Design of an optimal combination therapy with broadly neutralizing antibodies to suppress HIV-1

. 2022 Jul 19;11:e76004. doi: 10.7554/eLife.76004

Algorithm 4. Importance sampled log-likelihood given a single datapoint
procedure SigmaLikelihood ( $\hat{μ}, {\hat{μ}}^{†}, m, s, θ_{t s}, N$ ) ⊳ Takes mutant $m$ and susceptible $s$ counts for a particular site at a single timepoint. Returns an approximate log-likelihood function $l (\hat{σ}) = \log P (m, s \| θ_{t s}, \hat{σ}) + c$ , up to an additive constant. SigmaLikelihood is a closure that returns a one-parameter function. We used N =103 samples. $\begin{array}{ll} θ \leftarrow \hat{μ} θ_{t s} \\ θ^{†} \leftarrow {\hat{μ}}^{†} θ_{t s} \end{array}$ for $i \in 1 : N$ do $x_{i} \sim Beta (θ, θ^{†})$ ⊳ Sample from the neutral distribution $w_{i} \leftarrow \frac{B (θ + m, θ^{†} + s)}{B (θ, θ^{†})} \frac{1}{x_{i}^{m} (1 - x_{i})^{w}}$ ⊳ Importance weight ratio $y_{i} \sim Beta (θ + m, θ^{†} + s)$ ⊳ sample from the neutral distribution, conditioned on the observations $v_{i} \leftarrow \frac{B (θ + m, θ^{†} + s)}{B (θ, θ^{†})} \frac{1}{y_{i}^{m} (1 - y_{i})^{w}}$ ⊳ Importance weight ratio end for $Z_{0} (\hat{σ}) := \frac{1}{N} \sum_{i} e^{\hat{σ} θ_{t s} x_{i}} \frac{1}{1 + w_{i}} + \frac{1}{N} \sum_{i} e^{- \hat{σ} θ_{t s} y_{i}} \frac{1}{1 + v_{i}}$ ⊳ importance sampling mean $Z_{1} (\hat{σ}) := \frac{1}{N} \sum_{i} e^{\hat{σ} θ_{t s} x_{i}} \frac{1}{1 + w_{i}^{- 1}} + \frac{1}{N} \sum_{i} e^{- \hat{σ} θ_{t s} y_{i}} \frac{1}{1 + v_{i}^{- 1}}$ ⊳ Note the inversion in the weighting factor compared to Z₀. return $l (\hat{σ}) := \log Z_{1} (\hat{σ}) - \log Z_{0} (\hat{σ})$ ⊳ Return a log-likelihood function. The same random variable realizations $x_{1 : N}$ and $y_{1 : N}$ are cached in memory and used for each function evaluation, making $l (\hat{σ})$ continuous and differentiable. end procedure

Algorithm 4. Importance sampled log-likelihood given a single datapoint

procedure SigmaLikelihood (

\hat{μ}, {\hat{μ}}^{†}, m, s, θ_{t s}, N

) ⊳ Takes mutant

m

and susceptible

s

counts for a particular site at a single timepoint. Returns an approximate log-likelihood function

l (\hat{σ}) = \log P (m, s | θ_{t s}, \hat{σ}) + c

, up to an additive constant. SigmaLikelihood is a closure that returns a one-parameter function. We used N =103 samples.

\begin{array}{ll} θ \leftarrow \hat{μ} θ_{t s} \\ θ^{†} \leftarrow {\hat{μ}}^{†} θ_{t s} \end{array}

for

i \in 1 : N

x_{i} \sim Beta (θ, θ^{†})

⊳ Sample from the neutral distribution

w_{i} \leftarrow \frac{B (θ + m, θ^{†} + s)}{B (θ, θ^{†})} \frac{1}{x_{i}^{m} (1 - x_{i})^{w}}

⊳ Importance weight ratio

y_{i} \sim Beta (θ + m, θ^{†} + s)

⊳ sample from the neutral distribution, conditioned on the observations

v_{i} \leftarrow \frac{B (θ + m, θ^{†} + s)}{B (θ, θ^{†})} \frac{1}{y_{i}^{m} (1 - y_{i})^{w}}

⊳ Importance weight ratio
end for

Z_{0} (\hat{σ}) := \frac{1}{N} \sum_{i} e^{\hat{σ} θ_{t s} x_{i}} \frac{1}{1 + w_{i}} + \frac{1}{N} \sum_{i} e^{- \hat{σ} θ_{t s} y_{i}} \frac{1}{1 + v_{i}}

⊳ importance sampling mean

Z_{1} (\hat{σ}) := \frac{1}{N} \sum_{i} e^{\hat{σ} θ_{t s} x_{i}} \frac{1}{1 + w_{i}^{- 1}} + \frac{1}{N} \sum_{i} e^{- \hat{σ} θ_{t s} y_{i}} \frac{1}{1 + v_{i}^{- 1}}

⊳ Note the inversion in the weighting factor compared to Z₀.
return

l (\hat{σ}) := \log Z_{1} (\hat{σ}) - \log Z_{0} (\hat{σ})

⊳ Return a log-likelihood function. The same random variable realizations

x_{1 : N}

and

y_{1 : N}

are cached in memory and used for each function evaluation, making

l (\hat{σ})

continuous and differentiable.
end procedure