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Biehl et al (2020) present some interesting observations on an early formulation of the free energy principle in (Friston, 2013). We use these observations to scaffold a discussion of the technical arguments that underwrite the free energy principle. This discussion focuses on solenoidal coupling between various (subsets of) states in sparsely coupled systems that possess a Markov blanket - and the distinction between exact and approximate Bayesian inference, implied by the ensuing Bayesian mechanics., A response to a technical critique [arXiv:2001.06408] of the free energy principle as presented in "Life as we know it"

In this paper, the stability of the adaptive fading extended Kalman filter with the matrix forgetting factor when applied to the state estimation problem with noise terms in the non–linear discrete–time stochastic systems has been analysed. The analysis is conducted in a similar manner to the standard extended Kalman filter’s stability analysis based on stochastic framework. The theoretical results show that under certain conditions on the initial estimation error and the noise terms, the estimation error remains bounded and the state estimation is stable.The importance of the theoretical results and the contribution to estimation performance of the adaptation method are demonstrated interactively with the standard extended Kalman filter in the simulation part.

We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=\Psi_n(X_{n-1}^x)$, $X_0^x=x$, where $\Psi _n$ are i.i.d. random continuous transformations of some unbounded closed subset of ${\mathbb{R}}$. We assume here that $\Psi_n$ behaves asymptotically like $A_nx$, for some random positive number $A_n$ [the main example is the affine stochastic recursion $\Psi_n(x)=A_nx+B_n$]. Our aim is to describe invariant Radon measures of the process $X_n^x$ in the critical case, when ${\mathbb{E}}\log A_1=0$. We prove that those measures behave at infinity like $\frac{dx}{x}$. We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which include, for instance, reflected random walks, stochastic dynamical systems on the unit interval $[0,1]$, additive Markov processes and a variant of the Galton--Watson process., Comment: Published at http://dx.doi.org/10.1214/13-AOP903 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study random "perturbation" to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm $1$. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian motion scaled by ${\frac{8}{n(n-1)}}$ where $n$ is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion., Comment: Published at http://dx.doi.org/10.1214/14-AOP981 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Random diffeomorphisms with bounded absolutely continuous noise are knownto possess a finite number of stationary measures. We discuss thedependence of stationary measures on an auxiliary parameter, thusdescribing bifurcations of families of random diffeomorphisms. Abifurcation theory is developed under mild regularity assumptions on thediffeomorphisms and the noise distribution (e.g. smooth diffeomorphismswith uniformly distributed additive noise are included). We distinguishbifurcations where the density function of a stationary measure variesdiscontinuously or where the support of a stationary measure variesdiscontinuously. We establish that generic random diffeomorphisms arestable. The densities of stable stationary measures are shown to be smoothand to depend smoothly on an auxiliary parameter, except at bifurcationvalues. The bifurcation theory explains the occurrence of transients andintermittency as the main bifurcation phenomena in random diffeomorphisms.Quantitative descriptions by means of average escape times from sets asfunctions of the parameter are provided. Further quantitative propertiesare described through the speed of decay of correlations as a function ofthe parameter. Random differentiable maps which are not necessarilyinjective are studied in one dimension; we show that stableone-dimensional random maps occur open and dense and that in one-parameterfamilies bifurcations are typically isolated. We classify codimension-onebifurcations for one-dimensional random maps; we distinguish threepossible kinds, the random saddle node, the random homoclinic and therandom boundary bifurcation. The theory is illustrated on families ofrandom circle diffeomorphisms and random unimodal maps.

We evaluate the scale at which the multifractal structure of some random Gibbs measures becomes discernible. The value of this scale is obtained through what we call the growth speed in H\"older singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar Gibbs measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes.