A variety of questions in causal inference can be represented as probability distributions over hypothetical worlds where idealized randomized experiments known as interventions have taken place. Some such questions are really questions of causal effect of a particular intervention, while others are counterfactual and consider results of interventions which violate the state of affairs actually observed. Randomized experiments are expensive and often illegal. It is therefore imperative to find ways of evaluating, or identifying causal effect and counterfactual questions from available information, and causal assumptions. In this paper, we review the state of the art in identification of causal effects and related counterfactual quantities in the framework of graphical causal models, a formalism where a causal domain of interest is represented by directed acyclic graphs with vertices representing variables of interest, and arrows representing direct causal influences

We introduce a framework for estimating causal effects of binary and continuous treatments in high dimensions. We show how posterior distributions of treatment and outcome models can be used together with doubly robust estimators. We propose an approach to uncertainty quantification for the doubly robust estimator, which utilizes posterior distributions of model parameters and (1) results in good frequentist properties in small samples, (2) is based on a single run of a Markov chain Monte Carlo (MCMC) algorithm, and (3) improves over frequentist measures of uncertainty which rely on asymptotic properties. We consider a flexible framework for modeling the treatment and outcome processes within the Bayesian paradigm that reduces model dependence, accommodates nonlinearity, and achieves dimension reduction of the covariate space. We illustrate the ability of the proposed approach to flexibly estimate causal effects in high dimensions and appropriately quantify uncertainty. We show that our proposed variance estimation strategy is consistent when both models are correctly specified, and we see empirically that it performs well in finite samples and under model misspecification. Finally, we estimate the effect of continuous environmental exposures on cholesterol and triglyceride levels.

Intentional binding refers to the subjective compression of the time interval between an action and its consequence. While intentional binding has been widely used as a proxy for the sense of agency, its underlying mechanism has been largely veiled. Bayesian causal inference (BCI) has gained attention as a potential explanation, but currently lacks sufficient empirical support. Thus, this study implemented various computational models to describe the possible mechanisms of intentional binding, fitted them to individual observed data, and quantitatively evaluated their performance. The BCI models successfully isolated the parameters that potentially contributed to intentional binding (i.e., causal belief and temporal prediction) and generally better explained an observer’s time estimation than traditional models such as maximum likelihood estimation. The estimated parameter values suggested that the time compression resulted from an expectation that the actions would immediately cause sensory outcomes. Furthermore, I investigated the algorithm that realized this BCI and found probability-matching to be a plausible candidate; people might heuristically reconstruct event timing depending on causal uncertainty rather than optimally integrating causal and temporal posteriors. The evidence demonstrated the utility of computational modeling to investigate how humans infer the causal and temporal structures of events and individual differences in that process.

This tutorial provides a first general introduction to causal inference with the do-calculus, as introduced in the work of Judea Pearl and a hands-on programming example in R for an artificial data set instantiating Simpson's paradox, showcasing the use of Bayesian regression modeling to obtain causal effect estimates with quantified uncertainty.

Discrete Bayesian Networks (BNs) have been very successful as a framework both for inference and for expressing certain causal hypotheses. In this paper we present a class of graphical models called the chain event graph (CEG) models, that generalises the class of discrete BN models. This class is suited for representing conditional independence and sample space structures of asymmetric models. It retains many useful properties of discrete BNs, in particular admitting conjugate estimation. It provides a flexible and expressive framework for representing and analysing the implications of causal hypotheses, expressed in terms of the effects of a manipulation of the generating underlying system.We prove that, as for a BN, identifiability analyses of causal effects can be performed through examining the topology of the CEG graph, leading to theorems analogous to the Backdoor theorem for the BN

Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of that model. This “best model” approach is very useful but can become brittle if there are a large number of models to compare, and if different subjects use different models. To overcome this shortcoming we propose the combination of two further approaches: (i) family level inference and (ii) Bayesian model averaging within families. Family level inference removes uncertainty about aspects of model structure other than the characteristic of interest. For example: What are the inputs to the system? Is processing serial or parallel? Is it linear or nonlinear? Is it mediated by a single, crucial connection? We apply Bayesian model averaging within families to provide inferences about parameters that are independent of further assumptions about model structure. We illustrate the methods using Dynamic Causal Models of brain imaging data

Explanations in Bayesian networks are usually probabilistic measures of how well a hypothesis is supported by observations. This observational based approach does not fulfill all the properties one would expect from an explanation. In particular, it does not include any notion of causality. In this thesis, we investigate how causality can be used to allow for an interpretation of explanations as statements of cause and effect, between a hypothesis and observations. We detail how factor graphs can be used as an abstraction of Bayesian networks to perform efficient probabilistic inference in Bayesian network with the aim of using these methods to find explanations. We provide a critique of existing approaches to explanation, and list the requirements for explanations that consider cause-effect relationships. We propose an algorithm for finding explanations that allows for causal interpretation as well as probabilistic. The algorithm is based on the concept of causal information flow and the search for explanations is inspired by feature selection. The proposed approach is evaluated on two sample networks. The results are compared with those of existing approaches and we conclude that our proposed algorithm finds explanations that satisfy the requirements we defined for causal explanations

Pearl (2000), Spirtes et al (1993) and Lauritzen (2001) set up a new framework to encode the causal relationships between the random variables by a causal Bayesian network. The estimation of the conditional probabilities in a Bayesian network has received considerable attention by several investigators (e. g., Jordan (1998), Geiger and Heckerman (1997), Ileckerman et al (1995)), but, this issue has not been studied in a causal Bayesian network. In this thesis, we define the multicausal essential graph on the equivalence class of Bayesian networks in which each member of this class manifests a sort of strong type of invariance under (causal) manipulation called hypercausality. We then characterise the families of prior distributions on the parameters of the Bayesian networks which are consistent with hypercausality and show that their unmanipulated uncertain Bayesian networks must demonstrate the independence assumptions. As a result, such prior distributions satisfy a generalisation of the Geiger and lieckerman condition. In particular, when the corresponding essential graph is undirected, the mentioned class of prior distributions will reduce to the Hyper-Dirichlet family (see Chapter 6). In tile second part of this thesis, we will calculate certain local sensitivity measures and through them we are able to provide the solutions for the following questions: Is the network structure that is learned from data robust with respect to changes of the directionality of some specific arrows? Is the local conditional distributions associated with the specified node robust with respect to the changes to its prior distribution or with respect to the changes to the local conditional distribution of another node? Most importantly, is the posterior distribution associated with the parameters of any node robust with respect to the changes to the prior distribution associated with the parameters of one specific node? Finally, are the quantities mentioned above robust with respect to the changes in the independence assumptions described in Chapter 3? Most of the local sensitivity measures (particularly, local measures of the overall posteriors sensitivity), developed in the last decade, tend to diverge to infinity as the sample size becomes very large (Gustafson (1994) and Gustafson et al (1996)). This is in contrast to our knowledge that, starting from different priors, posteriors tend to agree as the data accumulate. Here we define a now class of metrics with more satisfactory asymptotic behaviour. The advantage of the corresponding local sensitivity measures is boundedness for large sample size

Bayesian models of human learning are becoming increasingly popular in cognitive science. We argue that their purported confirmation largely relies on a methodology that depends on premises that are inconsistent with the claim that people are Bayesian about learning and inference. Bayesian models in cognitive science derive their appeal from their normative claim that the modeled inference is in some sense rational. Standard accounts of the rationality of Bayesian inference imply predictions that an agent selects the option that maximizes the posterior expected utility. Experimental confirmation of the models, however, has been claimed because of groups of agents that “probability match” the posterior. Probability matching only constitutes support for the Bayesian claim if additional unobvious and untested (but testable) assumptions are invoked. The alternative strategy of weakening the underlying notion of rationality no longer distinguishes the Bayesian model uniquely. A new account of rationality—either for inference or for decision-making—is required to successfully confirm Bayesian models in cognitive science

Abstract Causal inference in observational studies with high-dimensional covariates presents significant challenges. We introduce CausalBGM, an AI-powered Bayesian generative modeling approach that captures the causal relationship among covariates, treatment, and outcome. The core innovation is to estimate the individual treatment effect (ITE) by learning the individual-specific distribution of a low-dimensional latent feature set (e.g., latent confounders) that drives changes in both treatment and outcome. This individualized posterior representation yields estimates of the individual treatment effect (ITE) together with well-calibrated posterior intervals while mitigating confounding effect. CausalBGM is fitted through an iterative algorithm to update the model parameters and the latent features until convergence. This framework leverages the power of AI to capture complex dependencies among variables while adhering to the Bayesian principles. Extensive experiments demonstrate that CausalBGM achieves superior or competitive performance compared to state-of-the-art methods, particularly in scenarios with high-dimensional covariates and large-scale datasets. By addressing key limitations of existing methods, CausalBGM emerges as a robust and promising framework for advancing causal inference in a wide range of modern applications. The code for CausalBGM is available at https://github.com/liuq-lab/bayesgm. The document for CausalBGM is available at https://bayesgm.readthedocs.io. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Bayesian causal inference in randomized experiments usually imposes model-based structure on potential outcomes. Yet causal inferences from randomized experiments are especially credible because they depend on a known assignment process, not a probability model of potential outcomes. In this article, I derive a randomization-based procedure for Bayesian inference of causal effects in a finite population setting. I formally show that this procedure satisfies Bayesian analogues of unbiasedness and consistency under weak conditions on a prior distribution. Unlike existing model-based methods of Bayesian causal inference, my procedure supposes neither probability models that generate potential outcomes nor independent and identically distributed random sampling. Unlike existing randomization-based methods of Bayesian causal inference, my procedure does not suppose that potential outcomes are discrete and bounded. Consequently, researchers can reap the benefits of Bayesian inference without sacrificing the properties that make inferences from randomized experiments especially credible in the first place.

The perception of the external world relies on integrating information from multiple sensory modalities. To do this effectively, the brain must determine whether sensory signals come from a common source and, if so, combine them to reduce perceptual uncertainty. While Bayesian observer models have been successful in accounting for multisensory causal inference decisions by humans, they typically rely on simplifying assumptions that may not reflect the true complexity of human perception. In this study, we challenge two assumptions common in Bayesian multisensory perception models: homoskedastic (constant across space) sensory noise and Gaussian priors. We collected an auditory-visual perceptual dataset featuring both unisensory and bisensory tasks, where participants must either provide stimulus location estimates or same-different source judgments. Subsequently, we developed a flexible semiparametric approach that allowed us to infer the sensory noise and prior shapes from participants' data, and subsequently `distill' them into new model classes through visual inspection of the semiparametrically fitted function shapes. We find that human multisensory perception is best described by an eccentricity-dependent sensory noise that plateaus in the periphery and a prior distribution with a narrow central peak and smoother tails. We also found evidence for auditory range recalibration and increased sensory noise in multisensory conditions, suggesting complex interactions between sensory modalities. These findings deviate substantially from traditional modeling assumptions and highlight the value of making Bayesian cognitive models more data-driven. Overall, our study demonstrates the value of systematically exploring model assumptions in multisensory research and provides a new set of modeling tools for perceptual causal inference

This study proposes a set of AI-driven frameworks for conducting visual modeling and probabilistic simulation of strategic pathways in cross-strait unification scenarios focusing on the Taiwan Strait issue. The model incorporates Bayesian networks to capture causal dependencies among key variables such as economic integration, political reconfiguration, military deterrence, and international intervention; employs Markov chains to simulate the evolution of long-term social opinion in Taiwan; and evaluates optimal unification strategies under multiple scenarios via a decision tree algorithm. The framework also introduces system dynamics techniques and advanced visualization tools to enhance the interpretability and decision support functions of the model.The quantitative simulation results show that economic dependence is the core driving force for peaceful unification. When Taiwan's economic dependence on China exceeds 60%, the probability of peaceful unification rises to over 70% within 20 years. Sustained military deterrence significantly reduces the likelihood of direct external intervention, and the probability of successful military unification is significantly higher in a scenario where the U.S. and Japan's military response is limited. The study also finds that a shift in the island's political ecology is crucial: the dominance of pro-continental parties increases the probability of reunification by more than 30%.In terms of short-term unification strategies (within 5 years), simulation results show that the probability of successful unification can be over 85% if high-intensity military pressure, economic blockade and political polarization are adopted, and the U.S. and Japan's military intervention is effectively blocked. However, such strategies come with significant geopolitical risks. To reduce the risk, the model suggests supplementing the tactics with strategic deception, diplomatic mediation, psychological warfare, and precision military operations.This study proposes a transferable modeling approach for simulating complex geopolitical scenarios, combining probabilistic insights with dynamic visualization analysis, to provide scientific support for strategic planning and policymaking.

Average human behavior in cue combination tasks is well predicted by Bayesian inference models. As this capability is acquired over developmental timescales, the question arises, how it is learned. Here we investigated whether reward dependent learning, that is well established at the computational, behavioral, and neuronal levels, could contribute to this development. It is shown that a model free reinforcement learning algorithm can indeed learn to do cue integration, i.e. weight uncertain cues according to their respective reliabilities and even do so if reliabilities are changing. We also consider the case of causal inference where multimodal signals can originate from one or multiple separate objects and should not always be integrated. In this case, the learner is shown to develop a behavior that is closest to Bayesian model averaging. We conclude that reward mediated learning could be a driving force for the development of cue integration and causal inference

Bayesian networks (BNs) are useful for coding conditional independence statements between a given set of measurement variables. On the other hand, event trees (ETs) are convenient for representing asymmetric structure and how situations unfold. In this paper we report the development of a new graphical framework for discrete probability models called the Chain Event Graph (CEG). The class of CEG models contains finite BNs as a special case. Unlike the BN, the CEG is equally appropriate for representing conditional independencies in asymmetric systems and does not need dependent variables to be specified in advance. As with the BN, it also provides a framework for learning relevant conditional probabilities and propagation. Furthermore, being a function of an ET, the CEG is a more exible way of representing various causal hypotheses than the BN. This new framework is illustrated throughout by a biological regulatory network: the tryptophan metabolic pathway in the bacterium E. coli