Multinomial Distribution Research Articles

Damage Effectiveness Calculation of Hitting Targets with Ammunition Based on Bayesian Multinomial Distribution

Owning to the fact that ammunition can cause varying degrees of damage to its target, this article presents a damage effectiveness calculation method of hitting targets with ammunition based on Bayesian multinomial distribution to solve the problems of complex processes, few trial times and difficult calculations of damage probability in target-hitting tests with high-tech ammunition, according to a calculation index of damage effectiveness about the occurrence probability of different damage. Based on the concept of symmetry, the idea of “divide damage level—determine distribution—integrate information—solve distribution” is adopted. Firstly, this paper describes the damage effectiveness test of ammunition attacking targets with multiple distributions; secondly, this paper integrates the damage effectiveness information of ammunition strike targets with Dempster–Shafer evidence theory (D–S evidence theory) and symmetry advantage; finally, this paper attempts to solve the symmetric posterior distribution of damage effectiveness parameters with Bayesian theory and the Markov chain Monte Carlo (MCMC) method. The result demonstrates that this method is very significant in improving the calculation accuracy of ammunition damage effectiveness, which could describe the damage effectiveness of ammunition in detail by integrating the prior information with multiple types of damage effectiveness.

Analysis and Allocation of Cancer-Related Genes Using Vague DNA Sequence Data.

To test the equality of several independent multinomial distributions, the chi-square test for count data is applied. The existing test can be applied when complete information about the data is available. The complex process, such as DNA count, the existing test under classical statistics may mislead. To overcome the issue, the modification of the chi-square test for multinomial distribution under neutrosophic statistics is presented in this paper. The modified form of the chi-square test statistic under indeterminacy/uncertainty is presented and applied using the DNA count data. From the DNA count data analysis, simulation, and comparative studies, the proposed test is found to be informative, springy, and good as compared with the existing tests.

A120 AUTOMATED BOWEL PREPARATION DETECTION WITH DEEP. CONVOLUTIONAL NEURAL NETWORKS

BackgroundIntroduction: Bowel preparation inadequacy has been shown to increase post-colonoscopy colorectal cancer. As such, the USMSTF recommends repeating colonoscopy within 1 year if bowel preparation is inadequate. However, bowel preparation documentation is variable in clinical practice, and physician recommendations adherent to USMTF guidelines are inconsistent.AimsAims: We present an automated computer assisted method using deep convolutional neural networks to detect bowel preparation and adequacy of bowel preparation with the Boston Bowel Preparation Scale (BBPS).MethodsMethods: We extracted 38523 images of colonic lumen between 2015 and 2017 from screening colonoscopies. Bowel preparation scores were assessed with BBPS. Adequate bowel preparation was defined as BBPS ≥2, and inadequate bowel preparation was defined as BBPS <2. The dataset was split into 26966 images for training, 7704 for validation, and 3853 for testing. Training data was sampled with replacement from a multinomial distribution to balance subclass distributions in each batch. We developed 2 convoluted neural networks (CNN) using PyTorch with a Densenet-169 backbone pre-trained on ImageNet and fine-tuned on our data for classifying adequacy of bowel preparation (binary) and for subclassification of BBPS (multi-class). We used Adam optimiser with an initial learning rate of 3x10-4 and a scheduler to decay the learning rate of each parameter group by 0.1 every 7 epochs along with focal loss as our criterion for both classifiers.ResultsResults: The overall accuracy on the test data set for BBPS subclassification was 0.91. The sensitivity for BBPS 0, 1, 2 and 3 were 0.84, 0.91, 0.86, and 0.96, respectively. The specificity for BBPS 0, 1, 2, and 3 were 1.00, 0.98, 0.95, and 0.93, respectively. The overall accuracy of the test data set for adequacy of bowel preparation was 0.97. The sensitivity for adequacy of bowel preparation for BBPS <2 and BBPS ≥2 was 0.92 and 0.99, respectively. The specificity for adequacy of bowel preparation for BBPS <2 and BBPS ≥2 was 0.99 and 0.92, respectively.Conclusions Conclusion: We present an automated computer-assisted detection method of bowel preparation with deep convolutional neural networks. The algorithm is capable of accurate classification of adequacy of bowel preparation (97%) and subclassification of bowel preparation (91%) with high sensitivity and specificity. This algorithm can be applied to automate documentation of bowel preparation and adequacy of bowel preparation. Additional studies will need to be conducted to demonstrate its applicability in real-time colonoscopy.Funding AgenciesNone

Improving estimations of life history parameters of small animals in mesocosm experiments: A case study on mosquitoes

AbstractMesocosm experiments enable researchers to study animal dynamics, but determining accurate estimates of survival and development rates of different life stages can be difficult, especially as the subjects may be hard to sample and mortality rates can be high. We propose a new methodology for estimating such parameters.We used an experimental set‐up with 48 aquatic mesocosms, each with 20 first instar mosquito larvae and under 1 of 12 treatments with varying temperatures and nutrient concentrations. We took daily subsamples of the aquatic life stages as well as counting the emerging adults. We developed a method to estimate the survival and development probabilities at each life stage, based on optimising a matrix population model. We used two different approaches, one assuming the difference between predictions and observations was normally distributed, and the other using a combination of a normal and a multinomial distribution. For each approach, the resulting optimisation problem had around 100 parameters, making conventional gradient descent ineffective with our limited number of data points. We solved this by computing the formal derivatives of our matrix model.Both approaches proved effective in predicting mosquito populations over time, also when compared against a separate validation dataset, and the two approaches produced similar results. They also both predicted similar trends in the survival and development probabilities for each life stage, although there were some differences in the actual values. The approach which only used the normal distribution was considerably more computationally efficient than the mixed distribution approach.This is an effective approach for determining the survival and development rates of small animals in mesocosm experiments. We have not found any other reliable methodology for estimating these parameters, especially not from incomplete data or when there are many different experimental treatments. This methodology enables researchers to gain a much more detailed understanding of the life cycles of small animals, potentially leading to advances in a wide range of areas, for example in mosquito‐borne disease risk or in considering the effects of biodiversity loss or climate change on different species.

Hierarchical Bayesian integrated model for estimating migratory bird harvest in Canada

AbstractThe Canadian Wildlife Service (CWS) requires reliable estimates of the harvest of migratory game birds, including waterfowl, to effectively manage populations of these hunted species. The National Harvest Survey is an annual survey of hunters who purchase Canada's mandatory migratory game bird hunting permit, integrating information from a survey of hunting activity with information from a separate survey of species composition in the harvest. We used these survey data to estimate the number of birds harvested for each species and hunting activity metrics (e.g., number of active hunters, days spent hunting). The analytical methods used to generate these estimates have not changed since the survey was first designed in the early 1970s. We describe a new hierarchical Bayesian integrated model, which replaces the series of ratio estimators that comprised the old model. We are using this new model to generate estimates for migratory bird harvests as of the 2019–2020 hunting season, and to generate updated estimates for all earlier years. The hierarchical Bayesian model uses over‐dispersed Poisson distributions to model mean hunter activity and harvest (zero‐inflated Poisson and zero‐truncated Poisson, respectively). It also includes multinomial distributions to model some key components (e.g., variation in harvest across periods of the hunting season, the species composition of the harvest within each of those periods, the age and sex composition in the harvests of a given species). We estimated the parameters of the Poisson and the multinomial distributions for each year as random effects using first‐difference time‐series. This time‐series component allows the model to share information across years and reduces the sensitivity of the estimates to annual sampling noise. The new model estimates are generally very similar to those from the old model, particularly for the species that occur most commonly in the harvest, so the results do not suggest any major changes to harvest management decisions and regulations. Estimates for all species from the new model are more precise and less susceptible to annual sampling error, particularly for species that occur less commonly in the harvest (e.g., sea ducks, other species of conservation concern). This new model, with its hierarchical Bayesian framework, will also facilitate future improvements and elaborations, allowing the incorporation of prior information from the rich literature and knowledge in game bird management and biology.

Low-rank matrix denoising for count data using unbiased Kullback-Leibler risk estimation

Many statistical studies are concerned with the analysis of observations organized in a matrix form whose elements are count data. When these observations are assumed to follow a Poisson or a multinomial distribution, it is of interest to focus on the estimation of either the intensity matrix (Poisson case) or the compositional matrix (multinomial case) when it is assumed to have a low rank structure. In this setting, it is proposed to construct an estimator minimizing the regularized negative log-likelihood by a nuclear norm penalty. Such an approach easily yields a low-rank matrix-valued estimator with positive entries which belongs to the set of row-stochastic matrices in the multinomial case. Then, as a main contribution, a data-driven procedure is constructed to select the regularization parameter in the construction of such estimators by minimizing (approximately) unbiased estimates of the Kullback-Leibler (KL) risk in such models, which generalize Stein's unbiased risk estimation originally proposed for Gaussian data. The evaluation of these quantities is a delicate problem, and novel methods are introduced to obtain accurate numerical approximation of such unbiased estimates. Simulated data are used to validate this way of selecting regularizing parameters for low-rank matrix estimation from count data. For data following a multinomial distribution, the performances of this approach are also compared to K-fold cross-validation. Examples from a survey study and metagenomics also illustrate the benefits of this methodology for real data analysis.

The value function of a transportation problem

We investigate the value of an optimal transportation problem with the maximization objective as a function of costs and vectors of production and consumption. The value is concave in production. For generic costs, the numbers of linearity domains and peak points are independent of costs and consumption. The peak points are determined by an auxiliary assignment problem. The volumes of the linearity domains are independent of costs while their dependence on consumption can be expressed via the multinomial distribution.

The Risk Measurement under the Variance-Gamma Process with Drift Switching

The paper discusses an extension of the variance-gamma process with stochastic linear drift coefficient. It is assumed that the linear drift coefficient may switch to a different value at the exponentially distributed time. The size of the drift jump is supposed to have a multinomial distribution. We have obtained the distribution function, the probability density function and the lower partial expectation for the considered process in closed forms. The results are applied to the calculation of the value at risk and the expected shortfall of the investment portfolio in the related multivariate stochastic model.

Birth Size and Maternal, Social, and Environmental Factors in the Province of Jujuy, Argentina.

Introduction: birth size is affected by diverse maternal, environmental, social, and economic factors. Aim: analyze the relationships between birth size—shown by the indicators small for gestational age (SGA) and large for gestational age (LGA)—and maternal, social, and environmental factors in the Argentine province of Jujuy, located in the Andean foothills. Methods: data was obtained from 49,185 mother-newborn pairs recorded in the Jujuy Perinatal Information System (SIP) between 2009 and 2014, including the following: newborn and maternal weight, length/height, and body mass index (BMI); gestational age and maternal age; mother’s educational level, nutritional status, marital status and birth interval; planned pregnancy; geographic-linguistic origin of surnames; altitudinal place of birth; and unsatisfied basic needs (UBN). The dataset was split into two groups, SGA and LGA, and compared with adequate for gestational age (AGA). Bivariate analysis (ANOVA) and general lineal modeling (GLM) with multinomial distribution were employed. Results: for SGA newborns, risk factors were altitude (1.43 [1.12–1.82]), preterm birth (5.33 [4.17–6.82]), older maternal age (1.59 [1.24–2.05]), and primiparous mothers (1.88 [1.06–3.34]). For LGA newborns, the risk factors were female sex (2.72 [5.51–2.95]), overweight (1.33 [1.22–2.46]) and obesity (1.85 [1.66–2.07]). Conclusions: the distribution of birth size and the factors related to its variability in Jujuy are found to be strongly conditioned by provincial terrain and the clinal variation due to its Andean location.

On the intermediate asymptotic efficiency of goodness-of-fit tests in multinomial distributions

We consider goodness-of-fit tests for uniformity of a multinomial distribution by means of tests based on a class of symmetric statistics, defined as the sum of some function of cell-frequencies. We are dealing with an asymptotic regime, where the number of cells grows with the sample size. Most attention is focused on the class of power divergence statistics. The aim of this article is to study the intermediate asymptotic relative efficiency of two tests, where the powers of the tests are asymptotically non-degenerate and the sequences of alternatives converge to the hypothesis, but not too fast. The intermediate asymptotic relative efficiency of the χ2 test wrt an arbitrary symmetric test is considered in details.

Sparse Count Data Clustering Using an Exponential Approximation to Generalized Dirichlet Multinomial Distributions.

Clustering frequency vectors is a challenging task on large data sets considering its high dimensionality and sparsity nature. Generalized Dirichlet multinomial (GDM) distribution is a competitive generative model for count data in terms of accuracy, yet its parameters estimation process is slow. The exponential-family approximation of the multivariate Polya distribution has shown to be efficient to train and cluster data directly, without dimensionality reduction. In this article, we derive an exponential-family approximation to the GDM distributions, and we call it (EGDM). A mixture model is developed based on the new member of the exponential-family of distributions, and its parameters are learned through the deterministic annealing expectation-maximization (DAEM) approach as a new clustering algorithm for count data. Moreover, we propose to estimate the optimal number of EGDM mixture components based on the minimum message length (MML) criterion. We have conducted a set of empirical experiments, concerning text, image, and video clustering, to evaluate the proposed approach performance. Results show that the new model attains a superior performance, and it is considerably faster than the corresponding method for GDM distributions.

Hybrid DNN-Dirichlet Anomaly Detection and Ranking: Case of Burned Areas Discovery

In the past decade, anomaly detection has experienced an expanding attraction in satellite data analysis. Monitoring wildfire dynamics plays a substantial part in global land management, i.e., to detect and determine the expansion of such areas, estimate the deterioration of forest regions, and assist the intervention plan. In this paper, we proposed an approach for Sentinel-2 scenes that detects anomalies in burned area contexts using a rank-ordered method on a single post-event image. We adopted a self-supervised paradigm in learning image representations by training a deep convolutional model to differentiate between a series of geometric transformations. Dirichlet distributions are selected as priors to characterize the variability of random multinomial distribution in multispectral data. Dirichlet precision parameters are computed from observed data and are used to construct a ranking function that quantifies the degree of anomaly in data based on softmax responses given by the trained classifier. We evaluated the performance of the proposed method on a cumulative effort of two remote sensing tasks, open-set detection (i.e., test datasets contain classes unseen at the training time) and location separation (i.e., test datasets include images from distinct spatial location than the training images). The experiments were performed on three different datasets, BigEarthNet and two actual burned area Sentinel-2 datasets, i.e., from predisposed zones to fire events, Australia, respectively Bolivia. We retained in all test datasets state-of-the-art performance, considering the substantial and diversified types of natural anomalies in multispectral data.

Proactive Radar Protection System in Shared Spectrum via Forecasting Secondary User Power Levels

Spectrum sharing in radar bands with interference forecasting for enhanced radar protection can help design proactive resource allocation solutions which can achieve high data rates for wireless communication networks on one hand and help protect the incumbent radar systems. We consider radar spectrum sharing in 5.6GHz where a weather radar operates as a primary system and the dominant secondary system is an enterprise network consisting of access points (APs) in a university campus. Our work models transmit the power of the APs as a time series with multinomial distribution based on real collected data. The aggregated interference due to the transmissions from the APs at the radar is forecasted using a long short-term memory (LSTM) based neural network. Monte Carlo dropout is utilized to generate prediction intervals that capture the uncertainties in the interference from the APs. Finally, by using both average and upper limits of predicted interference time series a cloud-assisted efficient sharing and radar protection algorithm is proposed. Tracking the rotating radar is not required in the proposed system. The results show that the proposed efficient sharing and radar protection system ensures better radar protection and increased throughput for wireless communication users.

Multinomial and Hypergeometric Distributions in Markov Categories

Markov categories, having tensors with copying and discarding, provide a setting for categorical probability. This paper uses finite colimits and what we call uniform states in such Markov categories to define a (fixed size) multiset functor, with basic operations for sums and zips of multisets, and a graded monad structure. Multisets can be used to represent both urns filled with coloured balls and also draws of multiple balls from such urns. The main contribution of this paper is the abstract definition of multinomial and hypergeometric distributions on multisets, as draws. It is shown that these operations interact appropriately with various operations on multisets.

Efficient Bayesian estimation of permutation entropy with Dirichlet priors

Estimation of permutation entropy (PE) using Bayesian statistical methods is presented for systems where the ordinal pattern sampling follows an independent, multinomial distribution. The desired PE posterior distribution is demonstrated to closely approximate a standard Beta distribution whose hyperparameters can be computed directly from ordinal pattern counts. It is further shown that this Bayesian approach is a generalization of previously-published frequentist methods. Because Bayesian posterior distributions can be estimated for very short time series, this method enables PE analysis on data sets currently not compatible with existing methods due to their limited size. To demonstrate the power and flexibility of this technique, the PE of a semiconductor laser with optical feedback (SLWOF) is shown to be time-variant by visualizing how the PE posterior distribution varies over sequences of small (1000 point) partitions of the output time-series.

A flexible framework for intervention analysis applied to credit-card usage during the coronavirus pandemic

We develop a variant of intervention analysis designed to measure a change in the law of motion for the distribution of individuals in a cross-section, rather than modeling the moments of the distribution. To calculate a counterfactual forecast, we discretize the distribution and employ a Markov model in which the transition probabilities are modeled as a multinomial logit distribution. Our approach is scalable and is designed to be applied to micro-level data. A wide panel often carries with it several imperfections that complicate the analysis when using traditional time-series methods; our framework accommodates these imperfections. The result is a framework rich enough to detect intervention effects that not only shift the mean, but also those that shift higher moments, while leaving lower moments unchanged. We apply this framework to document the changes in credit usage of consumers during the COVID-19 pandemic. We consider multinomial logit models of the dependence of credit-card balances, with categorical variables representing monthly seasonality, homeownership status, and credit scores. We find that, relative to our forecasts, consumers have greatly reduced their use of credit. This result holds for homeowners and renters as well as consumers with both high and low credit scores.

1064. Treatment Success in Reducing Recurrent Clostridioides difficile Infection with Investigational Live Biotherapeutic RBX2660 Is Associated with Microbiota Restoration: Consistent Evidence from a Phase 3 Clinical Trial

BackgroundSeveral investigational microbiota-based live biotherapeutics are in clinical development for reducing recurrence of Clostridioides difficile infection (rCDI), including RBX2660 a liquid suspension of a broad consortium of microbiota, which includes Bacteridetes and Firmicutes. RBX2660 has been evaluated in >600 participants in 6 clinical trials. Here we report that RBX2660 induced significant shifts to the intestinal microbiota of treatment-responsive participants in PUNCH CD3—a Phase 3 randomized, double-blinded, placebo-controlled trial.MethodsPUNCH CD3 participants received a single dose of RBX2660 or placebo between 24 to 72 hours after completing rCDI antibiotic treatment. Clinical response was the absence of CDI recurrence at eight weeks after treatment. Participants voluntarily submitted stool samples prior to blinded study treatment (baseline), 1, 4 and 8 weeks, 3 and 6 months after receiving study treatment. Samples were extracted and sequenced using shallow shotgun methods. Operational taxonomic unit (OTU) data were used to calculate relative taxonomic abundance, alpha diversity, and the Microbiome Health Index (MHI)—a biomarker of antibiotic-induced dysbiosis and restoration.ResultsClinically, RBX2660 demonstrated superior efficacy versus placebo (70.4% versus 58.1%). From before to after treatment, RBX2660-treated clinical responders’ microbiome diversity shifted significantly (Mann-Whitney), and so did microbiome composition (Generalized Wald Test). Post-treatment changes were characterized by increased Bacteroidia and Clostridia and decreased Gammaproteobacteria and Bacilli, changes and were durable to at least 6 months. Repeated measures analysis confirmed that shifts were greater among RBX2660 responders compared to placebo responders (DMRepeat). The majority of responders’ MHI values shifted from a range common to antibiotic dysbiosis to a range common in healthy populations.Figure 1 Left panel. Mean relative abundance taxonomic class level at timepoints for participants in PUNCH CD3 before and after RBX2660 treatment, and for doses of RBX2660 administered in PUNCH CD3. The four taxonomic classes that change most from before to after treatment are shown with the mean and confidence intervals based on fitting OTU data to a Dirichlet multinomial distribution. Right panel, MHI biomarker for the same time points and investigational product groups, shown as median (red) and individual samples. A previously calculated threshold of MHI = 7.2 is shown (dotted line), above which MHI values predict healthy, below which MHI values predict antibiotic-induced dysbiosis.ConclusionAmong PUNCH CD3 clinical responders, RBX2660 significantly restored microbiota from less to more healthy compositions, and this restoration was durable to at least 6 months. These clinically-correlated microbiome shifts are highly consistent with results from multiple prior trials of RBX2660.Disclosures Ken Blount, PhD, Rebiotix Inc., a Ferring Company (Employee) Dana M. Walsh, PhD, Rebiotix (Employee)

1039. Rapid Restoration of Bile Acid Compositions After Treatment with RBX2660 for Recurrent Clostridioides difficile Infection—Results from the PUNCH CD3 Phase 3 Trial

BackgroundMicrobiota-based treatments are increasingly evaluated as a strategy to reduce recurrence of Clostridioides difficile infection (rCDI), and their proposed mechanisms include restoration of the microbiota and microbiota-mediated functions, including bile acid metabolism. RBX2660—a broad-consortium investigational live biotherapeutic—has been evaluated in >600 participants in 6 clinical trials, with consistent reduction of rCDI recurrence. Here we report that fecal bile acid compositions were significantly restored in treatment-responsive participants in PUNCH CD3—a Phase 3 randomized, double-blinded, placebo-controlled trial of RBX2660.MethodsPUNCH CD3 participants received a single dose of RBX2660 or placebo between 24 to 72 hours after completing rCDI antibiotic treatment. Clinical response was the absence of CDI recurrence at eight weeks after treatment. Participants voluntarily submitted stool samples prior to blinded study treatment (baseline), 1, 4 and 8 weeks, 3 and 6 months after receiving study treatment. A liquid chromatography tandem mass spectrometry method was developed to extract and quantify 33 bile acids from all participant fecal samples received up to the 8-week time point. Mean bile acid compositions were fit to a Dirichlet multinomial distribution and compared across time points and between RBX2660- and placebo-treated participants.ResultsClinically, RBX2660 demonstrated superior efficacy versus placebo (70.4% versus 58.1%). RBX2660-treated clinical responders’ bile acid compositions shifted significantly from before to after treatment. Specifically, primary bile acids predominated before treatment, whereas secondary bile acids predominated after treatment (Figure 1A). These changes trended higher among RBX2660 responders compared to placebo responders. Importantly, median levels of lithocholic acid (LCA) and deoxycholic acid (DCA) showed large, significant increases after treatment (Figure 1B). A. Bile acid compositions before (BL) and up to 8 weeks after RBX2660 treatment among treatment responders. Compositions are shown as the fraction of total bile acids classified as primary or secondary conjugated or deconjugated bile acids. B. Concentrations of lithocholic acid (LCA) and deoxycholic acid (DCA) among RBX2660 treatment responders, shown with individual samples and time point group median with interquartile ranges.ConclusionAmong PUNCH CD3 clinical responders, RBX2660 significantly restored bile acids from less to more healthy compositions. These clinically correlated bile acid shifts are highly consistent with results from a prior trial of RBX2660.Disclosures Romeo Papazyan, PhD, Ferring Research Institute (Employee) Bryan Fuchs, PhD, Ferring Pharmaceuticals (Employee) Ken Blount, PhD, Rebiotix Inc., a Ferring Company (Employee)

Statistical Model of Correlation Difference and Related-Key Linear Cryptanalysis

The goal of this work is to propose a related-key model for linear cryptanalysis. We start by giving the mean and variance of the difference of sampled correlations of two Boolean functions when using the same sample of inputs to compute both correlations. This result is further extended to determine the mean and variance of the difference of correlations of a pair of Boolean functions taken over a random data sample of fixed size and over a random pair of Boolean functions. We use the properties of the multinomial distribution to achieve these results without independence assumptions. Using multivariate normal approximation of the multinomial distribution we obtain that the distribution of the difference of related-key correlations is approximately normal. This result is then applied to existing related-key cryptanalyses. We obtain more accurate right-key and wrong-key distributions and remove artificial assumptions about independence of sampled correlations. We extend this study to using multiple linear approximations and propose a Χ2-type statistic, which is proven to be Χ2 distributed if the linear approximations are independent. We further examine this statistic for multidimensional linear approximation and discuss why removing the assumption about independence of linear approximations does not work in the related-key setting the same way as in the single-key setting.

A First Approach to Closeness Distributions

Probabilistic graphical models allow us to encode a large probability distribution as a composition of smaller ones. It is oftentimes the case that we are interested in incorporating in the model the idea that some of these smaller distributions are likely to be similar to one another. In this paper we provide an information geometric approach on how to incorporate this information and see that it allows us to reinterpret some already existing models. Our proposal relies on providing a formal definition of what it means to be close. We provide an example on how this definition can be actioned for multinomial distributions. We use the results on multinomial distributions to reinterpret two already existing hierarchical models in terms of closeness distributions.