAR(:x), (:y), (:y_x) state/marginal rules and @average_energy,
  delegating to the tested AR rules via effective (q_θ, q_γ) derived from q(w) (no
  duplicated algebra).
• MvNormalGamma prior node [out, μ, Λ, α, β] with its :out message rule and
  @average_energy (so free_energy = true works for ConjugateAR models).

Math references

• Normal-Gamma density — https://statproofbook.github.io/P/ng-pdf
• BLR Normal-Gamma posterior — https://statproofbook.github.io/P/blr-post
• Normal-Gamma differential entropy — https://statproofbook.github.io/P/ng-dent

All formulas were cross-checked against these references and for internal consistency
(message ↔ marginal ↔ prod, entropy ↔ self cross-entropy, univariate ↔ NormalGamma).

Test plan

• ConjugateAR(:w) message params + prod(prior, message) == :w marginal
• (:x), (:y), (:y_x) delegation matches AR with effective (q_θ, q_γ)
• BLR reference (orders 1–3), univariate NormalGamma reduction, properness/PSD
• MvNormalGamma(:out) pass-through; average energy (self cross-entropy == entropy)
• End-to-end RxInfer AR(2): recovers transition precision (0.99 vs 1.0) and coefficients
• CI green on this branch

[github-actions]

🤖 Code Formatting

Your PR still has some code formatting issues. I've updated PR #613 with the necessary formatting changes.

You can merge that PR into this branch to fix the code style check.

Alternatively, you can run make format locally and push the changes yourself.

[wmkouw]

Looks great. Go for it.

My only comment is that the conjugate_autoregressive/y.jl message can already be cast as a Tdist. I realized after our call that the reason we don't have T-distributions in ExponentialFamily.jl is because T-distributions are not part of the exponential family. They do exist in Distributions.jl (https://juliastats.org/Distributions.jl/stable/univariate/#Distributions.TDist; although it may not be full location-scale parameterization).

[RetrospectiveRotations]

Looks great. Go for it.

My only comment is that the conjugate_autoregressive/y.jl message can already be cast as a Tdist. I realized after our call that the reason we don't have T-distributions in ExponentialFamily.jl is because T-distributions are not part of the exponential family. They do exist in Distributions.jl (https://juliastats.org/Distributions.jl/stable/univariate/#Distributions.TDist; although it may not be full location-scale parameterization).

I asked our mutual friend for the reason of the current implementation. Since TDist isn't exp-family it has no closed-form product with the downstream Gaussian message, so as an internal message we'd just project back to Gaussian, right? Do you want me to change this?

[wmkouw]

Well, the y message is for output predictions which tend not to be re-used. And there are multiple ways of doing the Gaussian approximation (e.g., mean-field VI for accurate mean, moment matching for accurate variance).

But this is fine for now. If we need more, we can always open up a new branch with some options under the ARmeta.

[github-actions]

🤖 Code Formatting

Your PR still has some code formatting issues. I've updated PR #613 with the necessary formatting changes.

You can merge that PR into this branch to fix the code style check.

Alternatively, you can run make format locally and push the changes yourself.

[codecov]

Codecov Report

❌ Patch coverage is 98.33333% with 1 line in your changes missing coverage. Please review.
✅ Project coverage is 82.88%. Comparing base (b410342) to head (9176063).
⚠️ Report is 1 commits behind head on main.

Files with missing lines	Patch %	Lines
src/nodes/predefined/conjugate_autoregressive.jl	88.88%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #612      +/-   ##
==========================================
+ Coverage   82.64%   82.88%   +0.24%     
==========================================
  Files         214      220       +6     
  Lines        6660     6720      +60     
==========================================
+ Hits         5504     5570      +66     
+ Misses       1156     1150       -6     

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