Autoregressive node

The AR node (also exported as Autoregressive) encodes a Bayesian autoregressive process of order p:

\[y_t \sim \mathcal{N}(\theta^\top x_t, \, \gamma^{-1})\]

where yₜ is the current observation, xₜ = (yₜ₋₁, …, yₜ₋ₚ) is the vector of p lagged values, θ is the vector of AR coefficients, and γ is the observation precision.

This node is the natural building block for time series models such as AR(p), latent AR processes, and state-space models with autoregressive dynamics.

Interfaces

Metadata

ARMeta is required and must be passed explicitly — the node has no default meta:

y[t] ~ AR(x[t], θ, γ) where { meta = ARMeta(Multivariate, order, ARsafe()) }

The constructor takes:

• Univariate or Multivariate — variate form (determines how x and y are interpreted).
• order — the AR order p (must equal 1 for Univariate).
• ARsafe() or ARunsafe() — numerical stability mode (ARsafe adds a small regularization to avoid singular matrices; ARunsafe is faster but may be numerically fragile).

Univariate vs multivariate

ARMeta{Univariate} treats y and the first element of x as scalars, with order forced to 1. This is an AR(1) model.

ARMeta{Multivariate} uses the full companion-matrix representation to handle AR(p) for p > 1. The state vector x has length p, and the AR process is embedded as a linear state-space model. See CompanionMatrix for the underlying algebraic structure.

State vector slicing

The ReactiveMP.ar_unit and ReactiveMP.ar_slice utilities extract specific parts of the joint state vector in the multivariate setting:

• ReactiveMP.ar_unit — returns an appropriately shaped zero vector or matrix for initializing accumulators.
• ReactiveMP.ar_slice — extracts a subvector or submatrix from a joint mean/covariance. This is used inside rules to separate the y part from the x part of the joint Gaussian q(y, x).

These are internal helpers that surface when writing custom rules for AR-based models.

ar_unit(::T, order)

ar_slice(::T, array, ranges...)