Recompute the root conditional likelihoods at the fitted rate, then multiply by the uniform root prior and normalize. The factors cancel between numerator and denominator:
(1)
prob = initialise_prob(leaves)
felsenstein(prob, mu_max, len(leaves), left, right, left_time, right_time)
root_posterior = prob[-1] / prob[-1].sum()
print(dict(zip("ACGT", root_posterior)))
At the refined optimum, the posterior distribution is
(2)
Thus G is the most probable ancestral base, but it is not inferred with certainty. This conditions on the fitted rate and does not additionally average over uncertainty in that rate.

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