Priors

polars_ts.bayesian_var.priors

Prior specifications for Bayesian VAR.

Provides the Minnesota (Litterman) and Normal-Wishart prior dataclasses, along with helper functions for building prior precision matrices.

MinnesotaPrior dataclass

Minnesota (Litterman) prior for BVAR.

Shrinks VAR coefficients toward a random walk: own first lag receives prior mean 1, all others 0. Tightness parameters control how strongly the prior pulls toward this structure.

Parameters

lambda1 Overall tightness. Smaller values shrink more aggressively. lambda2 Cross-variable tightness (relative to own-lag). Typically < 1 so cross-variable lags are shrunk harder. lambda3 Lag decay. Higher values shrink distant lags more aggressively. Prior variance for lag l is scaled by l^{-lambda3}. sigma_scale Per-variable residual variance estimates. If None, estimated from univariate AR(p) regressions.

NormalWishartPrior dataclass

Normal-Wishart conjugate prior for BVAR.

Places a matrix-normal prior on the coefficient matrix B and a Wishart prior on the precision matrix Sigma^{-1}.

Parameters

B0 Prior mean for the coefficient matrix, shape (k, k*p+1). If None, defaults to random walk (identity on first own-lag). V0 Prior precision (inverse covariance) for vec(B), shape (k*p+1, k*p+1). If None, uses Minnesota-style diagonal with the given tightness. S0 Prior scale matrix for Wishart, shape (k, k). If None, uses identity scaled by data variance. nu0 Degrees of freedom for Wishart. Must be >= k. If None, defaults to k + 2. tightness Diagonal tightness for automatic V0 construction.

_estimate_sigma_from_ar(data, p)

Estimate per-variable residual variance from univariate AR(p).

_minnesota_prior_precision(k, p, prior, sigma_scale)

Build Minnesota prior mean B0 and precision V0_inv.

Returns

B0 Prior mean, shape (k, k*p+1). V0_inv Prior precision diagonal, shape (k*p+1,).