Bayesian Mixture Model Research Articles

A Hierarchical Bayesian Mixture Model Approach for Analysis of Resting-State Functional Brain Connectivity: An Alternative to Thresholding.

This article proposes a Bayesian hierarchical mixture model to analyze functional brain connectivity where mixture components represent “positively connected” and “non-connected” brain regions. Such an approach provides a data-informed separation of reliable and spurious connections in contrast to arbitrary thresholding of a connectivity matrix. The hierarchical structure of the model allows simultaneous inferences for the entire population as well as for each individual subject. A new connectivity measure, the posterior probability of a given pair of brain regions of a specific subject to be connected given the observed correlation of regions' activity, can be computed from the model fit. The posterior probability reflects the connectivity of a pair of regions relative to the overall connectivity pattern of an individual, which is overlooked in traditional correlation analyses. This article demonstrates that using the posterior probability might diminish the effect of spurious connections on inferences, which is present when a correlation is used as a connectivity measure. In addition, simulation analyses reveal that the sparsification of the connectivity matrix using the posterior probabilities might outperform the absolute thresholding based on correlations. Therefore, we suggest that posterior probability might be a beneficial measure of connectivity compared with the correlation. The applicability of the introduced method is exemplified by a study of functional resting-state brain connectivity in older adults.

Water sources of riparian plants based on stable hydrogen and oxygen isotopes in Lanzhou section of the Yellow River, China

As plant species for riparian ecological restoration in northern China, Tamarix ramosissima and Salix matsudana play an important role in river protection, flood control, regional climate regulation, and landscape construction of riparian vegetation. Two sampling sites were selected in the riparian zones along the Lanzhou section of Yellow River, where plant xylems and potential water sources were collected. The direct comparison method, Bayesian mixture model MixSIAR and the proportional similarity index (PS index) were used to determine the proportions of water utilization for each potential water source and the relationship of two species in water utilization. The results showed that shallow soil (0-30 cm) was the main water source during growing season, with utilization ratio being 28.3% for T. ramosissima and 24.4% for S. matsudana. For T. ramosissima, river water had the lowest contribution (16.6%), and for S. matsudana, groundwater contributed the least (17.9%). In the months with low soil moisture, plants increased the utilization ratios of river water and groundwater. The PS index at the sampling site S1 and S2 was 91.0% and 87.7%, respectively. On a monthly basis, the index in May was the highest, indicating an inter-month divergence in water use relationship. At the floodplain, there were even utilization ratios for each potential water source, which is an optimal strategy to obtain water from each potential source to the maximum extent. Our results provided theoretical basis for riparian tourism development along the Lanzhou section of the Yellow River and plant water management in environment protection in the Yellow River Basin.

Using Artificial Intelligence for Pattern Recognition in a Sports Context.

Optimizing athlete’s performance is one of the most important and challenging aspects of coaching. Physiological and positional data, often acquired using wearable devices, have been useful to identify patterns, thus leading to a better understanding of the game and, consequently, providing the opportunity to improve the athletic performance. Even though there is a panoply of research in pattern recognition, there is a gap when it comes to non-controlled environments, as during sports training and competition. This research paper combines the use of physiological and positional data as sequential features of different artificial intelligence approaches for action recognition in a real match context, adopting futsal as its case study. The traditional artificial neural networks (ANN) is compared with a deep learning method, Long Short-Term Memory Network, and also with the Dynamic Bayesian Mixture Model, which is an ensemble classification method. The methods were used to process all data sequences, which allowed to determine, based on the balance between precision and recall, that Dynamic Bayesian Mixture Model presents a superior performance, with an F1 score of 80.54% against the 33.31% achieved by the Long Short-Term Memory Network and 14.74% achieved by ANN.

Modeling neuroaffective biomarkers of drug addiction: A Bayesian nonparametric approach using dirichlet process mixtures

BackgroundThe properties of neurophysiological processes related to addiction have received much attention in the literature. However, empirical evidence of meaningful and useful characterization of these processes is limited. Recent studies have found that electrophysiological responses to emotional and drug-related cues can be used to create profiles that reliably predict smoking relapse. New methodThis paper evaluates the validity of classifying electrophysiological responses into distinct profiles using a Bayesian dirichlet process mixture (DPM) model. The DPM is a Bayesian nonparametric (BNP) method to modeling unknown number of profiles characterized by uncertainty in cluster membership and in cluster number. ResultsThe DPM model confirmed previously identified neuroaffective reactivity profiles, but also revealed a finer level of granularity in the clustering. Specifically, in addition to the two clusters previously identified in the literature, the BNP methods identified a cluster of individuals showing similar responses to smoking, pleasant, neutral and unpleasant cues. Comparison with existing methodsBNP models provide an alternative to the k-mean clustering approach to modeling EEG-based neuroaffective profiles. Unlike k-means clustering, BNP models compute the probability that a subject belongs to a cluster while taking into consideration uncertainty in the number of clusters. ConclusionsOur results confirm the reliability of the two clusters previously identified in these data, but also provide new insights by revealing a cluster that presented similar responses to stimuli with different contents. This finding may be related to the uncertainty in classification or overlapping brain-reactivity profiles.

BREM-SC: a bayesian random effects mixture model for joint clustering single cell multi-omics data.

Droplet-based single cell transcriptome sequencing (scRNA-seq) technology, largely represented by the 10× Genomics Chromium system, is able to measure the gene expression from tens of thousands of single cells simultaneously. More recently, coupled with the cutting-edge Cellular Indexing of Transcriptomes and Epitopes by Sequencing (CITE-seq), the droplet-based system has allowed for immunophenotyping of single cells based on cell surface expression of specific proteins together with simultaneous transcriptome profiling in the same cell. Despite the rapid advances in technologies, novel statistical methods and computational tools for analyzing multi-modal CITE-Seq data are lacking. In this study, we developed BREM-SC, a novel Bayesian Random Effects Mixture model that jointly clusters paired single cell transcriptomic and proteomic data. Through simulation studies and analysis of public and in-house real data sets, we successfully demonstrated the validity and advantages of this method in fully utilizing both types of data to accurately identify cell clusters. In addition, as a probabilistic model-based approach, BREM-SC is able to quantify the clustering uncertainty for each single cell. This new method will greatly facilitate researchers to jointly study transcriptome and surface proteins at the single cell level to make new biological discoveries, particularly in the area of immunology.

A Bayesian sparse finite mixture model for clustering data from a heterogeneous population

In this paper, we introduce a Bayesian approach for clustering data using a sparse finite mixture model (SFMM). The SFMM is a finite mixture model with a large number of components $k$ previously fixed where many components can be empty. In this model, the number of components $k$ can be interpreted as the maximum number of distinct mixture components. Then, we explore the use of a prior distribution for the weights of the mixture model that take into account the possibility that the number of clusters $k_{\mathbf{c}}$ (e.g., nonempty components) can be random and smaller than the number of components $k$ of the finite mixture model. In order to determine clusters we develop a MCMC algorithm denominated Split-Merge allocation sampler. In this algorithm, the split-merge strategy is data-driven and was inserted within the algorithm in order to increase the mixing of the Markov chain in relation to the number of clusters. The performance of the method is verified using simulated datasets and three real datasets. The first real data set is the benchmark galaxy data, while second and third are the publicly available data set on Enzyme and Acidity, respectively.

Use of Strong Habitat–Abundance Relationships in Assessing Population Status of Cryptic Fishes: An Example Using the Harlequin Darter

AbstractUnderstanding trends in abundance is important to fisheries conservation, but techniques for estimating streamwide abundance of cryptic fishes with strong habitat–abundance relationships are not well established and need further development. We developed techniques for addressing this need using the Harlequin Darter Etheostoma histrio, a small, cryptic freshwater fish associated with submerged wood in streams. Our objectives were to (1) determine how Harlequin Darter abundance and the amount of submerged wood were related at sampled sites and (2) use this relationship to estimate Harlequin Darter abundance at unsampled sites and extrapolate Harlequin Darter abundance estimates and associated uncertainty streamwide. We conducted a mark–recapture study to estimate abundance of Harlequin Darters in 25‐m stream reaches at 24 sites in Big Escambia Creek (BEC) and 18 sites in Pine Barren Creek (PBC) (Escambia River tributaries in northwestern Florida). The number of wood pieces (submerged wood ≥1.5 m long and ≥0.25 m in circumference) in both creeks was counted and mapped using side‐scan sonar and a geographic information system. Harlequin Darter and wood data were used in a Bayesian multinomial mixture model to estimate site abundance of Harlequin Darters, to determine the relationship between wood and Harlequin Darter abundance, and to extrapolate Harlequin Darter abundance streamwide. We found a positive relationship between wood and Harlequin Darter abundance in both creeks, and there were more wood pieces in PBC than in BEC. Streamwide abundance of Harlequin Darters was greater in PBC than in BEC. The extrapolated streamwide abundance estimates were 9,369 Harlequin Darters (95% credible interval = 6,668–13,402) in PBC and 7,439 Harlequin Darters (95% credible interval = 4,493–11,226) in BEC. Our methods effectively estimated abundance of a small, cryptic fish that uses complex wood habitat. In addition, our findings may assist in the conservation of the Harlequin Darter.

Application of VBGMM for pitch type classification: analysis of TrackMan's pitch tracking data

In the game of baseball, each pitcher throws various types of pitches, such as cutter, curve ball, slider, and splitter. Although the type of a given pitch may be inferred by audience and/or obtained from the TrackMan data, the actual pitch type (i.e., the pitch type declared by the pitcher) may not be known. Classification of pitch types is a challenging task, as pitched baseballs may have different kinematic characteristics across pitchers even if the self-declared pitch types are the same. In addition, there is a possibility that the kinematic characteristics of pitched baseballs are identical even if the self-declared pitch types are different. In this study, we aimed to classify TrackMan data of pitched baseballs into pitch types by applying the Variational Bayesian Gaussian Mixture Models technique. We also aimed to analyze the kinematic characteristics of the classified pitch types and indices related to batting performance while pitching each pitch type. The results showed that the pitch types could not be accurately classified solely by kinematic characteristics, but with consideration of the characteristics of the fastball the accuracy improves substantially. This study could provide a basis for the development of a more accurate automatic pitch type classification system.

Approximating Predictive Probabilities of Gibbs-Type Priors

Gibbs-type random probability measures, or Gibbs-type priors, are arguably the most “natural” generalization of the celebrated Dirichlet prior. Among them the two parameter Poisson–Dirichlet prior certainly stands out in terms of mathematical tractability and interpretability of its predictive probabilities, which made it the natural candidate in a plethora of applications. Given a random sample of size n from an arbitrary Gibbs-type prior, we show that the corresponding predictive probabilities admit a large n approximation, with an error term vanishing as o(1/n), which maintains the same desirable features as the predictive probabilities of the two parameter Poisson–Dirichlet prior. Our result is illustrated through an extensive simulation study, which includes an application in the context of Bayesian nonparametric mixture modeling.

Identifying strong lenses with unsupervised machine learning using convolutional autoencoder

ABSTRACTIn this paper, we develop a new unsupervised machine learning technique comprised of a feature extractor, a convolutional autoencoder, and a clustering algorithm consisting of a Bayesian Gaussian mixture model. We apply this technique to visual band space-based simulated imaging data from the Euclid Space Telescope using data from the strong gravitational lenses finding challenge. Our technique promisingly captures a variety of lensing features such as Einstein rings with different radii, distorted arc structures, etc., without using predefined labels. After the clustering process, we obtain several classification clusters separated by different visual features which are seen in the images. Our method successfully picks up ∼63 per cent of lensing images from all lenses in the training set. With the assumed probability proposed in this study, this technique reaches an accuracy of 77.25 ± 0.48 per cent in binary classification using the training set. Additionally, our unsupervised clustering process can be used as the preliminary classification for future surveys of lenses to efficiently select targets and to speed up the labelling process. As the starting point of the astronomical application using this technique, we not only explore the application to gravitationally lensed systems, but also discuss the limitations and potential future uses of this technique.

Bayesian central statistical monitoring using finite mixture models in multicenter clinical trials

BackgroundCentral monitoring (CM), in which data across all clinical sites are monitored, has an important role in risk-based monitoring. Several statistical methods have been proposed to compare patient outcomes among the sites for detecting atypical sites that have different trends in observed data. These methods assume that the number of clinical sites is not small, e.g., 100 or more. In addition, the proportion of atypical sites is assumed to be relatively small. However, in actuality, the central statistical monitoring (CSM) has to be implemented in small or moderate sized clinical trials such as small phase II clinical trials. The number of sites is no longer large in such situations. Therefore, it is of concern that existing methods may not work efficiently in CM of small or moderate sized clinical trials. In the light of this problem, we propose a Bayesian CSM method to detect atypical sites as the robust method against the existence of atypical sites. MethodsWe use Bayesian finite mixture models (FMM) to model patient outcome values of both atypical and typical sites. In the method, the distributions of outcome values in normal sites are determined by choosing the body distribution, which has the largest mixture parameter value of finite mixture models based on the assumption that normal sites are in the majority. Atypical sites are detected by the criterion based on the posterior predictive distribution of normal site's outcome values derived from only the chosen body distribution. ResultsProposed method is evaluated by cumulative detection probability and type I error averaged over sites every round of CSM under the various scenarios, being compared with the conventional type analysis. If the total number of patients enrolled is 48, the proposed method is superior at least 10% for any shift sizes at the 2nd and the 3rd rounds. If the total number of patients is 96, both methods show similar detection probability for only one atypical site and large shift size. However, the proposed method is superior for the other scenarios. It is observed that all the type I errors averaged over sites are little difference between the methods at all the scenarios. ConclusionWe propose a Bayesian CSM method which works efficiently in a practical use of CM. It is shown that our method detects atypical sites with high probability regardless of the proportion of the atypical sites under the small clinical trial settings which is the target of our proposed method.

Probability density function of ocean noise based on a variational Bayesian Gaussian mixture model.

Extensive ocean noise records have kurtoses markedly different from the Gaussian distribution and therefore exhibit non-Gaussianity, which influences the performance of many sonar signal processing methods. To model the amplitude distribution, this paper studies a Bayesian Gaussian mixture model (BGMM) and its associated learning algorithm, which exploits the variational inference method. The most compelling feature of the BGMM is that it automatically selects a suitable number of effective components and then can approximate a sophisticated distribution in practical applications. The probability density functions (PDFs) of three types of noise in different frequency bands collected in the South China Sea-ambient noise, ship noise, and typhoon noise-are modeled and the goodness of fit is examined by applying the one-sample Kolmogorov-Smirnov test. The results demonstrate that: (i) Ambient noise in the low-frequency band may be slightly non-Gaussian, ship noise in each considered band is apparently non-Gaussian, and typhoons affect the noise in the low-frequency band to make it apparently non-Gaussian, while the noise in the high-frequency band is less affected and appears to be Gaussian. (ii) BGMM has higher goodness of fit than the Gaussian or Gaussian mixture model. (iii) In the non-Gaussian case, despite some components having small mixing coefficients, they are of great significance for describing the PDF.

Implementation of Bayesian Mixture Models in identifying subpopulation of breast cancer patients based on blood test measurements

A complete blood test is one of a series of initial examinations of cancer patients that is relatively easy. The use of blood measurement components in analysing patient conditions is commonly used. However, it is not the case for the ratio and inter-ratio components of blood measurements, and this is what is proposed in this study. The built hypothesis is that the ratio and inter-ratio components of blood tests that can explain the condition of cancer patients are better than the blood test’s own components. An analysis will also be conducted to develop a patient profile based on these measurements, and those that can clearly distinguish between patient groups will be identified. The Finite Mixture Model is a method for modelling heterogeneous data that may originate from different subpopulations, where subpopulations represent groups of patients based on a particular latent condition. This model takes the form of a superposition of several distributions, which in this study, a Gaussian distribution is used. The parameter estimation used in the Bayesian method, which determines the prior distribution of the model parameters, and it is combined with the likelihood which will produce a posterior distribution. Then, the Markov Chain Monte Carlo-Gibbs Sampler is used to draw samples on the parameters of the posterior distribution. By using the breast cancer patient blood test data from the Oncology Department of a hospital in Jakarta, with 100,000 iterations as burn-in, and 200,000 iterations for sampling, based on Deviance Information Criterion values, the optimal grouping is two subpopulations using blood ratio and inter-ratio measurements. Two subpopulations were identified, with the first population is characterized by low distribution value and the second subpopulation with the opposite characteristics. The explanatory factors of ratio data are ratio neutrophils to lymphocytes, ratio platelets to lymphocytes, and ratio lymphocytes to monocytes.

A Joint Mass–Radius–Period Distribution of Exoplanets

Abstract The radius–period distribution of exoplanets has been characterized by the Kepler survey, and the empirical mass–radius relation by the subset of Kepler planets with mass measurements. We combine the two in order to constrain the joint mass–radius–period distribution of Kepler transiting planets. We employ hierarchical Bayesian modeling and mixture models to formulate four models with varying complexity and fit these models to the data. We find that the most complex models that treat planets with significant gaseous envelopes, evaporated core planets, and intrinsically rocky planets as three separate populations are preferred by the data and provide the best fit to the observed distribution of Kepler planets. We use these models to calculate occurrence rates of planets in different regimes and to predict masses of Kepler planets, revealing the model-dependent nature of both. When using models with envelope mass loss to calculate η ⊕, we find nearly an order of magnitude drop, indicating that many Earth-like planets discovered with Kepler may be evaporated cores which do not extrapolate out to higher orbital periods. This work provides a framework for higher-dimensional studies of planet occurrence and for using mixture models to incorporate different theoretical populations of planets.

Inferring B cell specificity for vaccines using a Bayesian mixture model

BackgroundVaccines have greatly reduced the burden of infectious disease, ranking in their impact on global health second only after clean water. Most vaccines confer protection by the production of antibodies with binding affinity for the antigen, which is the main effector function of B cells. This results in short term changes in the B cell receptor (BCR) repertoire when an immune response is launched, and long term changes when immunity is conferred. Analysis of antibodies in serum is usually used to evaluate vaccine response, however this is limited and therefore the investigation of the BCR repertoire provides far more detail for the analysis of vaccine response.ResultsHere, we introduce a novel Bayesian model to describe the observed distribution of BCR sequences and the pattern of sharing across time and between individuals, with the goal to identify vaccine-specific BCRs. We use data from two studies to assess the model and estimate that we can identify vaccine-specific BCRs with 69% sensitivity.ConclusionOur results demonstrate that statistical modelling can capture patterns associated with vaccine response and identify vaccine specific B cells in a range of different data sets. Additionally, the B cells we identify as vaccine specific show greater levels of sequence similarity than expected, suggesting that there are additional signals of vaccine response, not currently considered, which could improve the identification of vaccine specific B cells.

A Bayesian sample selection model based on normal mixture to investigate household car ownership and usage behavior

Selection bias is an important issue in analyzing household car ownership and usage behavior. If it is not well considered in the modeling process, estimates will be biased. In this paper, we use a Bayesian sample selection model, which accounts for the selection bias, to investigate household car ownership and usage behavior. Employing the approach of normal mixture, the new established model relaxes the bivariate normal assumption in the traditional sample selection model and can capture flexible coupling relationship between car ownership and usage behavior. Moreover, the model does not require specifying any marginal distribution. Three cross validation experiments using simulated data suggest that the new model is effective in revealing parameters’ true values and in capturing actual error distribution. Considering overfitting issue, various tests are proposed to determine the most likely number of normal components. After testing, the new model with 3 components has a stronger explanation power in analyzing the interdependence between household car ownership and usage behavior in terms of goodness of fit and generalization ability. By comparison, estimates from the traditional normal model are seriously biased regarding magnitude, significance level, even the sign. Lastly, to test the efficiency of Bayesian normal mixture model, performance of the Copula model is evaluated. The result indicates that the normal mixture model with two components already has a strong power in capturing the general pattern of error distribution, and its goodness of fit has been impressively improved compared with the traditional normal model and the Copula model.

Use of Bayesian Mixture Models in Analyzing Heterogeneous Survival Data: A Simulation Study

Background and Aim: One of the statistical methods used to analyze the time-to-event medical data is survival analysis. In survival models, the response variable is time to the occurrence of an event. The main characteristic of survival data is the existence of censored data. When we have the distribution of survival time, we can use parametric methods. Among the important and popular distributions that can be used, we can mention the Weibull distribution. If the data derives from a heterogeneous population, simple parametric models (such as Weibull) would not fit the data appropriately. One of the methods which have been introduced to overcome this problem is the use of mixture models. Methods: To assess the validity of the two-component Weibull mixture model, we use a simulation method on heterogeneous survival data. For this purpose, data with different sample sizes were produced in a batch of 1000. Then, the validity of the model is checked using root mean square error (RMSE) criterion Results: It is obtained that increasing the sample size would decrease the RMSE in the parameters. However the maximum observed RMSE in all the parameters was negligible. Conclusion: The Bayesian Weibull mixture model was a proper fit for the heterogeneous survival data

A Classification of Streamflow Patterns Across the Coastal Gulf of Alaska

AbstractStreamflow controls many freshwater and marine processes, including salinity profiles, sediment composition, fluxes of nutrients, and the timing of animal migrations. Watersheds that border the Gulf of Alaska (GOA) comprise over 400,000 km2 of largely pristine freshwater habitats and provide ecosystem services such as reliable fisheries for local and global food production. Yet no comprehensive watershed‐scale description of current temporal and spatial patterns of streamflow exists within the coastal GOA. This is an immediate need because the spatial distribution of future streamflow patterns may shift dramatically due to warming air temperature, increased rainfall, diminishing snowpack, and rapid glacial recession. Our primary goal was to describe variation in streamflow patterns across the coastal GOA using an objective set of descriptors derived from flow predictions at the downstream‐most point within each watershed. We leveraged an existing hydrologic runoff model and Bayesian mixture model to classify 4,140 watersheds into 13 classes based on seven streamflow statistics. Maximum discharge timing (annual phase shift) and magnitude relative to mean discharge (amplitude) were the most influential attributes. Seventy‐six percent of watersheds by number showed patterns consistent with rain or snow as dominant runoff sources, while the remaining watersheds were driven by rain‐snow, glacier, or low‐elevation wetland runoff. Streamflow classes exhibited clear mechanistic links to elevation, ice coverage, and other landscape features. Our classification identifies watersheds that might shift streamflow patterns in the near future and, importantly, will help guide the design of studies that evaluate how hydrologic change will influence coastal GOA ecosystems.

A hierarchical Bayesian latent class mixture model with censorship for detection of linear temporal changes in antibiotic resistance.

Identifying and controlling the emergence of antimicrobial resistance (AMR) is a high priority for researchers and public health officials. One critical component of this control effort is timely detection of emerging or increasing resistance using surveillance programs. Currently, detection of temporal changes in AMR relies mainly on analysis of the proportion of resistant isolates based on the dichotomization of minimum inhibitory concentration (MIC) values. In our work, we developed a hierarchical Bayesian latent class mixture model that incorporates a linear trend for the mean log2MIC of the non-resistant population. By introducing latent variables, our model addressed the challenges associated with the AMR MIC values, compensating for the censored nature of the MIC observations as well as the mixed components indicated by the censored MIC distributions. Inclusion of linear regression with time as a covariate in the hierarchical structure allowed modelling of the linear creep of the mean log2MIC in the non-resistant population. The hierarchical Bayesian model was accurate and robust as assessed in simulation studies. The proposed approach was illustrated using Salmonella enterica I,4,[5],12:i:- treated with chloramphenicol and ceftiofur in human and veterinary samples, revealing some significant linearly increasing patterns from the applications. Implementation of our approach to the analysis of an AMR MIC dataset would provide surveillance programs with a more complete picture of the changes in AMR over years by exploring the patterns of the mean resistance level in the non-resistant population. Our model could therefore serve as a timely indicator of a need for antibiotic intervention before an outbreak of resistance, highlighting the relevance of this work for public health. Currently, however, due to extreme right censoring on the MIC data, this approach has limited utility for tracking changes in the resistant population.

A fully Bayesian mixture model approach for identifying noncompliance in a regulatory tobacco clinical trial.

Identifying noncompliance in a randomized trial is challenging, but could be improved by leveraging biomarker data to identify participants that did not comply with their assigned treatment. For randomized trials of very low nicotine content (VLNC) cigarettes, the biomarker of total nicotine equivalents (TNE) could be used to identify noncompliance. Compliant participants should have lower levels of TNEs than participants that did not comply and smoked normal nicotine content cigarettes, resulting in a mixture of compliant and noncompliant participants at each dose level. Thresholds of TNE could then be identified from the compliant groups at each dose level and used to determine which study participants were compliant. Furthermore, proposed biological relationships of TNE with nicotine dose could be incorporated into improve the efficiency of estimation, but may introduce bias if misspecified. To account for multiple modeling assumptions across dose levels, we explore model averaging via reversible jump markov chain monte carlo (MCMC) within each dose level to take advantage of improvements in efficiency when the proposed relationship is true and to downweight the biological model when it is misspecified. In simulation studies, we demonstrate that model averaging in the presence of a correct biological relationship results in a decrease in the mean square error (MSE) of up to 85%, but downweights the model in dose levels where the relationship is not appropriate. We apply our approach to data from a randomized trial of VLNC cigarettes to estimate TNE thresholds and probability of compliance curves as a function of TNEs for each nicotine dose used in the trial.