statespace_score

The analytic score (DK §7.3.3) from one smoother pass.

By the Fisher identity, \(\partial \log L / \partial\psi = E[\partial \log p(Y, \alpha; \psi) / \partial\psi \mid Y]\) (DK eqs. 7.13-7.15). For parameters in \(H_t\), \(R_t Q_t R_t'\) and the \(P_*\) part of the initialization this gives (DK eq. 7.16; Koopman and Shephard 1992)

\[\frac{\partial \log L}{\partial \psi_i} = \tfrac12 \sum_t \mathrm{tr}(\dot H_t G^\varepsilon_t) + \tfrac12 \sum_{t<n} \mathrm{tr}\big(\dot W_t (r_t r_t' - N_t)\big) + \tfrac12 \mathrm{tr}(\dot P_* G^1),\]

with \(W_t = R_t Q_t R_t'\), \(G^\varepsilon_t = H_t^{-1}(\hat\varepsilon_t \hat\varepsilon_t' + Var(\varepsilon_t | Y) - H_t) H_t^{-1}\) and \(G^1 = P_*^+ (V_1 + (\hat\alpha_1 - a_1)(\hat\alpha_1 - a_1)' - P_*) P_*^+\).

The derivatives \(\dot H, \dot W, \dot P_*\) are central differences of what the model’s update sets, so no extra code is needed. The identity holds for the diffuse log likelihood (DK §7.3.5) and for a concentrated scale, but not with a burn-in. With missing data it needs the conditional moments of the missing elements of \(\varepsilon_t\), which the smoother provides.

analytic_score

subroutine analytic_score(model, params, score, llf, info, analytic)
  class(ssm_model_t), intent(inout) :: model
  real(dp), intent(in) :: params(:)
  real(dp), intent(out) :: score(:)
  real(dp), intent(out) :: llf
  integer, intent(out) :: info
  logical, intent(out), optional :: analytic(:)

The score at constrained params and the log likelihood from the same filter run. A parameter that moves Z, T, c, d, \(a_1\) or \(P_\infty\), or a singular \(H_t\), is outside the formula: without analytic the call returns SS_ERR_UNSUPPORTED; with it, such parameters are marked false (score 0) and the others are computed, for the hybrid gradient of fit. An indefinite H or Q returns SS_ERR_NOT_PD.