Conjugate IG update on sigma_G^2 given the AR(1) innovations
eta_t = g_t - rho * g_{t-1}, t = 2..N, plus the stationary
density at t = 1. Default prior is IG(2, 1) on sigma_G^2.
sample_sigma_G2(
vecG_path,
rho_G,
prior_shape = 2,
prior_rate = 1,
s2_obs = 1,
v_cap_mult = 4
)observation-noise variance (1 for probit/binary).
cap on the stationary G-state variance in units of
s2_obs (default 4).
Scale identification. The model identifies only the product
\(U_t G_t V_t'\), not \(G_t\) alone, so the overall scale of
vec(G_t) is free: left unchecked the chain finds a degenerate
mode where sigma_G^2 (and hence G_t) inflates by orders
of magnitude while U,V collapse to compensate, leaving the
linear predictor unchanged but the reported G_cube meaningless.
Clamping rho_G alone does not bound this because the stationary
state variance is sigma_G^2 / (1 - rho_G^2). We therefore cap
the implied stationary variance at v_cap_mult * s2_obs (a few
observation-noise units), which forces the multiplicative scale onto
U,V -- where the regularising N(0, s2) prior pins it -- and
keeps G_t on a scale comparable to a static-G fit. The cap is
loose enough never to bind on a genuinely small-variation G_t.