Fig 1.
Illustration of the nested sampling approximation with a uniform prior on [0, 1].
A: The integral over the parameter space ∫Ω l(θ)dθ. B: The transformed integral over the prior volume x.
Fig 2.
Illustration of likelihood approximation with a particle filter.
A: Top: Likelihood for different parameters k (red) and contour lines of the joint distribution Π(k, log(k) of the parameter k and its likelihood approximation , based on 106 samples of the likelihood approximation obtained with a particle filter with 100 particles. Bottom: The constrained priors π(k|l(k) > ϵ) and
for ϵ = 1e − 24. B: Example distributions
(blue) for k = 1, 1.2 and 1.4 and the true likelihood l(k) red.
Fig 3.
Inference on the birth-death model.
A: Histogram of the posterior estimate obtained with LF-NS using N = 100 and H = 100. The true posterior is indicated in black. B: Development of the estimation of the Bayesian evidence using the estimation based solely on the dead points
, the estimate approximation from the live points
and the estimation based on both
. The corresponding standard errors are indicated as the shaded areas. The true Bayesian evidence is indicated with the dashed red line. C: Estimate of the current variance estimate
and the lower bounds for the lowest achievable variance
. D: Developments of the different error estimations for each iteration.
Fig 4.
Inference on the Lac-Gfp model.
A: Development of the estimation of the Bayesian evidence using the estimation based solely on the dead points , the estimate approximation from the live points
and the estimation that uses both
. The corresponding standard errors are indicated as the shaded areas. B: Estimate of the current variance estimate
and the lower bounds for the lowest achievable variance
. C: The acceptance rate of the LF-NS algorithm for each iteration (blue) and the cumulative time needed for each iteration in hours (red). The computation was performed on 48 cores in parallel on the Euler cluster of the ETH Zurich. D: Marginals of the inferred posterior distributions of the parameters based on one simulated trajectory. The blue lines indicate the parameters used for the simulation of the data.
Fig 5.
Inference on the transcription model.
A: Schematic representation of the gene expression model. The model consists of a gene that switches between an “on” and an “off” state with rates kon and koff. When “on” the gene is getting transcribed at rate kr. The transcription process is modelled through n RNA species that sequentially transform from one to the next at rate λ. The observed species are all of the intermediate RNAi species. B: The marginal posterior distribution of the parameters of the system. The histogram indicates the posterior obtained through LF-NS, the blue line indicates the posterior obtained from a long pMCMC run and the shaded areas indicate the parameter ranges that were considered in [58]. The scale of the x-axis does not represent the range of the prior and has been chosen for presentational purposes. C: The five trajectories used for the parameter inference.
Fig 6.
Runtime comparison between LF-NS, pMCMC and ABC.
A: The estimated Bayesian evidence for the different LF-NS runs. The error bars indicate the standard deviation of the final BE estimate. B: The estimated mean and standard deviation of the marginal posterior for c3 for the different algorithm runs. The solid red line indicates the mean and standard deviation of the true marginal posterior for c3.