Intrinsic Information-Theoretic Models.

With this follow-up paper, we continue developing a mathematical framework based on information geometry for representing physical objects. The long-term goal is to lay down informational foundations for physics, especially quantum physics. We assume that we can now model information sources as univariate normal probability distributions N (µ, σ0), as before, but with a constant σ0 not necessarily equal to 1. Then, we also relaxed the independence condition when modeling m sources of information. Now, we model m sources with a multivariate normal probability distribution Nm(µ,Σ0) with a constant variance-covariance matrix Σ0 not necessarily diagonal, i.e., with covariance values different to 0, which leads to the concept of modes rather than sources. Invoking Schrödinger's equation, we can still break the information into m quantum harmonic oscillators, one for each mode, and with energy levels independent of the values of σ0, altogether leading to the concept of "intrinsic". Similarly, as in our previous work with the estimator's variance, we found that the expectation of the quadratic Mahalanobis distance to the sample mean equals the energy levels of the quantum harmonic oscillator, being the minimum quadratic Mahalanobis distance at the minimum energy level of the oscillator and reaching the "intrinsic" Cramér-Rao lower bound at the lowest energy level. Also, we demonstrate that the global probability density function of the collective mode of a set of m quantum harmonic oscillators at the lowest energy level still equals the posterior probability distribution calculated using Bayes' theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. While these new assumptions certainly add complexity to the mathematical framework, the results proven are invariant under transformations, leading to the concept of "intrinsic" information-theoretic models, which are essential for developing physics.

Information-Theoretic Models for Physical Observables.

This work addresses J.A. Wheeler's critical idea that all things physical are information-theoretic in origin. In this paper, we introduce a novel mathematical framework based on information geometry, using the Fisher information metric as a particular Riemannian metric, defined in the parameter space of a smooth statistical manifold of normal probability distributions. Following this approach, we study the stationary states with the time-independent Schrödinger's equation to discover that the information could be represented and distributed over a set of quantum harmonic oscillators, one for each independent source of data, whose coordinate for each oscillator is a parameter of the smooth statistical manifold to estimate. We observe that the estimator's variance equals the energy levels of the quantum harmonic oscillator, proving that the estimator's variance is definitively quantized, being the minimum variance at the minimum energy level of the oscillator. Interestingly, we demonstrate that quantum harmonic oscillators reach the Cramér-Rao lower bound on the estimator's variance at the lowest energy level. In parallel, we find that the global probability density function of the collective mode of a set of quantum harmonic oscillators at the lowest energy level equals the posterior probability distribution calculated using Bayes' theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. Interestingly, the opposite is also true, as the prior is constant. Altogether, these results suggest that we can break the sources of information into little elements: quantum harmonic oscillators, with the square modulus of the collective mode at the lowest energy representing the most likely reality, supporting A. Zeilinger's recent statement that the world is not broken into physical but informational parts.

Multi-site reproducibility of prefrontal-hippocampal connectivity estimates by stochastic DCM.

This study examined the reproducibility of prefrontal-hippocampal connectivity estimates obtained by stochastic dynamic causal modeling (sDCM). 180 healthy subjects were measured by functional magnetic resonance imaging (fMRI) during a standard working memory N-Back task at three different sites (Mannheim, Bonn, Berlin; each with 60 participants). The reproducibility of regional activations in key regions for working memory (dorsolateral prefrontal cortex, DLPFC; hippocampal formation, HF) was evaluated using conjunction analyses across locations. These analyses showed consistent activation of right DLPFC and deactivation of left HF across all three different sites. The effective connectivity between DLPFC and HF was analyzed using a simple two-region sDCM. For each subject, we evaluated sixty-seven alternative sDCMs and compared their relative plausibility using Bayesian model selection (BMS). Across all locations, BMS consistently revealed the same winning model, with the 2-Back working memory condition as driving input to both DLPFC and HF and with a connection from DLPFC to HF. Statistical tests on the sDCM parameter estimates did not show any significant differences across the three sites. The consistency of both the BMS results and model parameter estimates indicates the reliability of sDCM in our paradigm. This provides a basis for future genetic and clinical studies using this approach.