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Licensed Unlicensed Requires Authentication Published by De Gruyter May 10, 2014

Efficient parametric inference for stochastic biological systems with measured variability

Iain G. Johnston


Stochastic systems in biology often exhibit substantial variability within and between cells. This variability, as well as having dramatic functional consequences, provides information about the underlying details of the system’s behavior. It is often desirable to infer properties of the parameters governing such systems given experimental observations of the mean and variance of observed quantities. In some circumstances, analytic forms for the likelihood of these observations allow very efficient inference: we present these forms and demonstrate their usage. When likelihood functions are unavailable or difficult to calculate, we show that an implementation of approximate Bayesian computation (ABC) is a powerful tool for parametric inference in these systems. However, the calculations required to apply ABC to these systems can also be computationally expensive, relying on repeated stochastic simulations. We propose an ABC approach that cheaply eliminates unimportant regions of parameter space, by addressing computationally simple mean behavior before explicitly simulating the more computationally demanding variance behavior. We show that this approach leads to a substantial increase in speed when applied to synthetic and experimental datasets.

Corresponding author: Iain G. Johnston, Department of Mathematics, Imperial College London, London SW7 2AZ, UK, e-mail:


We wish to show that if the discrepancy arising from the deterministic mean ρ^m exceeds ϵ, it is likely that the combined discrepancies from the sample mean and sample variance ρm+ρv also exceed ϵ. We will assume that all measurements, and therefore the mean, are non-negative. Due to this non-negativity, the relation ρ^m>ϵρm+ρv>ϵ holds if ρ^mρmρv. Consider the case in which the deterministic mean μ^ differs by an amount δ from the expected value of the sample mean μ. The magnitude of δ is limited by the standard error on the mean: for a reasonably well-characterized mean measurement with low standard error, we assume that ∣δ/μ∣<1. Expanding Equations 5 and 6 gives

(13)ρ^mρm=(log(μ+δ)log(m))2(log(μ)log(m))2 (13)
(14)(log(μ)log(m))2+2(log(μ)log(m))δμ+O((δ/μ)2)(log(μ)log(m))2 (14)
(15)=2(log(μ)log(m))δμ+O((δ/μ)2). (15)

If the discrepancy associated with mean measurements (log(μ)–log(m)) is of a similar magnitude or less than the discrepancy from variance measurements (log(σ2)–log(ν) then, neglecting higher-order terms, as ∣δ/μ∣≪1, 2(log(μ)–log(m))δ/μ≪(log(σ2)–log(ν))2 and hence ρ^mρm<ρv, the condition required for the validity of our threshold assumption.


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Published Online: 2014-5-10
Published in Print: 2014-6-1

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