Hierarchical Inference Research Articles

To integrate or not to integrate: Temporal dynamics of hierarchical Bayesian causal inference.

To form a percept of the environment, the brain needs to solve the binding problem—inferring whether signals come from a common cause and are integrated or come from independent causes and are segregated. Behaviourally, humans solve this problem near-optimally as predicted by Bayesian causal inference; but the neural mechanisms remain unclear. Combining Bayesian modelling, electroencephalography (EEG), and multivariate decoding in an audiovisual spatial localisation task, we show that the brain accomplishes Bayesian causal inference by dynamically encoding multiple spatial estimates. Initially, auditory and visual signal locations are estimated independently; next, an estimate is formed that combines information from vision and audition. Yet, it is only from 200 ms onwards that the brain integrates audiovisual signals weighted by their bottom-up sensory reliabilities and top-down task relevance into spatial priority maps that guide behavioural responses. As predicted by Bayesian causal inference, these spatial priority maps take into account the brain’s uncertainty about the world’s causal structure and flexibly arbitrate between sensory integration and segregation. The dynamic evolution of perceptual estimates thus reflects the hierarchical nature of Bayesian causal inference, a statistical computation, which is crucial for effective interactions with the environment.

Water Supply Pipeline Risk Index Assessment Based on Cohesive Hierarchical Fuzzy Inference System

As populations grow, facilities such as roads, bridges, railways lines, commercial and residential buildings, etc., must be expanded and maintained. There are extensive networks of underground facilities to fulfil the demand, such as water supply pipelines, sewage pipelines, metro structures, etc. Hence, a method to regularly assesses the risk of the underground facility failures is needed to decrease the chance of accidental loss of service or accidents that endanger people and facilities. In the proposed work, a cohesive hierarchical fuzzy inference system (CHFIS) was developed. A novel method is proposed for membership function (MF) determination called the heuristic based membership functions determination (HBMFD) method to determine an appropriate MF set for each fuzzy logic method in CHFIS. The proposed model was developed to decrease the number of rules for the full structure fuzzy inference system with all rule implementation. Four very crucial parameters were considered in the proposed work that are inputs to the proposed CHFIS model in order to calculate the risk of water supply pipelines. In order to fully implement the proposed CHFIS just 85 rules are needed while using the traditional Mamdani fuzzy inference system, 900 rules are required. The novel method greatly reduces implementation time and rule design sets that are extremely time consuming to develop and difficult to maintain.

Coarse-to-fine road scene segmentation via hierarchical graphical models

The road scene segmentation is an important problem which is helpful for a higher level of the scene understanding. This article presents a novel approach for image semantic segmentation of road scenes via a hierarchical graph-based inference. A deep encoder–decoder network is first applied for a fast pixel-wise classification. Then, hierarchical graph-based inference is performed to get an accurate segmentation result. In the inference process, all the object classes are grouped into fewer categories which contains at least one class. The category labels are assigned to image superpixels using Markov random field model. For each category, a pixel-level labeling based on fully connected conditional random fields is performed to divide image into different classes. After the inference for all categories, the results are integrated together to get the final segmentation. In additional to low-level affinity functions, the feature maps from network are integrated in pairwise potentials of the graphical models. This hierarchical inference scheme can alleviate the confusion of classes belonging to different categories. It performs well for small objects without adding more computational burden. Both qualitative and quantitative assessments are adopted to evaluate the proposed method. The results on benchmark data sets prove the effectiveness of the proposed hierarchical scheme, and the performance is competitive with the state-of-the-art methods.

Attention promotes the neural encoding of prediction errors.

The encoding of sensory information in the human brain is thought to be optimised by two principal processes: ‘prediction’ uses stored information to guide the interpretation of forthcoming sensory events, and ‘attention’ prioritizes these events according to their behavioural relevance. Despite the ubiquitous contributions of attention and prediction to various aspects of perception and cognition, it remains unknown how they interact to modulate information processing in the brain. A recent extension of predictive coding theory suggests that attention optimises the expected precision of predictions by modulating the synaptic gain of prediction error units. Because prediction errors code for the difference between predictions and sensory signals, this model would suggest that attention increases the selectivity for mismatch information in the neural response to a surprising stimulus. Alternative predictive coding models propose that attention increases the activity of prediction (or ‘representation’) neurons and would therefore suggest that attention and prediction synergistically modulate selectivity for ‘feature information’ in the brain. Here, we applied forward encoding models to neural activity recorded via electroencephalography (EEG) as human observers performed a simple visual task to test for the effect of attention on both mismatch and feature information in the neural response to surprising stimuli. Participants attended or ignored a periodic stream of gratings, the orientations of which could be either predictable, surprising, or unpredictable. We found that surprising stimuli evoked neural responses that were encoded according to the difference between predicted and observed stimulus features, and that attention facilitated the encoding of this type of information in the brain. These findings advance our understanding of how attention and prediction modulate information processing in the brain, as well as support the theory that attention optimises precision expectations during hierarchical inference by increasing the gain of prediction errors.

Dielectric permittivity-water content relationships in woodchips: Particle size and temperature effects

Information on the amount of water held by woodchips in the field is needed to assess environmental risks when they are kept in stockpiles or for inferring their efficiency as denitrifying bioreactors. Reflectometry is a widely used method for estimating volumetric water content (θ) in permeable media, but material-specific calibration functions should be developed. This study aimed at developing relationships between the bulk dielectric permittivity (εb) and θ in woodchips considering their particle size distribution (PSD), temperature (T) and the dielectric permittivity of the dry material plus air (εsa). The εb of fine, medium and coarse PSDs were measured with CS616-L water content reflectometers up to θ of 0.83 cm3/cm3 and at T (°C) of 10°, 24°, 37°, 55° and 70°. Calibration equations were developed using a multivariate power law function fitted to the data with a hierarchical Bayesian inference procedure. The best fit (R2 = 0.96; RMSE=0.04 cm3/cm3) was obtained for the fine PSD (εsa ∼ 1.09); while medium (εsa ∼ 1.09) and coarse (εsa ∼ 1.03) PSDs were more influenced by T and the presence of large air voids (R2 ∼ 0.92; RMSE∼0.06 cm3/cm3). A modified calibration equation independent of PSD (εsa = 1) reduced only slightly the goodness-of-fit (R2 = 0.90; RMSE=0.07 cm3/cm3). The equations developed in this study are well suited for field applications as they cover a wider range of θ than previously published functions and they are applicable within the range of T found in field applications.

Hyper Suprime-Cam view of the CMASS galaxy sample

Aims. We wish to determine the distribution of dark matter halo masses as a function of the stellar mass and the stellar mass profile for massive galaxies in the Baryon Oscillation Spectroscopic Survey (BOSS) constant-mass (CMASS) sample. Methods. We used grizy photometry from the Hyper Suprime-Cam (HSC) to obtain Sérsic fits and stellar masses of CMASS galaxies for which HSC weak-lensing data are available. This sample was visually selected to have spheroidal morphology. We applied a cut in stellar mass, log M*/M⊙ > 11.0, and selected ∼10 000 objects thus. Using a Bayesian hierarchical inference method, we first investigated the distribution of Sérsic index and size as a function of stellar mass. Then, making use of shear measurements from HSC, we measured the distribution of halo mass as a function of stellar mass, size, and Sérsic index. Results. Our data reveal a steep stellar mass-size relation Re ∝ M*βR, with βR larger than unity, and a positive correlation between Sérsic index and stellar mass: n ∝ M*0.46. The halo mass scales approximately with the 1.7 power of the stellar mass. We do not find evidence for an additional dependence of halo mass on size or Sérsic index at fixed stellar mass. Conclusions. Our results disfavour galaxy evolution models that predict significant differences in the size growth efficiency of galaxies living in low- and high-mass halos.

Digging the population of compact binary mergers out of the noise

Coalescing compact binaries emitting gravitational wave (GW) signals, as recently detected by the Advanced LIGO-Virgo network, constitute a population over the multi-dimensional space of component masses and spins, redshift, and other parameters. Characterizing this population is a major goal of GW observations and may be approached via parametric models. We demonstrate hierarchical inference for such models with a method that accounts for uncertainties in each binary merger's individual parameters, for mass-dependent selection effects, and also for the presence of a second population of candidate events caused by detector noise. Thus, the method is robust to potential biases from a contaminated sample and allows us to extract information from events that have a relatively small probability of astrophysical origin.

Fuzzy signaling game of deception between ant-inspired deceptive robots with interactive learning

In this study the robotic deception phenomenon is raised in the framework of a signaling game which utilizes fuzzy logic and game theory along with inspirations from nature. Accomplishing the fuzzy signaling strategy set for deceptive players serves as a great part of our contribution and on this aim, hierarchical fuzzy inference systems support receiver’s actions and sender’s ant-inspired deceptive signals (track and pheromone). In addition, special deceptive robots and visually-supported experimental environment are also provided. The fuzzy behavior of robots defines the strategy type of players. The final result of deception process depends on this strategy type which leads to proposing a payoff matrix in which each cell of mutual costs is defined with special supporting logic related to our deception game with pursuit–evasion applications. Furthermore, motivated by animal signaling, through applying mixed strategies on deceiver’s honesty level and rival’s trust level, the corresponding learning dynamics are investigated and the conceptual discussion put forward serves as a proof to the smart human-like behavior that occurs between the robots: the interactive learning. Simulation results show that robots are capable of interactive learning within deceptive interaction and finally change their strategies to adopt themselves to new situation occurred due to opponent’s strategy change. Because of repetitive change in strategies as a result of learning, the conditions of a persistent deception without breakdown holds for this game where deceiver can frequently benefit from deception without leaving rival to lose its trust totally. The change in strategy will happen after a short time needed to learn the new situation. In rival’s learning process, this short time, which we call the ignorance time, exactly is the period that deceiver can benefit from deception while its evil intends are still concealed. Moreover, in this study an algorithm is given for the proposed signaling game of deception and an illustrative experiment in the introduced experimental environment demonstrates the process of a successful deception. The paper also gives solution to the proposed game by analyzing mixed Nash equilibrium which turns out to be the interior center fixed point of the learning dynamics.

Improved estimation of cancer dependencies from large-scale RNAi screens using model-based normalization and data integration

The availability of multiple datasets comprising genome-scale RNAi viability screens in hundreds of diverse cancer cell lines presents new opportunities for understanding cancer vulnerabilities. Integrated analyses of these data to assess differential dependency across genes and cell lines are challenging due to confounding factors such as batch effects and variable screen quality, as well as difficulty assessing gene dependency on an absolute scale. To address these issues, we incorporated cell line screen-quality parameters and hierarchical Bayesian inference into DEMETER2, an analytical framework for analyzing RNAi screens (https://depmap.org/R2-D2). This model substantially improves estimates of gene dependency across a range of performance measures, including identification of gold-standard essential genes and agreement with CRISPR/Cas9-based viability screens. It also allows us to integrate information across three large RNAi screening datasets, providing a unified resource representing the most extensive compilation of cancer cell line genetic dependencies to date.

Optimum Soybean Seeding Rates by Yield Environment in Southern Brazil

Core Ideas Soybean yield response to seeding rate was dependent on yield environment. Optimum seeding rate increased as yield environments were reduced. Seeding rate could be reduced by 18% for high‐yielding relative to low‐yielding environments, without penalizing yields. Planting date interacts with seed yield response to seeding rate, optimum seeding rates increase with late planting. For high‐yielding environment, late planting time decreased yields regardless of the seeding rate. Optimizing seed inputs while increasing farming profit is the main purpose of variable rate seeding (VRS) technology adoption. Previous studies in corn (Zea mays L.) suggested that optimal seeding rates increase as yield productivity level increased. For soybean [Glycine max (L.) Merr.], optimal yield‐to‐seeding rate by yield level has not been fully investigated, representing a scientific knowledge gap. Therefore, a dataset was collected from 109 replicated field trials from Southern Brazil (2180 experimental units) presenting the following objectives: (i) identify the optimum seeding rate at varying yield levels (herein termed as yield environments), and (ii) explore the contribution of management factors (i.e., seeding rate, planting date, row spacing, maturity groups, growing season, yield environment, and ecological region) on soybean seed yield. Hierarchical modeling and Bayesian statistical inference were used to predict optimum seeding rate at varying yield environments, while conditional inference tree analysis was explored to identify and rank factors contributing to yield variation. The main results were: (i) soybean seeding rate increased from high‐ to low‐yielding environments; (ii) seeding rate could be reduced by 18% in high‐yielding (>5 Mg ha−1) relative to the low‐yielding (<4 Mg ha−1) environments, without penalizing yields. For improving site‐specific soybean seeding rate prescriptions, future studies should focus on the physiological mechanisms underpinning yield formation and on understanding the main factors (soil × plant × weather) contributing to the differential optimum seeding rate response.

Evolution of Angiosperm Pollen. 6. The Celastrales, Oxalidales, and Malpighiales (Com) Clade and Zygophyllales

Analyzing pollen morphological data on a contemporary phylogenetic framework can enhance our understanding of the distribution, diversity, and evolution of palynological characters. In this paper, the sixth in a series detailing pollen morphological characters across angiosperms, we focus on the Celastrales, Oxalidales, and Malpighiales (COM) clade and Zygophyllales, together comprising ca. 20,000 species in 47 families within fabids. We first examined pollen grains from 21 species with light, scanning electron, and transmission electron microscopy, to illustrate pollen diversity within the COM clade and Zygophyllales. Second, based on a reexamination of previously published pollen data and our new observations, we optimized 21 pollen characters on a maximum likelihood (ML) tree of 169 genera in 45 families representing all four orders, using Fitch parsimony, ML, and hierarchical Bayesian inference. The pollen morphology of this group displays great diversity, particularly in size, aperture number, supratectal element shape, and tectum sculpture. Plesiomorphic states for 18 characters were inferred unambiguously under all methods for the COM clade, and over 300 character state changes were hypothesized by each analysis on lineages at different levels within the group. Changes of state were found to occur most frequently in the characters outline in polar view, pollen size, and tectum sculpture; changes of state occurred least frequently in the characters dispersal unit, symmetry, and tectum presence/absence. We identified diagnostic character states for several monophyletic clades and explored palynological evidence to shed light on some unresolved relationships. For example, the previously poorly resolved Malpighiales were found to be distinguished by a single pollen character state change (from annulus absent to present). Patterns of evolutionary change in several notable pollen characters, such as the number of pollen apertures (from three to other states) and tectum sculpture (from perforate to reticulate), were further confirmed, and their possible adaptive functions are proposed. Reflecting current interest in the relationship between pollen morphology and pollination syndrome, we conducted tests of correlated evolution between pollen morphology and pollination syndrome, with results demonstrating significant correlations between a thin exine and anemophily and, unexpectedly, between presence of supratectal elements and anemophily. Evidence for previously postulated evolutionary trends in aperture number (increasing), tectum sculpture (from perforate to reticulate), and infratectum structure (from columellate to granulate) is discussed. The relatively high frequency of state changes in presence or absence of supratectal elements may be linked to switches in pollination syndrome within this group.

Forget-me-some: General versus special purpose models in a hierarchical probabilistic task.

Humans build models of their environments and act according to what they have learnt. In simple experimental environments, such model-based behaviour is often well accounted for as if subjects are ideal Bayesian observers. However, more complex probabilistic tasks require more sophisticated forms of inference that are sufficiently computationally and statistically taxing as to demand approximation. Here, we study properties of two approximation schemes in the context of a serial reaction time task in which stimuli were generated from a hierarchical Markov chain. One, pre-existing, scheme was a generically powerful variational method for hierarchical inference which has recently become popular as an account of psychological and neural data across a wide swathe of probabilistic tasks. A second, novel, scheme was more specifically tailored to the task at hand. We show that the latter model fit significantly better than the former. This suggests that our subjects were sensitive to many of the particular constraints of a complex behavioural task. Further, the tailored model provided a different perspective on the effects of cholinergic manipulations in the task. Neither model fit the behaviour on more complex contingencies that competently. These results illustrate the benefits and challenges that come with the general and special purpose modelling approaches and raise important questions of how they can advance our current understanding of learning mechanisms in the brain.

Mining gravitational-wave catalogs to understand binary stellar evolution: A new hierarchical Bayesian framework

Catalogs of stellar-mass compact binary systems detected by ground-based gravitational-wave instruments (such as Advanced LIGO and Advanced Virgo) will offer insights into the demographics of progenitor systems and the physics guiding stellar evolution. Existing techniques approach this through phenomenological modeling, discrete model selection, or model mixtures. Instead, we explore a novel technique that mines gravitational-wave catalogs to directly infer posterior probability distributions of the hyper-parameters describing formation and evolutionary scenarios (e.g. progenitor metallicity, kick parameters, and common-envelope efficiency). We use a bank of compact-binary population synthesis simulations to train a Gaussian-process emulator that acts as a prior on observed parameter distributions (e.g. chirp mass, redshift, rate). This emulator slots into a hierarchical population inference framework to extract the underlying astrophysical origins of systems detected by Advanced LIGO and Advanced Virgo. Our method is fast, easily expanded with additional simulations, and can be adapted for training on arbitrary population synthesis codes, as well as different detectors like LISA.

The material-weight illusion is a Bayes-optimal percept under competing density priors.

The material-weight illusion (MWI) is one example in a class of weight perception illusions that seem to defy principled explanation. In this illusion, when an observer lifts two objects of the same size and mass, but that appear to be made of different materials, the denser-looking (e.g., metal-look) object is perceived as lighter than the less-dense-looking (e.g., polystyrene-look) object. Like the size-weight illusion (SWI), this perceptual illusion occurs in the opposite direction of predictions from an optimal Bayesian inference process, which predicts that the denser-looking object should be perceived as heavier, not lighter. The presence of this class of illusions challenges the often-tacit assumption that Bayesian inference holds universal explanatory power to describe human perception across (nearly) all domains: If an entire class of perceptual illusions cannot be captured by the Bayesian framework, how could it be argued that human perception truly follows optimal inference? However, we recently showed that the SWI can be explained by an optimal hierarchical Bayesian causal inference process (Peters, Ma & Shams, 2016) in which the observer uses haptic information to arbitrate among competing hypotheses about objects’ possible density relationship. Here we extend the model to demonstrate that it can readily explain the MWI as well. That hierarchical Bayesian inference can explain both illusions strongly suggests that even puzzling percepts arise from optimal inference processes.

GPU-accelerated hierarchical Bayesian estimation of luminosity functions using flux-limited observations with photometric noise

We describe a computational framework for hierarchical Bayesian inference with simple (typically single-plate) parametric graphical models that uses graphics processing units (GPUs) to accelerate computations, enabling deployment on very large datasets. Its C++ implementation, CUDAHM (CUDA for Hierarchical Models) exploits conditional independence between instances of a plate, facilitating massively parallel exploration of the replication parameter space using the single instruction, multiple data architecture of GPUs. It provides support for constructing Metropolis-within-Gibbs samplers that iterate between GPU-accelerated robust adaptive Metropolis sampling of plate-level parameters conditional on upper-level parameters, and Metropolis-Hastings sampling of upper-level parameters on the host processor conditional on the GPU results. CUDAHM is motivated by demographic problems in astronomy, where density estimation and linear and nonlinear regression problems must be addressed for populations of thousands to millions of objects whose features are measured with possibly complex uncertainties. We describe a thinned latent point process framework for modeling such demographic data. We demonstrate accurate GPU-accelerated parametric conditional density deconvolution for simulated populations of up to 300,000 objects in ~1 hour using a single NVIDIA Tesla K40c GPU. Supplementary material provides details about the CUDAHM API and the demonstration problem.

A Bayesian Account of Psychopathy: A Model of Lacks Remorse and Self-Aggrandizing.

This article proposes a formal model that integrates cognitive and psychodynamic psychotherapeutic models of psychopathy to show how two major psychopathic traits called lacks remorse and self-aggrandizing can be understood as a form of abnormal Bayesian inference about the self. This model draws on the predictive coding (i.e., active inference) framework, a neurobiologically plausible explanatory framework for message passing in the brain that is formalized in terms of hierarchical Bayesian inference. In summary, this model proposes that these two cardinal psychopathic traits reflect entrenched maladaptive Bayesian inferences about the self, which defend against the experience of deep-seated, self-related negative emotions, specifically shame and worthlessness. Support for the model in extant research on the neurobiology of psychopathy and quantitative simulations are provided. Finally, we offer a preliminary overview of a novel treatment for psychopathy that rests on our Bayesian formulation.

Multi-Person Pose Estimation via Multi-Layer Fractal Network and Joints Kinship Pattern.

We propose an effective method to boost the accuracy of multi-person pose estimation in images. Initially, the three-layer fractal network was constructed to regress multi-person joints location heatmap that can help to enhance an image region with receptive field and capture more joints local-contextual feature information, thereby producing keypoints heatmap intermediate prediction to optimize human body joints regression results. Subsequently, the hierarchical bi-directional inference algorithm was proposed to calculate the degree of relatedness (call it Kinship) for adjacent joints, and it combines the Kinship between adjacent joints with the spatial constraints, which we refer to as joints kinship pattern matching mechanism, to determine the best matched joints pair. We iterate the above-mentioned joints matching process layer by layer until all joints are assigned to a corresponding individual. Comprehensive experiments demonstrate that the proposed approach outperforms the state-of-the-art schemes and achieves about 1% and 0.6% increase in mAP on MPII multi-person subset and MSCOCO 2016 keypoints challenge.

DOA Estimation for a Mixture of Uncorrelated and Coherent Sources Based on Hierarchical Sparse Bayesian Inference with a Gauss-Exp-Chi2 Prior

Direction of arrival (DOA) estimation algorithms based on sparse Bayesian inference (SBI) can effectively estimate coherent sources without recurring to extra decorrelation techniques, and their estimation performance is highly dependent on the selection of sparse prior. Specifically, the specified sparse prior is expected to concentrate its mass on the zero and distribute with heavy tails; otherwise, these algorithms may suffer from performance degradation. In this paper, we introduce a new sparse-encouraging prior, referred to as “Gauss-Exp-Chi2” prior, and develop an efficient DOA estimation algorithm for a mixture of uncorrelated and coherent sources under a hierarchical SBI framework. The Gauss-Exp-Chi2 prior distribution exhibits a sharp peak at the origin and heavy tails, and this property makes it an appropriate prior to encourage sparse solutions. A three-layer hierarchical sparse Bayesian model is established. Then, by exploiting variational Bayesian approximation, the model parameters are estimated by alternately updating until Kullback-Leibler (KL) divergence between the true posterior and the variational approximation becomes zero. By constructing the source power spectra with the estimated model parameters, the number and locations of the highest peaks are extracted to obtain source number and DOA estimates. In addition, some implementation details for algorithm optimization are discussed and the Cramér-Rao bound (CRB) of DOA estimation is derived. Simulation results demonstrate the effectiveness and favorable performance of the proposed algorithm as compared with the state-of-the-art sparse Bayesian algorithms.

A Hierarchical Inference Model for Internet-of-Things

Internet-of-Things (IoT) has connected billions of devices to the Internet. These devices are already collecting zettabytes ( $10^{21}$ ) of data. However, the current IoT framework suffers from limited sensor energy, communication bandwidth, and server storage. These limitations impede the ability to send all the sensor data to the server all the time. Compact smart sensors provide a way to address this challenge. As opposed to the conventional sense-and-transmit sensors, emerging smart sensors can collect data, extract features, derive local inferences, and transmit only inference outcomes and possibly some raw data associated with rare events instead of all the raw data. This can dramatically cut down on the amount of sensor data transmitted, and hence its communication energy and network traffic. However, edge or server inference models trained with conventional machine learning approaches do not account for the fact that the smart sensors in the system have already performed a local inference. These approaches need all the sensor data and hence only cater to the traditional sense-and-transmit paradigm. This undoes the energy benefits brought about by smart sensors. In this paper, we propose a hierarchical inference model for IoT applications based on hierarchical learning and local inferences. Our model is able to take advantage of inference already performed on smart sensors, while at the same time accommodating conventional sense-and-transmit sensors in the IoT system. It also generalizes sensor-level inference to inference at other edge nodes by exploiting the intrinsically sensor/edge-grouped IoT data structure. We train classifiers hierarchically, aligned with the sensor-edge-server IoT paradigm. We verify our approach with seven IoT applications, demonstrating that the model is accurate, efficient, and generally applicable. We derive four edge-level inference models and four server-level inference models for these applications. For the four edge-level inference models, we reduce the number of bits transmitted from the sensor by $3.2\times$ - $42.7\times$ while at the same time also improving the classification accuracy by 0.3-6.7 percent. For the four server-level inference models, we reduce the number of edge-to-server bits transmitted by $17\times$ - $60\times$ , with classification accuracy change in the $-0.4$ - $+0.1$ percent range.

Towards Bayesian hierarchical inference of equilibrium climate sensitivity from a combination of CMIP5 climate models and observational data

Despite decades of research, large multi-model uncertainty remains about the Earth’s equilibrium climate sensitivity to carbon dioxide forcing as inferred from state-of-the-art Earth system models (ESMs). Statistical treatments of multi-model uncertainties are often limited to simple ESM averaging approaches. Sometimes models are weighted by how well they reproduce historical climate observations. Here, we propose a novel approach to multi-model combination and uncertainty quantification. Rather than averaging a discrete set of models, our approach samples from a continuous distribution over a reduced space of simple model parameters. We fit the free parameters of a reduced-order climate model to the output of each member of the multi-model ensemble. The reduced-order parameter estimates are then combined using a hierarchical Bayesian statistical model. The result is a multi-model distribution of reduced-model parameters, including climate sensitivity. In effect, the multi-model uncertainty problem within an ensemble of ESMs is converted to a parametric uncertainty problem within a reduced model. The multi-model distribution can then be updated with observational data, combining two independent lines of evidence. We apply this approach to 24 model simulations of global surface temperature and net top-of-atmosphere radiation response to abrupt quadrupling of carbon dioxide, and four historical temperature data sets. Our reduced order model is a 2-layer energy balance model. We present probability distributions of climate sensitivity based on (1) the multi-model ensemble alone and (2) the multi-model ensemble and observations.