Posterior Probability Estimates Research Articles

A petrophysically driven seismic inversion method for the total organic carbon content of source rocks

Organic carbon content is the main index for evaluating organic matter abundance of source rocks, and it is still difficult to quantitatively predict total organic carbon (TOC) in source rocks based on seismic data. The study of seismic fluid identification driven by petrophysics can help to understand the fluid characteristics and distribution patterns of subsurface oil and gas reservoirs. First, this paper clarifies the sweet spot parameters (parameters characterizing hydrocarbon enrichment) and sensitive elastic parameters (a parameter characterizing the nature of an ideal elastic body) of source rocks through theoretical petrophysical modeling. Next, the paper establishes the relationship between sensitive elastic parameters and sweet spot parameters TOC and constructs a statistical petrophysical model that can characterize the relationship between the two on this basis. And then we construct the joint distribution of TOC and elastic impedance through the Bayesian theoretical framework to obtain the maximum posterior probability estimate as the final TOC inversion results of source rocks. Our method successfully predicts the spreading of high-quality source rocks in the Wen4 Section of Lufeng 13 Subsag, and the inversion results are within an uncertainty range of ±14 m for well data, which proves the reliability of the method. The prediction results show that the organic matter abundance of source rocks in the Wen4 Section is high, and the organic carbon content is generally higher than 2%, which provides a reliable basis for the further implementation of the resource scale of the depression and the clarification of the hydrocarbon-rich area, which provides technical support for the evaluation of the source rocks of the new depression in the new area.

Reducing semantic ambiguity in domain adaptive semantic segmentation via probabilistic prototypical pixel contrast

Domain adaptation aims to reduce the model degradation on the target domain caused by the domain shift between the source and target domains. Although encouraging performance has been achieved by combining contrastive learning with the self-training paradigm, they suffer from ambiguous scenarios caused by scale, illumination, or overlapping when deploying deterministic embedding. To address these issues, we propose probabilistic prototypical pixel contrast (PPPC), a universal adaptation framework that models each pixel embedding as a probability via multivariate Gaussian distribution to fully exploit the uncertainty within them, eventually improving the representation quality of the model. In addition, we derive prototypes from probability estimation posterior probability estimation which helps to push the decision boundary away from the ambiguity points. Moreover, we employ an efficient method to compute similarity between distributions, eliminating the need for sampling and reparameterization, thereby significantly reducing computational overhead. Further, we dynamically select the ambiguous crops at the image level to enlarge the number of boundary points involved in contrastive learning, which benefits the establishment of precise distributions for each category. Extensive experimentation demonstrates that PPPC not only helps to address ambiguity at the pixel level, yielding discriminative representations but also achieves significant improvements in both synthetic-to-real and day-to-night adaptation tasks. It surpasses the previous state-of-the-art (SOTA) by +5.2% mIoU in the most challenging daytime-to-nighttime adaptation scenario, exhibiting stronger generalization on other unseen datasets. The code and models are available at https://github.com/DarlingInTheSV/Probabilistic-Prototypical-Pixel-Contrast.

Bayesian Uncertainty Estimation by Hamiltonian Monte Carlo: Applications to Cardiac MRI Segmentation

Deep learning (DL)-based methods have achieved state-of-the-art performance for many medical image segmentation tasks. Nevertheless, recent studies show that deep neural net- works (DNNs) can be miscalibrated and overconfident, leading to ”silent failures” that are risky for clinical applications. Bayesian DL provides an intuitive approach to DL failure de- tection, based on posterior probability estimation. However, the posterior is intractable for large medical image segmentation DNNs. To tackle this challenge, we propose a Bayesian learning framework using Hamiltonian Monte Carlo (HMC), tempered by cold posterior (CP) to accommodate medical data augmentation, named HMC-CP. For HMC compu- tation, we further propose a cyclical annealing strategy, capturing both local and global geometries of the posterior distribution, enabling highly efficient Bayesian DNN training with the same computational budget as training a single DNN. The resulting Bayesian DNN outputs an ensemble segmentation along with the segmentation uncertainty. We evaluate the proposed HMC-CP extensively on cardiac magnetic resonance image (MRI) segmentation, using in-domain steady-state free precession (SSFP) cine images as well as out-of-domain datasets of quantitative T1 and T2 mapping. Our results show that the proposed method improves both segmentation accuracy and uncertainty estimation for in- and out-of-domain data, compared with well-established baseline methods such as Monte Carlo Dropout and Deep Ensembles. Additionally, we establish a conceptual link between HMC and the commonly known stochastic gradient descent (SGD) and provide general insight into the uncertainty of DL. This uncertainty is implicitly encoded in the training dynamics but often overlooked. With reliable uncertainty estimation, our method provides a promising direction toward trustworthy DL in clinical applications. We release our code in <a href='https://gitlab.tudelft.nl/yidongzhao/hmc_uncertainty'>https://gitlab.tudelft.nl/yidongzhao/hmc_uncertainty</a>

A Software Tool for Estimating Uncertainty of Bayesian Posterior Probability for Disease.

The role of medical diagnosis is essential in patient care and healthcare. Established diagnostic practices typically rely on predetermined clinical criteria and numerical thresholds. In contrast, Bayesian inference provides an advanced framework that supports diagnosis via in-depth probabilistic analysis. This study's aim is to introduce a software tool dedicated to the quantification of uncertainty in Bayesian diagnosis, a field that has seen minimal exploration to date. The presented tool, a freely available specialized software program, utilizes uncertainty propagation techniques to estimate the sampling, measurement, and combined uncertainty of the posterior probability for disease. It features two primary modules and fifteen submodules, all designed to facilitate the estimation and graphical representation of the standard uncertainty of the posterior probability estimates for diseased and non-diseased population samples, incorporating parameters such as the mean and standard deviation of the test measurand, the size of the samples, and the standard measurement uncertainty inherent in screening and diagnostic tests. Our study showcases the practical application of the program by examining the fasting plasma glucose data sourced from the National Health and Nutrition Examination Survey. Parametric distribution models are explored to assess the uncertainty of Bayesian posterior probability for diabetes mellitus, using the oral glucose tolerance test as the reference diagnostic method.

RSS-Bagging: Improving Generalization Through the Fisher Information of Training Data.

The bagging method has received much application and attention in recent years due to its good performance and simple framework. It has facilitated the advanced random forest method and accuracy-diversity ensemble theory. Bagging is an ensemble method based on simple random sampling (SRS) method with replacement. However, SRS is the most foundation sampling method in the field of statistics, where exists some other advanced sampling methods for probability density estimation. In imbalanced ensemble learning, down-sampling, over-sampling, and SMOTE methods have been proposed for generating base training set. However, these methods aim at changing the underlying distribution of data rather than simulating it better. The ranked set sampling (RSS) method uses auxiliary information to get more effective samples. The purpose of this article is to propose a bagging ensemble method based on RSS, which uses the ordering of objects related to the class to obtain more effective training sets. To explain its performance, we give a generalization bound of ensemble from the perspective of posterior probability estimation and Fisher information. On the basis of RSS sample having a higher Fisher information than SRS sample, the presented bound theoretically explains the better performance of RSS-Bagging. The experiments on 12 benchmark datasets demonstrate that RSS-Bagging statistically performs better than SRS-Bagging when the base classifiers are multinomial logistic regression (MLR) and support vector machine (SVM).

Adversarial pre-optimized graph representation learning with double-order sampling for cross-modal retrieval

Cross-modal retrieval aims at retrieving relevant information across different media data, such as images and text. However, inconsistent distribution of different modal data causes a heterogeneity gap, which becomes an intractable issue. Previous researches aim at finding a semantic share space for various modalities, in which the similarity of them can be calculated by a uniform measurement. Graph representation learning has shown excellent performance in many tasks, like information retrieval. However, it is difficult to construct the most proper cross-modal graph which can better present the semantics of multi-modal data. Besides, the projected representations of various modal data optimized by the supervised class labels and unsupervised graph context learning are insufficiently aligned. To address these issues, this paper presents an Adversarial Pre-optimized Graph Representation Learning (AP-GRL) for cross-modal retrieval. Similar to mainstream methods, AP-GRL aims to learn a shared representation for various modalities. We design an adaptive pre-optimization mechanism to optimize the cross-modal data and their relationships, which is different from previously fixed graph. The raw constructed cross-modal graph is optimized by maximum posterior probability estimation. To fully exploit the graph, we proposed a double-order sampling strategy to take the width and depth into representation learning. Besides, we adopt adversarial learning with Wasserstein distance to reduce domain discrepancy in the domain-invariant representation learning. Extensive experiments on five widely used datasets demonstrate that AP-GRL outperforms state-of-the-art cross-modal retrieval methods.

Importance sampling and Bayesian model comparison in ecology and evolution

Abstract Bayesian approaches to the modelling of ecological systems are increasingly popular, but there are competing methods for formal model comparisons. Here, we focus on the task of performing multimodel inference through estimating posterior model weights, which encompasses uncertainties in the choice of competing model structure into the inference outputs. Model‐based approaches such as reversible‐jump Markov chain Monte Carlo (RJ‐MCMC) are flexible and allow multimodel inference, but can be complex to implement and optimise, and so we translate a model‐based approach for ecological applications using Importance Sampling to estimate the marginal likelihood of the data given a particular model. This approach allows for model comparison through the estimation of Bayes' Factors or interpretable posterior model probabilities, yielding model weights that facilitate multimodel inference through Bayesian model averaging. We demonstrate Importance Sampling with two case study investigations in animal demography: censused analysis of banded mongoose (Mungos mungo) survival where missing data are uncommon, and capture–mark–recapture analysis of European badger (Meles meles) survival where data are commonly missing. We compare outcomes of the model comparison using the Importance Sampling approach to those obtained through single‐model inference approaches using Deviance information criteria and the Watanabe–Akaike information criteria. The results of the Importance Sampling method aligns with RJ‐MCMC model comparisons while often being more straightforward to fit and optimise, particularly if the competing models are non‐nested.

An Hybrid Integration Method-Based Track-before-Detect for High-Speed and High-Maneuvering Targets in Ubiquitous Radar

Due to the limited transmission gain of ubiquitous radar systems, it has become necessary to use a long-time coherent integration method for range-Doppler (RD) analysis. However, when the target exhibits high-speed and high-maneuver capabilities, it introduces challenges, such as range migration (RM), Doppler frequency migration (DFM), and velocity ambiguity (VA) in the RD domain, thus posing significant difficulties in target detection and tracking. Moreover, the presence of VA further complicates the problem. To address these complexities while maintaining integration efficiency, this study proposes a hybrid integration approach. First, methods called Keystone-transform (KT) and matched filtering processing (MFP) are proposed for compensating for range migration (RM) and velocity ambiguity (VA) in Radar Detection (RD) images. The KT approach is employed to compensate for RM, followed by the generation of matched filters with varying ambiguity numbers. Subsequently, MFP enables the production of multiple RD images covering different but contiguous Doppler frequency ranges. These RD images can be compiled into an extended RD (ERD) image that exhibits an expanded Doppler frequency range. Second, an improved particle-filter (IPF) algorithm is raised to perform incoherent integration among ERD images and to achieve track-before-detect (TBD) for a target. In the IPF, the target state vector is augmented with ambiguous numbers, which are estimated via maximum posterior probability estimation. Then, to compensate for the DFM, a line spread model (LSM) is proposed instead of the point spread model (PSM) used in traditional PF. To evaluate the efficacy of the proposed method, a radar simulator is devised, encompassing comprehensive radar signal processing. The findings demonstrate that the proposed approach achieves a harmonious equilibrium between integration efficiency and computational complexity when it comes to detecting and tracking high-speed and high-maneuvering targets with intricate maneuvers. Furthermore, the algorithm’s effectiveness is authenticated by exploiting ubiquitous radar data.

Efficient reliability updating methods based on Bayesian inference and sequential learning Kriging

Reliability updating is an effective tool for reappraising reliability level of system when new observation information is obtained. The adaptive Kriging based reliability updating method (RUAK) inserts the adaptive Kriging into traditional simulation method to improve the efficiency of reliability updating. However, an identical candidate sampling pool is used to simultaneously estimate the prior failure probability and the posterior one in RUAK, which leads to a waste of computational resources in case of significant difference between the importance regions in estimation of prior and posterior failure probabilities. To overcome this disadvantage, an efficient reliability updating framework based on Bayesian inference and sequential learning Kriging is proposed in this paper. In the proposed method, two candidate sampling pools respectively for estimating the prior and posterior failure probabilities are separately constructed by prior probability density function (PDF) and posterior PDF obtained by Bayesian inference. Then, the Kriging model is established and sequentially refined in these two candidate sampling pools to accurately estimate the corresponding failure probabilities. Through combining different simulation methods with the proposed framework, the Monte Carlo simulation based and importance sampling based sequential learning Kriging methods are respectively developed for reliability updating.

Classification and feature selection methods based on fitting logistic regression to PU data

In our work, we examine the classification methods where the positive and unlabeled data are considered and where the conditional distribution of the true class label given the feature vector is governed by the model of logistic regression. Our first objective is to compute and compare the selected metrics allowing for the quality assessment of these methods. In this context, we investigate four methods of the posterior probability estimation, where the risk of logistic loss function is optimized: the naive approach, the weighted likelihood approach, as well as the quite recently proposed methods – the joint approach, and the LassoJoint method. The corresponding evaluations are basically performed for 13 machine learning models on some chosen – both low- and high-dimensional – datasets. Some of the mentioned machine learning model schemes have been directly borrowed from literature and some have been obtained through some modifications in the existing procedures. Our second goal is to establish the most stable and efficient approach for the posterior probability estimation. Moreover, we use the AdaSampling scheme for comparison of the considered classification methods. We also conducted comparisons of feature selection procedures – the Mutual Information-Based feature selection method and the LassoJoint approach. The current article is an enhancement of the conference paper Furmańczyk et al. (2022).

An improved dynamic programming tracking-before-detection algorithm based on LSTM network value function

The detection and tracking of small and weak maneuvering radar targets in complex electromagnetic environments is still a difficult problem to effectively solve. To address this problem, this paper proposes a dynamic programming tracking-before-detection method based on long short-term memory (LSTM) network value function(VL-DP-TBD). With the help of the estimated posterior probability provided by the designed LSTM network, the calculation of the posterior value function of the traditional DP-TBD algorithm can be more accurate, and the detection and tracking effect achieved for maneuvering small and weak targets is improved. Utilizing the LSTM network to model the posterior probability estimation of the target motion state, the posterior probability moving features of the maneuvering target can be learned from the noisy input data. By incorporating these posterior probability estimation values into the traditional DP-TBD algorithm, the accuracy and robustness of the calculation of the posterior value function can be enhanced, so that the improved architecture is capable of effectively recursively accumulating the movement trend of the target. Simulation results show that the improved architecture is able to effectively reduce the aggregation effect of a posterior value function and improve the detection and tracking ability for non-cooperative nonlinear maneuvering dim small target. Abbreviations LSTM: Long short-term memory; DP-TBD: Dynamic programming-based tracking before detection; DBT: Detection before tracking; TBD: Tracking before detection; HT-TBD: Tracking-before-detection algorithm based on the Hough transform; PF-TBD: Tracking-before-detection algorithm based on particle filtering; RFS-TBD: Tracking-before-detection algorithm based on random finite sets; SNR: Signal-to-noise ratio; DP: Dynamic programming; EVT: Extreme value theory; EVT: Generalized extreme value theory; GLRT: Generalized likelihood ratio detection; KT: Keystone transformation; PGA: Phase gradient autofocusing; CFAR: Constant false-alarm rate; J-CA-CFAR: Joint intensity-spatial CFAR; MF: Merit function; CP-DP-TBD: Candidate plot-based DP-TBD; CIT: Coherent integration time; RNN: Recurrent neural network; CS: Current statistical; Pd: Detection probability; Pt: Tracking probability.

Previous beliefs affect Bayesian reasoning in conditions fostering gist comprehension

It has been shown that Bayesian reasoning is affected by the believability of the data, but it is unknown which conditions could potentiate or reduce such belief effect. Here, we tested the hypothesis that the belief effect would mainly be observed in conditions fostering a gist comprehension of the data. Accordingly, we expected to observe a significant belief effect in iconic rather than in textual presentations and, in general, when nonnumerical estimates were requested. The results of three studies showed more accurate Bayesian estimates, either expressed numerically or nonnumerically, for icons than for text descriptions of natural frequencies. Moreover, in line with our expectations, nonnumerical estimates were, in general, more accurate for believable rather than for unbelievable scenarios. In contrast, the belief effect on the accuracy of the numerical estimates depended on the format and on the complexity of the calculation. The present findings also showed that single-event posterior probability estimates based on described frequencies were more accurate when expressed nonnumerically rather than numerically, opening new avenues for the development of interventions to improve Bayesian reasoning.

Antithrombotic regimens for the prevention of major adverse cardiac events in chronic coronary syndrome: A systematic review and network meta-analysis.

Antithrombotic therapy is the cornerstone of chronic coronary syndrome (CCS) management. However, the best treatment option that optimally balances bleeding risk and efficacy remains undefined. Our objective was to evaluate the effectiveness and safety of antithrombotic options and identify the optimal treatment option for patients with CCS. We used the MEDLINE, CENTRAL and Embase databases to search for randomized controlled trials with follow-up periods longer than 12 months that compared aspirin (ASA) monotherapy with other antithrombotic therapies in patients with CCS. The Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines were used. Extracted data [hazard ratios (HR)] were pooled using Bayesian fixed-effect models, allowing the estimation of credible intervals (CrI) and posterior probabilities of benefit, harm, and practical equivalence. Confidence in the results was assessed with the Confidence In Network Meta-Analysis (CINeMA) tool. The primary efficacy and safety outcomes were major adverse cardiovascular events (MACE) and primary bleeding, respectively. Secondary outcomes were acute myocardial infarction, ischemic stroke, all-cause, and cardiovascular-specific mortality. Five trials with a total of 80,605 patients were included. Mean patient age ranged from 61 to 69 years, while 20.3% to 31.4% were women. The reference treatment was ASA monotherapy. ASA + prasugrel 10 mg and clopidogrel 75 mg monotherapy presented the greatest benefit for MACE [HR 0.52 (95% CrI, 0.39-0.71); and 0.68 (95% CrI, 0.54-0.88)]. There was a probability of 98.8% that ASA + ticagrelor was practically equivalent to ASA monotherapy. Regarding the primary bleeding outcome, clopidogrel 75 mg monotherapy performed best [HR 0.64 (0.42, 0.99)]. There was a probability of 97.4% that ASA + Prasugrel 10 mg increases bleeding (HR > 1.0). Secondary outcome results followed a similar treatment ranking pattern as in primary outcomes. Overall, CINeMA confidence ratings were judged as either low or very low. These results revealed that clopidogrel monotherapy might provide the best risk-benefit balance in treating CCS. However, low CINeMA confidence ratings may preclude more forceful conclusions. Our analysis suggests that current guidelines recommending ASA as first-line therapy for CCS management need to be revised to include additional pharmacological options.

Study on passive location method of shallow water acoustic source with single hydrophone

In order to meet the needs of target detection in shallow water with concealed and low energy consumption platform, the Bayesian estimation theory was applied to underwater acoustic location. The underwater acoustic location model was established based on probability density function as the description of sound source state, so as to overcome the problems of unstable shallow water environment and mismatching of sound field model. Histogram filtering method is used to solve the integral solution in the process of posterior probability estimation of sound source state. The hierarchical grid histogram filtering method is proposed for the first time, which effectively improves the efficiency of histogram filtering iterative algorithm. The measured data and simulation results of SWelleX-96 show that the depth positioning accuracy can reach 35 m and the distance positioning accuracy can reach 0.69 km within the range of 200 m deep and 10 km long in shallow water environment, and the efficiency of the algorithm can be improved by N1/2 times.

Extending the Fellegi-Sunter record linkage model for mixed-type data with application to the French national health data system

Probabilistic record linkage is a process of combining data from different sources, when such data refer to common entities and identifying information is not available. A probabilistic record linkage framework that takes into account multiple non-identifying information that this is limited to simple binary comparison between matching variables has been previously proposed. An extension of this method is proposed for mixed-type comparison vectors. A mixture model for handling comparison values of low prevalence categorical matching variables, and a mixture of hurdle gamma distribution for handling comparison values of continuous matching variables have been developed. The parameters are estimated by means of the Expectation Conditional Maximization (ECM) algorithm. Through a Monte Carlo simulation study, both the posterior probability estimation for a record pair to be a match and the prediction of matched record pairs are evaluated. The simulation results indicate that the proposed methods outperform existing ones in most considered cases. The proposed methods are applied on a real dataset, to perform linkage between a registry of patients suffering from venous thromboembolism in the Brest district area (GETBO) and the French national health information system (SNDS).

Intelligent fault diagnosis of bevel gearboxes using semi-supervised probability support matrix machine and infrared imaging

Fault diagnosis is of great significance to ensure the reliability and safety of complex bevel gearbox systems. Most existing intelligent fault diagnosis approaches of bevel gearboxes are designed with vibration monitoring. However, the collected vibration data are vulnerable to noise pollution and machinery operating conditions. Besides, traditional fault diagnosis models highly rely on numerous labeled samples, and neglect the high cost of label annotation in real-world applications. Therefore, a novel fault diagnosis approach based on semi-supervised probability support matrix machine (SPSMM) and infrared imaging is proposed for bevel gearboxes in this paper, which has the following properties. Firstly, SPSMM classifies 2D matrix data directly without vectorization, thus fully utilizing the spatial information in infrared images. Secondly, a probability output strategy is designed for SPSMM to calculate the posterior class probability estimation of matrix inputs, and consequently enhance the diagnostic accuracy and robustness of the model. Thirdly, a semi-supervised learning (SSL) framework is proposed for SPSMM to carry out sample transfer from the unlabeled sample pool to the labeled sample pool, which can effectively alleviate the problem of insufficient labeled samples. The superiority of the proposed diagnosis approach is demonstrated with an infrared imaging dataset of a bevel gearbox.

Cautious weighted random forests

Random forest is an efficient and accurate classification model, which makes decisions by aggregating a set of trees, either by voting or by averaging class posterior probability estimates. However, tree outputs may be unreliable in presence of scarce data. The imprecise Dirichlet model (IDM) provides workaround, by replacing point probability estimates with interval-valued ones. This paper investigates a new tree aggregation method based on the theory of belief functions to combine such probability intervals, resulting in a cautious random forest classifier. In particular, we propose a strategy for computing tree weights based on the minimization of a convex cost function, which takes both determinacy and accuracy into account and makes it possible to adjust the level of cautiousness of the model. The proposed model is evaluated on 25 UCI datasets and is demonstrated to be more adaptive to the noise in training data and to achieve a better compromise between informativeness and cautiousness.

A pragmatic, performance‐based approach to levee safety assessments

Flood prone areas are often protected against flooding by an extensive network of flood defenses. To ensure their structural integrity, these flood defenses are periodically assessed. Many levees have been functioning well for decades, and have survived several relatively high hydraulic loads within their lifetime. However, information on survived load conditions is seldom included in levee safety assessments. Observed degradation from levee inspections is also not taken into account. That way, information that is useful to improve the accuracy of estimations of the actual strength of the levee remains unexploited. This study proposes a pragmatic approach to include observations of survived loads and levee degradation in the levee safety assessment. This approach consists of three steps: (1) a prior estimation of the failure probability, based on levee characteristics, (2) a posterior estimation of the failure probability, based on observed hydraulic loads, and (3) correction of the posterior failure probability estimation, based on levee inspections. In a case study, the estimated failure probabilities using this approach were much lower than when information on levee performance was not included. This study demonstrates the value of levee performance observations and how they could be included to improve levee safety assessments.

Resilient Filtering of Nonlinear Complex Dynamical Networks Under Randomly Occurring Faults and Hybrid Cyber-Attacks

This paper investigates the problem of recursive resilient filtering for discrete-time nonlinear dynamical networks with randomly switching topologies under randomly occurring faults and hybrid cyber-attacks. The attacks considered are composed of DoS attacks describing by binary random sequences and deception attacks, the amplitude of which obeys the Gaussian mixing distribution. A recursive resilient filter dependent on DoS attack sequences is designed to simultaneously estimate system states and occurred faults, where the desired gain is parameterized via the matrix iterative formula of filtering error covariance. Furthermore, the maximum posterior probability estimation is calculated to identify DoS attack sequences with the help of the classical Bayesian rule. In particular, a modified recursive filtering algorithm is established by replacing the corresponding parameters in the previously derived formulas about filter gains and covariance matrices. Finally, the effectiveness of the proposed recursive filtering algorithm is demonstrated by a simulation example.

Simplicity as a Cue to Probability: Multiple Roles for Simplicity in Evaluating Explanations.

People often face the challenge of evaluating competing explanations. One approach is to assess the explanations' relative probabilities-for example, applying Bayesian inference to compute their posterior probabilities. Another approach is to consider an explanation's qualities or "virtues," such as its relative simplicity (i.e., the number of unexplained causes it invokes). The current work investigates how these two approaches are related. Study 1 found that simplicity is used to infer the inputs to Bayesian inference (explanations' priors and likelihoods). Studies 1 and 2 found that simplicity is also used as a direct cue to the outputs of Bayesian inference (the posterior probability of an explanation), such that simplicity affects estimates of posterior probability even after controlling for elicited (Study 1) or provided (Study 2) priors and likelihoods, with simplicity having a larger effect in Study 1, where posteriors are more uncertain and difficult to compute. Comparing Studies 1 and 2 also suggested that simplicity plays additional roles unrelated to approximating probabilities, as reflected in simplicity's effect on how "satisfying" (vs. probable) an explanation is, which remained largely unaffected by the difficulty of computing posteriors. Together, these results suggest that the virtue of simplicity is used in multiple ways to approximate probabilities (i.e., serving as a cue to priors, likelihoods, and posteriors) when these probabilities are otherwise uncertain or difficult to compute, but that the influence of simplicity also goes beyond these roles.