Bayesian Estimation for Stochastic Gene Expression Using Multifidelity Models

Abstract

We will derive ergodicity results in the following sections based on Theorem 1 in the paper of Roberts and Rosenthal, 1 and we will use some proof techniques of Theorem 1 from Cui et al. 2 for part of our analysis. All random variables we will discuss below will be of the form X : Ω → X where X is a metric space with the associated Borel σ-algebra B(X ).Let X be the parameter space, assumed to have a metric space topology, and π : B(X ) → [0, 1] the target distribution to be sampled from by an adaptive MCMC algorithm. We will assume that π has a density f : X → [0, ∞). Let K γ denote a transition kernel that depends on an adaptation index γ ∈ Y, and assume that each K γ has π as an invariant distribution.We assume that for each fixed γ, an MCMC algorithm with K γ as the Markov transition kernel will eventually converge to π, that is lim n→∞ K n (x, .) − π T V = 0 where µ − ν T V = sup B∈B(X ) |µ(B) − ν(B)| is the total variation distance between two probability measures on X . Let X n be the random variable representing the state of the adaptive MCMC at iteration n, and let Γ n be the random variable representing the choice of kernel for updating from X n to X n+1 . The state of the algorithm is then modeled by the discrete-time stochastic process {(X n , Γ n )}, whose transition between steps is determined by the underlying rules of the algorithm. Finally, let G n = σ ({X 0 , . . . , X n , Γ 0 , . . . , Γ n }) denote the filtration generated by {(X n , Γ n )}. Thus, each Γ n+1 is a G n+1 -measurable random variable.Roberts and Rosenthal proved the following important result, which gives sufficient conditions for ergodicity of an adaptive MCMC.Theorem 1 (Theorem 1 in Roberts and Rosenthal 1 ). Consider an adaptive MCMC algo-

Inferring gene expression models from snapshot RNA data

“…For simplicity, we denote the RNA counts from all cells at all time points m = {{m j t k } j=1:J k } k=1:K . Using this information, our goal is to predict the transcriptional gene output, i.e., we infer both the gene states (that is, perform model selection) as well as infer the associated rate parameters and thus gene state connectivity (parameter inference) [10,11,26,38,39,[49][50][51].…”

“…A Poisson statistical expectation often disagrees with observation [9][10][11][12][13][14][15][16], and, as such, the perennial two-state model (Fig. 1 a ii) is conjectured [1,10,11,[13][14][15][16][17][18][19][20][21]. The two-state, or telegraph, model allows the gene regulatory network to transition between an inactive and active state.…”

“…Mass action formulations are fundamentally insufficient for smFISH data, as RNA copies may be present at low numbers, rendering information on their copy number fluctuations vital toward extracting kinetic parameters [21]. Even methods which resolve this issue using the forward Kolmogorov equation (chemical master equation or CME), which gives the probabilistic temporal evolution of single-cell RNA counts [8,35,[37][38][39][40][41][42][43], still lack robust methods to learn gene states.…”

Systematic comparison of modeling fidelity levels and parameter inference settings applied to negative feedback gene regulation

“…We also investigated the impact of reducing the number of samples per time point as in [ 39 ]. We present the results of this in S5 Fig .…”

Sampling methods for solving Bayesian model updating problems: A tutorial

“…While the case-studies presented in this paper are set in the context whereby measurement data set are considered as a single piece of information made available to make inferences on time-invariant uncertain model parameter(s), such conditions are specific and may not necessarily be true at all times. In general, the recorded measurements or data can come at different time-steps, especially when they are obtained from a system that is evolving with time (see [71,147,50,166,194,195]). These time-evolving data are related to external factors evolving with time such as the measurement noise or environmental loading conditions.…”