Causal relationships play a fundamental role in understanding the world around us. The ability to identify and understand cause-effect relationships is critical to making informed decisions, predicting outcomes, and developing effective strategies. However, deciphering causal relationships from observational data is a difficult task, as correlations alone may not provide definitive evidence of causality. In recent years, the field of machine learning (ML) has emerged as a powerful tool, offering new opportunities for uncovering hidden causal mechanisms and better understanding complex systems. In this work, we address the issue of detecting the intrinsic causal links of a large class of complex systems in the framework of the response theory in physics. We develop some theoretical ideas put forward by [1], and technically we use state-of-the-art ML techniques to build up models from data. We consider both linear stochastic and non-linear systems. Finally, we compute the asymptotic efficiency of the linear response based causal predictor in a case of large scale Markov process network of linear interactions.

We introduce a novel interpretable Neural Network (NN) model designed to perform precision bulk reconstruction under the AdS/CFT correspondence. According to the correspondence, a specific condensed matter system on a ring is holographically equivalent to a gravitational system on a bulk disk, through which tabletop quantum gravity experiments may be possible as reported in arXiv:2211.13863. The purpose of this paper is to reconstruct a higher-dimensional gravity metric from the condensed matter system data via machine learning using the NN. Our machine reads spatially and temporarily inhomogeneous linear response data of the condensed matter system, and incorporates a novel layer that implements the Runge-Kutta method to achieve better numerical control. We confirm that our machine can let a higher-dimensional gravity metric be automatically emergent as its interpretable weights, using a linear response of the condensed matter system as data, through supervised machine learning. The developed method could serve as a foundation for generic bulk reconstruction, i.e., a practical solution to the AdS/CFT correspondence, and would be implemented in future tabletop quantum gravity experiments.

Belief propagation on cyclic graphs is an efficient algorithm for comput- ing approximate marginal probability distributions over single nodes and neighboring nodes in the graph. In this paper we propose two new al- gorithms for approximating joint probabilities of arbitrary pairs of nodes and prove a number of desirable properties that these estimates fulfill. The first algorithm is a propagation algorithm which is shown to con- verge if belief propagation converges to a stable fixed point. The second algorithm is based on matrix inversion. Experiments compare a number of competing methods.

The receptive fields of simple cells in the visual cortex can be understood as linear filters. These filters can be modelled by Gabor functions, or by Gaussian derivatives. Gabor functions can also be combined in an `energy model' of the complex cell response. This paper proposes an alternative model of the complex cell, based on Gaussian derivatives. It is most important to account for the insensitivity of the complex response to small shifts of the image. The new model uses a linear combination of the first few derivative filters, at a single position, to approximate the first derivative filter, at a series of adjacent positions. The maximum response, over all positions, gives a signal that is insensitive to small shifts of the image. This model, unlike previous approaches, is based on the scale space theory of visual processing. In particular, the complex cell is built from filters that respond to the \twod\ differential structure of the image. The computational aspects of the new model are studied in one and two dimensions, using the steerability of the Gaussian derivatives. The response of the model to basic images, such as edges and gratings, is derived formally. The response to natural images is also evaluated, using statistical measures of shift insensitivity. The relevance of the new model to the cortical image representation is discussed.

Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the variational parameters, and in so doing inference of marginal probability distributions is improved. For the Bethe approximation and its generalizations, improvements are achieved with simple choices of the constraints. The approximations are presented as variational frameworks; iterative procedures related to message passing are provided for finding the minima.

The first algorithm is a propagation algorithm which is shown to converge if belief propagation converges to a stable fixed point.

Belief propagation on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph. In this paper we propose two new algorithms for approximating joint probabilities of arbitrary pairs of nodes and prove a number of desirable properties that these estimates fulfill. The first algorithm is a propagation algorithm which is shown to converge if belief propagation converges to a stable fixed point. The second algorithm is based on matrix inversion. Experiments compare a number of competing methods.

Belief propagation on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph. In this paper we propose two new algorithms for approximating joint probabilities of arbitrary pairs of nodes and prove a number of desirable properties that these estimates fulfill. The first algorithm is a propagation algorithm which is shown to converge if belief propagation converges to a stable fixed point. The second algorithm is based on matrix inversion. Experiments compare a number of competing methods.

A general linear response method for deriving improved estimates of correlations in the variational Bayes framework is presented. Three applications are given and it is discussed how to use linear response as a general principle for improving mean field approximations.

A general linear response method for deriving improved estimates of correlations in the variational Bayes framework is presented. Three applications are given and it is discussed how to use linear response as a general principle for improving mean field approximations.