Every quantity on this page is conditional on the supplied model, prior and
observed record. The model supplies the likelihood and is not estimated
from the record, so a positive value is influence under the dynamics that
were supplied: it is not an empirically identified causal effect, an
intervention effect or a significance statement. Gaussian relative entropy
is oriented as smoother relative to filter, and its per-time value is a
pointwise information gain in nats; summed over the record with the step it
is a time-integrated value in nats times model time. The value carries the
prior and the integration step as well as the model; aci() gives the
structurally independent null's return at two steps and at a step coarse
against the prior, each with the prior variance, horizon, scheme and
regularization status it was measured under. A normal aci() call
uses the supplied-code backward-ODE headline smoother, including its
correlated-noise correction, independently of keep. lag_table() and
aci(table = ...) instead use the complete online Theorem 3 smoother;
their finite-grid diagonal can therefore differ from headline ACI.
aci_range() summarizes the duration of influence on the discrete time
grid. A finite adaptive table is labelled objective_on_truncated_table;
its tail_bound field is a heuristic tail estimate and must not be
interpreted as a certified error bound. It is a diagnostic under the
retained record, not a guarantee about the cells the truncation dropped. The
l1_linf estimator is a ratio, integrated with composite Simpson by
default, following the ACI reference code; quadrature = "sum" uses the L1
grid-function sum instead. The exact objective is a finite threshold sum
with no time-axis quadrature, so it is unaffected by that choice.
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