Use exact intercept term A^-1 (exp(A t) - I) b

[ErikRingen]

Problem

When branch-length caching was added (efcb253, v1.0.0), A_solve_cache = A^-1 (exp(A t) - I) was symmetrised before being multiplied by b:

for (i in 1:J) for (j in 1:i) {
  real val = 0.5 * (A_solve_cache[u][i, j] + A_solve_cache[u][j, i]);
  ...
}

That matrix is only symmetric when A is, so models with asymmetric cross-selection effects used an incorrect expected change along each branch. The JAX backend copied the same step. Every branch uses the cached path, because all branch lengths are cached. The uncached fallback was always exact.

Change

• Remove the symmetrisation in 05-transformed-parameters.stan (Stan) and _compute_caches() (JAX). The VCV_cache symmetrisation stays, since that matrix is a covariance.
• Add tests/testthat/test-ou_mean.R. It sets drift to zero, then checks that the tip means equal the exact OU solution mu + exp(A)(eta_anc - mu) on an ultrametric tree with asymmetric A and non-zero b. The test fails on main (e.g. 0.52 vs 0.86) and passes with this change. None of the existing tests could catch this:
  • the recovery test fixes b = 0;
  • the fixture tests only check that fits run;
  • the Stan/JAX comparisons check two implementations that shared the step.

Impact

b is weakly identified on an ultrametric tree, and it absorbs most of the error.

Comparison (current vs fixed code)	A and Q	b
Authority example	≤ 0.1 posterior SD	≤ 0.1 posterior SD
Simulated normal + Bernoulli, 2 scenarios	indistinguishable	indistinguishable
Synthetic cichlid example from Ringen et al. (2026), data simulated with b = 0	≤ 0.08 posterior SD	0.39–0.63 posterior SD; all three means move closer to the true value of 0, intervals slightly wider (SD 0.50–0.72 → 0.60–0.83). One simulated dataset.

Checks

• Default test suite (NOT_CRAN=true): 0 failures.
• Extended suite (COEVOLVE_EXTENDED_TESTS=true): direction, recovery, scaling, prior and the full Stan/JAX log-density suite (21 configurations), all passing.

This is independent of #125 and applies cleanly on top of it.

🤖 Generated with Claude Code

[codecov]

Codecov Report

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