T. A. Enßlin

Gordian Edenhofer, Reimar H. Leike, Philipp Frank et al.

Gaussian Processes (GPs) are highly expressive, probabilistic models. A major limitation is their computational complexity. Naively, exact GP inference requires $\mathcal{O}(N^3)$ computations with $N$ denoting the number of modeled points. Current approaches to overcome this limitation either rely on sparse, structured or stochastic representations of data or kernel respectively and usually involve nested optimizations to evaluate a GP. We present a new, generative method named Iterative Charted Refinement (ICR) to model GPs on nearly arbitrarily spaced points in $\mathcal{O}(N)$ time for decaying kernels without nested optimizations. ICR represents long- as well as short-range correlations by combining views of the modeled locations at varying resolutions with a user-provided coordinate chart. In our experiment with points whose spacings vary over two orders of magnitude, ICR's accuracy is comparable to state-of-the-art GP methods. ICR outperforms existing methods in terms of computational speed by one order of magnitude on the CPU and GPU and has already been successfully applied to model a GP with $122$ billion parameters.

Torsten Enßlin, Carolin Weidinger, Philipp Frank

The concept of attention, numerical weights that emphasize the importance of particular data, has proven to be very relevant in artificial intelligence. Relative entropy (RE, aka Kullback-Leibler divergence) plays a central role in communication theory. Here we combine these concepts, attention and RE. RE guides optimal encoding of messages in bandwidth-limited communication as well as optimal message decoding via the maximum entropy principle (MEP). In the coding scenario, RE can be derived from four requirements, namely being analytical, local, proper, and calibrated. Weighted RE, used for attention steering in communications, turns out to be improper. To see how proper attention communication can emerge, we analyze a scenario of a message sender who wants to ensure that the receiver of the message can perform well-informed actions. If the receiver decodes the message using the MEP, the sender only needs to know the receiver's utility function to inform optimally, but not the receiver's initial knowledge state. In case only the curvature of the utility function maxima are known, it becomes desirable to accurately communicate an attention function, in this case a by this curvature weighted and re-normalized probability function. Entropic attention communication is here proposed as the desired generalization of entropic communication that permits weighting while being proper, thereby aiding the design of optimal communication protocols in technical applications and helping to understand human communication. For example, our analysis shows how to derive the level of cooperation expected under misaligned interests of otherwise honest communication partners.

Gordian Edenhofer, Philipp Frank, Jakob Roth et al.

Imaging is the process of transforming noisy, incomplete data into a space that humans can interpret. NIFTy is a Bayesian framework for imaging and has already successfully been applied to many fields in astrophysics. Previous design decisions held the performance and the development of methods in NIFTy back. We present a rewrite of NIFTy, coined NIFTy.re, which reworks the modeling principle, extends the inference strategies, and outsources much of the heavy lifting to JAX. The rewrite dramatically accelerates models written in NIFTy, lays the foundation for new types of inference machineries, improves maintainability, and enables interoperability between NIFTy and the JAX machine learning ecosystem.

Torsten Enßlin

Information field theory (IFT), the information theory for fields, is a mathematical framework for signal reconstruction and non-parametric inverse problems. Artificial intelligence (AI) and machine learning (ML) aim at generating intelligent systems including such for perception, cognition, and learning. This overlaps with IFT, which is designed to address perception, reasoning, and inference tasks. Here, the relation between concepts and tools in IFT and those in AI and ML research are discussed. In the context of IFT, fields denote physical quantities that change continuously as a function of space (and time) and information theory refers to Bayesian probabilistic logic equipped with the associated entropic information measures. Reconstructing a signal with IFT is a computational problem similar to training a generative neural network (GNN) in ML. In this paper, the process of inference in IFT is reformulated in terms of GNN training. In contrast to classical neural networks, IFT based GNNs can operate without pre-training thanks to incorporating expert knowledge into their architecture. Furthermore, the cross-fertilization of variational inference methods used in IFT and ML are discussed. These discussions suggests that IFT is well suited to address many problems in AI and ML research and application.

Johannes Zacherl, Philipp Frank, Torsten A. Enßlin

Neural Networks play a growing role in many science disciplines, including physics. Variational Autoencoders (VAEs) are neural networks that are able to represent the essential information of a high dimensional data set in a low dimensional latent space, which have a probabilistic interpretation. In particular the so-called encoder network, the first part of the VAE, which maps its input onto a position in latent space, additionally provides uncertainty information in terms of a variance around this position. In this work, an extension to the Autoencoder architecture is introduced, the FisherNet. In this architecture, the latent space uncertainty is not generated using an additional information channel in the encoder, but derived from the decoder, by means of the Fisher information metric. This architecture has advantages from a theoretical point of view as it provides a direct uncertainty quantification derived from the model, and also accounts for uncertainty cross-correlations. We can show experimentally that the FisherNet produces more accurate data reconstructions than a comparable VAE and its learning performance also apparently scales better with the number of latent space dimensions.

Torsten Enßlin, Viktoria Kainz, Céline Bœhm

Reputation is a central element of social communications, be it with human or artificial intelligence (AI), and as such can be the primary target of malicious communication strategies. There is already a vast amount of literature on trust networks addressing this issue and proposing ways to simulate these networks dynamics using Bayesian principles and involving Theory of Mind models. The main issue for these simulations is usually the amount of information that can be stored and is usually solved by discretising variables and using hard thresholds. Here we propose a novel approach to the way information is updated that accounts for knowledge uncertainty and is closer to reality. In our game, agents use information compression techniques to capture their complex environment and store it in their finite memories. The loss of information that results from this leads to emergent phenomena, such as echo chambers, self-deception, deception symbiosis, and freezing of group opinions. Various malicious strategies of agents are studied for their impact on group sociology, like sycophancy, egocentricity, pathological lying, and aggressiveness. Even though our modeling could be made more complex, our set-up can already provide insights into social interactions and can be used to investigate the effects of various communication strategies and find ways to counteract malicious ones. Eventually this work should help to safeguard the design of non-abusive AI systems.

Matteo Guardiani, Philipp Frank, Andrija Kostić et al.

The viral load of patients infected with SARS-CoV-2 varies on logarithmic scales and possibly with age. Controversial claims have been made in the literature regarding whether the viral load distribution actually depends on the age of the patients. Such a dependence would have implications for the COVID-19 spreading mechanism, the age-dependent immune system reaction, and thus for policymaking. We hereby develop a method to analyze viral-load distribution data as a function of the patients' age within a flexible, non-parametric, hierarchical, Bayesian, and causal model. The causal nature of the developed reconstruction additionally allows to test for bias in the data. This could be due to, e.g., bias in patient-testing and data collection or systematic errors in the measurement of the viral load. We perform these tests by calculating the Bayesian evidence for each implied possible causal direction. The possibility of testing for bias in data collection and identifying causal directions can be very useful in other contexts as well. For this reason we make our model freely available. When applied to publicly available age and SARS-CoV-2 viral load data, we find a statistically significant increase in the viral load with age, but only for one of the two analyzed datasets. If we consider this dataset, and based on the current understanding of viral load's impact on patients' infectivity, we expect a non-negligible difference in the infectivity of different age groups. This difference is nonetheless too small to justify considering any age group as noninfectious.

Jakob Knollmüller, Torsten Enßlin

We showed how to use trained neural networks to perform Bayesian reasoning in order to solve tasks outside their initial scope. Deep generative models provide prior knowledge, and classification/regression networks impose constraints. The tasks at hand were formulated as Bayesian inference problems, which we approximately solved through variational or sampling techniques. The approach built on top of already trained networks, and the addressable questions grew super-exponentially with the number of available networks. In its simplest form, the approach yielded conditional generative models. However, multiple simultaneous constraints constitute elaborate questions. We compared the approach to specifically trained generators, showed how to solve riddles, and demonstrated its compatibility with state-of-the-art architectures.

Jakob Knollmüller, Torsten A. Enßlin

Solving Bayesian inference problems approximately with variational approaches can provide fast and accurate results. Capturing correlation within the approximation requires an explicit parametrization. This intrinsically limits this approach to either moderately dimensional problems, or requiring the strongly simplifying mean-field approach. We propose Metric Gaussian Variational Inference (MGVI) as a method that goes beyond mean-field. Here correlations between all model parameters are taken into account, while still scaling linearly in computational time and memory. With this method we achieve higher accuracy and in many cases a significant speedup compared to traditional methods. MGVI is an iterative method that performs a series of Gaussian approximations to the posterior. We alternate between approximating the covariance with the inverse Fisher information metric evaluated at an intermediate mean estimate and optimizing the KL-divergence for the given covariance with respect to the mean. This procedure is iterated until the uncertainty estimate is self-consistent with the mean parameter. We achieve linear scaling by avoiding to store the covariance explicitly at any time. Instead we draw samples from the approximating distribution relying on an implicit representation and numerical schemes to approximately solve linear equations. Those samples are used to approximate the KL-divergence and its gradient. The usage of natural gradient descent allows for rapid convergence. Formulating the Bayesian model in standardized coordinates makes MGVI applicable to any inference problem with continuous parameters. We demonstrate the high accuracy of MGVI by comparing it to HMC and its fast convergence relative to other established methods in several examples. We investigate real-data applications, as well as synthetic examples of varying size and complexity and up to a million model parameters.

Maximilian Kurthen, Torsten A. Enßlin

We address the problem of two-variable causal inference without intervention. This task is to infer an existing causal relation between two random variables, i.e. $X \rightarrow Y$ or $Y \rightarrow X$ , from purely observational data. As the option to modify a potential cause is not given in many situations only structural properties of the data can be used to solve this ill-posed problem. We briefly review a number of state-of-the-art methods for this, including very recent ones. A novel inference method is introduced, Bayesian Causal Inference (BCI), which assumes a generative Bayesian hierarchical model to pursue the strategy of Bayesian model selection. In the adopted model the distribution of the cause variable is given by a Poisson lognormal distribution, which allows to explicitly regard the discrete nature of datasets, correlations in the parameter spaces, as well as the variance of probability densities on logarithmic scales. We assume Fourier diagonal Field covariance operators. The model itself is restricted to use cases where a direct causal relation $X \rightarrow Y$ has to be decided against a relation $Y \rightarrow X$ , therefore we compare it other methods for this exact problem setting. The generative model assumed provides synthetic causal data for benchmarking our model in comparison to existing State-of-the-art models, namely LiNGAM , ANM-HSIC , ANM-MML , IGCI and CGNN . We explore how well the above methods perform in case of high noise settings, strongly discretized data and very sparse data. BCI performs generally reliable with synthetic data as well as with the real world TCEP benchmark set, with an accuracy comparable to state-of-the-art algorithms. We discuss directions for the future development of BCI .

Johannes Oberpriller, T. A. Enßlin

Fitting a simplifying model with several parameters to real data of complex objects is a highly nontrivial task, but enables the possibility to get insights into the objects physics. Here, we present a method to infer the parameters of the model, the model error as well as the statistics of the model error. This method relies on the usage of many data sets in a simultaneous analysis in order to overcome the problems caused by the degeneracy between model parameters and model error. Errors in the modeling of the measurement instrument can be absorbed in the model error allowing for applications with complex instruments.

Jakob Knollmüller, Torsten A. Enßlin

The inference of deep hierarchical models is problematic due to strong dependencies between the hierarchies. We investigate a specific transformation of the model parameters based on the multivariate distributional transform. This transformation is a special form of the reparametrization trick, flattens the hierarchy and leads to a standard Gaussian prior on all resulting parameters. The transformation also transfers all the prior information into the structure of the likelihood, hereby decoupling the transformed parameters a priori from each other. A variational Gaussian approximation in this standardized space will be excellent in situations of relatively uninformative data. Additionally, the curvature of the log-posterior is well-conditioned in directions that are weakly constrained by the data, allowing for fast inference in such a scenario. In an example we perform the transformation explicitly for Gaussian process regression with a priori unknown correlation structure. Deep models are inferred rapidly in highly and slowly in poorly informed situations. The flat model show exactly the opposite performance pattern. A synthesis of both, the deep and the flat perspective, provides their combined advantages and overcomes the individual limitations, leading to a faster inference.

Torsten A. Enßlin, Jakob Knollmüller

The inference of correlated signal fields with unknown correlation structures is of high scientific and technological relevance, but poses significant conceptual and numerical challenges. To address these, we develop the correlated signal inference (CSI) algorithm within information field theory (IFT) and discuss its numerical implementation. To this end, we introduce the free energy exploration (FrEE) strategy for numerical information field theory (NIFTy) applications. The FrEE strategy is to let the mathematical structure of the inference problem determine the dynamics of the numerical solver. FrEE uses the Gibbs free energy formalism for all involved unknown fields and correlation structures without marginalization of nuisance quantities. It thereby avoids the complexity marginalization often impose to IFT equations. FrEE simultaneously solves for the mean and the uncertainties of signal, nuisance, and auxiliary fields, while exploiting any analytically calculable quantity. Finally, FrEE uses a problem specific and self-tuning exploration strategy to swiftly identify the optimal field estimates as well as their uncertainty maps. For all estimated fields, properly weighted posterior samples drawn from their exact, fully non-Gaussian distributions can be generated. Here, we develop the FrEE strategies for the CSI of a normal, a log-normal, and a Poisson log-normal IFT signal inference problem and demonstrate their performances via their NIFTy implementations.

Reimar H. Leike, Torsten A. Enßlin

In Bayesian statistics probability distributions express beliefs. However, for many problems the beliefs cannot be computed analytically and approximations of beliefs are needed. We seek a loss function that quantifies how "embarrassing" it is to communicate a given approximation. We reproduce and discuss an old proof showing that there is only one ranking under the requirements that (1) the best ranked approximation is the non-approximated belief and (2) that the ranking judges approximations only by their predictions for actual outcomes. The loss function that is obtained in the derivation is equal to the Kullback-Leibler divergence when normalized. This loss function is frequently used in the literature. However, there seems to be confusion about the correct order in which its functional arguments, the approximated and non-approximated beliefs, should be used. The correct order ensures that the recipient of a communication is only deprived of the minimal amount of information. We hope that the elementary derivation settles the apparent confusion. For example when approximating beliefs with Gaussian distributions the optimal approximation is given by moment matching. This is in contrast to many suggested computational schemes.

Sebastian Dorn, Torsten A. Enßlin, Maksim Greiner et al.

The calibration of a measurement device is crucial for every scientific experiment, where a signal has to be inferred from data. We present CURE, the calibration uncertainty renormalized estimator, to reconstruct a signal and simultaneously the instrument's calibration from the same data without knowing the exact calibration, but its covariance structure. The idea of CURE, developed in the framework of information field theory, is starting with an assumed calibration to successively include more and more portions of calibration uncertainty into the signal inference equations and to absorb the resulting corrections into renormalized signal (and calibration) solutions. Thereby, the signal inference and calibration problem turns into solving a single system of ordinary differential equations and can be identified with common resummation techniques used in field theories. We verify CURE by applying it to a simplistic toy example and compare it against existent self-calibration schemes, Wiener filter solutions, and Markov Chain Monte Carlo sampling. We conclude that the method is able to keep up in accuracy with the best self-calibration methods and serves as a non-iterative alternative to it.

Torsten A. Enßlin, Henrik Junklewitz, Lars Winderling et al.

Response calibration is the process of inferring how much the measured data depend on the signal one is interested in. It is essential for any quantitative signal estimation on the basis of the data. Here, we investigate self-calibration methods for linear signal measurements and linear dependence of the response on the calibration parameters. The common practice is to augment an external calibration solution using a known reference signal with an internal calibration on the unknown measurement signal itself. Contemporary self-calibration schemes try to find a self-consistent solution for signal and calibration by exploiting redundancies in the measurements. This can be understood in terms of maximizing the joint probability of signal and calibration. However, the full uncertainty structure of this joint probability around its maximum is thereby not taken into account by these schemes. Therefore better schemes -- in sense of minimal square error -- can be designed by accounting for asymmetries in the uncertainty of signal and calibration. We argue that at least a systematic correction of the common self-calibration scheme should be applied in many measurement situations in order to properly treat uncertainties of the signal on which one calibrates. Otherwise the calibration solutions suffer from a systematic bias, which consequently distorts the signal reconstruction. Furthermore, we argue that non-parametric, signal-to-noise filtered calibration should provide more accurate reconstructions than the common bin averages and provide a new, improved self-calibration scheme. We illustrate our findings with a simplistic numerical example.