aci_filter(), aci_smoother() and aci_online() reconstruct hidden
states: from the record up to each time, from the whole record, and from
the record up to a fixed number of steps ahead of each time. lag_table()
stores the finite-lag divergences used by the CIR estimators. The lt_*
helpers access a table without depending on its storage representation.
aci_conditional() describes the conditional question, and
aci_conditional_reduce() carries out the model reduction its
method = "reduce" asks for. The historical lt_tail_bound() name is
retained for compatibility, but its value is a
heuristic tail estimate, not a certified mathematical error bound. A lag
table uses the complete online smoother of andreou2026cir Theorem 3
(Appendix G.1) as its reference. That
reference costs O(N) time-point work; table construction then costs work
proportional to the retained lag cells, with O(N^2) cells for a full table
in the worst case. aci_online() costs O(N) whatever the lag.
Scheme
Two discretizations of the same continuous-time smoothing problem are in
use, and meta$scheme on a path says which one produced it.
"backward_ode_euler" is aci_smoother(): the continuous backward smoothing
equations integrated with an Euler step. "theorem3_discrete" is
aci_online() and the lag table's reference smoother: the exact conditional
law of the hidden state given the observed increments on the sampling grid,
under the explicit single-step discretization. They agree only to first
order in the step, so aci_online() at lag = Inf does not reproduce
aci_smoother(), and the gap grows with the length of the record rather than
settling to a constant. See the Scheme section of aci_online() for the
measured size on the packaged ENSO partition. aci() reports the scheme its
own result was built under in meta$smoother_scheme.
References
Andreou, M. and Chen, N. (2026). Bridging prediction and attribution: identifying forward and backward causal influence ranges using assimilative causal inference. arXiv:2510.21889v2, 4 August 2026. doi:10.48550/arXiv.2510.21889
Andreou, M., Chen, N. and Li, Y. (2026). An adaptive online smoother with closed-form solutions and information-theoretic lag selection for conditional Gaussian nonlinear systems. Journal of Nonlinear Science 36(4), 71. doi:10.1007/s00332-026-10271-x