Bayesian inference with information content model check for Langevin equations

The Bayesian data analysis framework has been proven to be a systematic and effective method of parameter inference and model selection for stochastic processes. In this work, we introduce an information content model check that may serve as a goodness-of-fit, like the ${\chi }^2$ procedure, to complement conventional Bayesian analysis. We demonstrate this extended Bayesian framework on a system of Langevin equations, where coordinate-dependent mobilities and measurement noise hinder the normal mean-squared displacement approach.

Figure 1

(a) Sample track along with a deterministic fit. (b–e) Likelihoods for variations of single parameters (right) around the maximum likelihood point for the best candidate model, which in this case is $M_{\text{pull,clean}}$. The dashed lines represent the true values.

Figure 2

The inferred mean information ${\mathcal{H}}_n$ for the data set in Fig. 1 assuming the model $M_{\text{pull,clean}}$ along with the information $h^{*}=h({\mathit{x}}_n^{*},{\mathbf{\theta }}^{*})$ for replicated tracks from the same model and corresponding $p$ values.

Figure 3

Information measures as in Fig. 2 for the data set in Fig. 1 assuming the model $M_{\text{free}}$.

Figure 4

(a) A clean sample track in solid blue along with the same track with measurement noise pollution in orange circles. (b–e) Likelihoods for variations of single parameters around the maximum likelihood point for the best candidate model ($M_{\text{free,mn}}$). The true values are marked by the green dashes. Note that the likelihood curves are one dimensional slices of a four-dimensional function.

Figure 5

(a) The data from Fig. 1 is reproduced in solid blue while the orange circles represent data with a large added measurement noise (${\sigma }_{\mathrm{mn}}^2=40$). (b–e) Likelihoods for variations of single parameters around the maximum likelihood point for the best candidate model ($M_{\text{pull,mn}}$) are shown, displaying inaccuracy caused by the large measurement noise.

Figure 6

Information measures as in Fig. 2 for the data set in Fig. 5 assuming the model $M_{\text{pull,mn}}$.

Figure 7

Information measures as in Fig. 2 for the data set in Fig. 5 assuming the model $M_{\text{pull,clean}}$.