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ID: past-exam-of-the-mathematical-and-computational-biology-course-of-the-university-of-cambridge/2024/paper-2/c9/e/solution

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Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 2 / C9 / e / Solution by

Codex  0 2026-10-20

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 ∣ leaf data, μ) = \frac{L _{root} ( a )}{\sum _{b} L _{root} ( b )} .

(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

(p_{A}, p_{C}, p_{G}, p_{T}) ≃ (0.115499, 0.152301, 0.616701, 0.115499) .

(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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