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We evaluate the information geometric complexity of entropic motion on low-dimensional Gaussian statistical manifolds in order to quantify how difficult it is to make macroscopic predictions about systems in the presence of limited information. Specifically, we observe that the complexity of such entropic inferences not only depends on the amount of available pieces of information but also on the manner in which such pieces are correlated. Finally, we uncover that, for certain correlational structures, the impossibility of reaching the most favorable configuration from an entropic inference viewpoint seems to lead to an information geometric analog of the well-known frustration effect that occurs in statistical physics.

One of the main efforts in physics is modeling and predicting natural phenomena using relevant information about the system under consideration. Theoretical physics has had a general measure of the uncertainty associated with the behavior of a probabilistic process for more than 100 years: the Shannon entropy [

Along a similar avenue we may set Entropic Dynamics [

This quantity might help uncover connections between microscopic dynamics and experimentally observable macroscopic dynamics which is a fundamental issue in physics [

The investigation of the macroscopic behavior of complex systems in terms of the underlying statistical structure of its microscopic degrees of freedom also reveals effects due to the presence of microcorrelations [

The layout of the article is as follows. In

Given

The mapping

The values

Let

Given the Riemannian manifold

The idea of curvature is the fundamental tool to understand the geometry of the manifold

The recipe to compute some curvatures at a point

The sectional curvature is directly related to the topology of the manifold; along this direction the

We can consider upon the statistical manifold

Taking into consideration these facts, we single out as suitable indicator of dynamical (temporal) complexity, the information geometric complexity defined as the average dynamical statistical volume [

Relevant properties, concerning complexity of geodesic paths on curved statistical manifolds, of the quantity (

In the following we devote our attention to a Gaussian statistical model

We may further notice that the function

We now start to apply the concepts of the previous section to a Gaussian statistical model of Equation (

Consider the monovariate model with only

Consider now the monovariate Gaussian statistical model of Equation(

The take home message of the previous cases is that we have to account for both mean

Hence, comparing Equations (

Consider now the Gaussian statistical model

Thanks to Equation (

While the temporal behavior of the IGC (

Intuitively, this is due to the fact that for anticorrelated variables, an increase in one variable implies a decrease in the other one (different directional change): variables become more distant, thus more distinguishable in the Fisher–Rao information metric sense. Similarly, for positively correlated variables, an increase or decrease in one variable always predicts the same directional change for the second variable: variables do not become more distant, thus more distinguishable in the Fisher–Rao information metric sense. This may lead us to guess that in the presence of anticorrelations, motion on curved statistical manifolds via the Maximum Entropy updating methods becomes less complex.

In this section we consider a Gaussian statistical model

The manifold

First, we consider the trivariate Gaussian statistical model of Equation (

Comparing (

Notice that

When the trivariate Gaussian statistical model of Equation (

Then, comparing (

This is a function of the argument

Last, we consider the trivariate Gaussian statistical model of the Equation (

The comparison of (

The behaviors of

Ratio

The

Specifically, carrying out entropic inferences on the higher-dimensional manifold in the presence of anti-correlations, that is for

In summary, we considered low dimensional Gaussian statistical models (up to a trivariate model) and have investigated their dynamical (temporal) complexity. This has been quantified by the volume of geodesics for parameters characterizing the probability distribution functions. To the best of our knowledge, there is no

For the sake of completeness, we also point out that the relevance of volumes in quantifying the

In this work we uncover that in order to have a reduction in time of the complexity one has to consider both mean and variance as macro-variables. This leads to different topological structures of the parameter space in (

Specifically, recall that a geometrically frustrated system cannot simultaneously minimize all interactions because of geometric constraints [

In this article, we observe a significant change in the behavior of the information geometric complexity ratios as a function of the correlation coefficient in the trivariate Gaussian statistical models. Specifically, in the fully connected trivariate case, no peak arises and a monotonic behavior in

These considerations lead us to conclude that we have uncovered a very interesting information geometric resemblance of the more standard geometric frustration effect in Ising spin models. However, for a conclusive claim of the existence of an information geometric analog of the frustration effect, we feel we have to further deepen our understanding. A forthcoming research project along these lines will be a detailed investigation of both arbitrary triangular and square configurations of correlated Gaussian random variables where we take into consideration both the presence of different intensities and signs of pairwise interactions (

Domenico Felice and Stefano Mancini acknowledge the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under the FET-Open grant agreement TOPDRIM, number FP7-ICT-318121.

The authors have equally contributed to the paper. All authors read and approved the final manuscript.

The authors declare no conflict of interest.