Nested sampling for parameter inference in systems biology: application to an exemplar circadian model

Abstract

BackgroundModel selection and parameter inference are complex problems that have yet to be fully addressed in systems biology. In contrast with parameter optimisation, parameter inference computes both the parameter means and their standard deviations (or full posterior distributions), thus yielding important information on the extent to which the data and the model topology constrain the inferred parameter values.ResultsWe report on the application of nested sampling, a statistical approach to computing the Bayesian evidence Z, to the inference of parameters, and the estimation of log Z in an established model of circadian rhythms. A ten-fold difference in the coefficient of variation between degradation and transcription parameters is demonstrated. We further show that the uncertainty remaining in the parameter values is reduced by the analysis of increasing numbers of circadian cycles of data, up to 4 cycles, but is unaffected by sampling the data more frequently. Novel algorithms for calculating the likelihood of a model, and a characterisation of the performance of the nested sampling algorithm are also reported. The methods we develop considerably improve the computational efficiency of the likelihood calculation, and of the exploratory step within nested sampling.ConclusionsWe have demonstrated in an exemplar circadian model that the estimates of posterior parameter densities (as summarised by parameter means and standard deviations) are influenced predominately by the length of the time series, becoming more narrowly constrained as the number of circadian cycles considered increases. We have also shown the utility of the coefficient of variation for discriminating between highly-constrained and less-well constrained parameters.

The circadian dynamics of small nucleolar RNA in the mouse liver

“…The Bayesian evidence for each of the three models was calculated using nested sampling [27,28] and time courses were designated circadian where the evidence for the 24 h cycle was 10 times that for the alternative models. The nested sampling algorithm infers the phase and its standard deviation, both of which are of interest in assessing rhythmic behaviour.…”

polychord: next-generation nested sampling

“…PolyChord implements several novel features compared to Aitken & Akman's (2013) slice-based nested sampling. It utilises slice sampling in a manner that uses the information present in the live and phantom points to deal with correlated posteriors.…”

The circadian dynamics of small nucleolar RNA

“…This modified cosine model with period 24 hrs, with period 12 hrs and a linear model were fitted to each time series to assess the fit of a true circadian rhythm, a rapidly oscillating signal (likely noise given the sampling frequency of this data, but potentially due to transcription factor binding [22]) and a gradual change in expression respectively. The Bayesian evidence for each of the three models was calculated using nested sampling [23,24] and time courses were designated circadian where the evidence for the 24 hr cycle was ten times that for the alternative models. The nested sampling algorithm infers the phase and its standard deviation, both of which are of interest in assessing rhythmic behaviour.…”

“…The fit between the cosine models and expression data was assessed using the nested sampling algorithm to calculate the log of Bayesian evidence (also known as the marginal likelihood), log Z [23] from the likelihood function and the prior. All priors were selected uniformly from a range bounded by maximum and minimum values given above.…”