= Solution
Recompute the root conditional <likelihoods> at the fitted rate, then multiply by the uniform root prior and normalize. The factors $1/4$ cancel between numerator and denominator:
$$
p(a\mid\text{leaf data},\widehat\mu)
=\frac{L_{\mathrm{root}}(a)}{\sum_bL_{\mathrm{root}}(b)}.
$$
``
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
$$
\boxed{(p_A,p_C,p_G,p_T)\simeq(0.115499,\ 0.152301,\ 0.616701,\ 0.115499).}
$$
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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