# Bayesian Binary Review

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Contents

## Bayesian Binary Review

Applied researchers interested in Bayesian statistics are increasingly attracted to R because of the ease of which one can code algorithms to sample from posterior distributions as well as the significant number of packages contributed to the Comprehensive R Archive Network (CRAN) that provide tools for Bayesian inference. This task view catalogs these tools. In this task view, we divide those packages into four groups based on the scope and focus of the packages. We first review R packages that provide Bayesian estimation tools for a wide range of models. We then discuss packages that address specific Bayesian models or specialized methods in Bayesian statistics. This is followed by a description of packages used for post-estimation analysis. Finally, we review packages that link R to other Bayesian sampling engines such as JAGS , OpenBUGS , WinBUGS , and Stan .

Bayesian packages for general model fitting

• The arm package contains R functions for Bayesian inference using lm, glm, mer and polr objects.
• BACCO is an R bundle for Bayesian analysis of random functions. BACCO contains three sub-packages: emulator, calibrator, and approximator, that perform Bayesian emulation and calibration of computer programs.
• bayesm provides R functions for Bayesian inference for various models widely used in marketing and micro-econometrics. The models include linear regression models, multinomial logit, multinomial probit, multivariate probit, multivariate mixture of normals (including clustering), density estimation using finite mixtures of normals as well as Dirichlet Process priors, hierarchical linear models, hierarchical multinomial logit, hierarchical negative binomial regression models, and linear instrumental variable models.
• LaplacesDemon seeks to provide a complete Bayesian environment, including numerous MCMC algorithms, Laplace Approximation with multiple optimization algorithms, scores of examples, dozens of additional probability distributions, numerous MCMC diagnostics, Bayes factors, posterior predictive checks, a variety of plots, elicitation, parameter and variable importance, and numerous additional utility functions.
• MCMCpack provides model-specific Markov chain Monte Carlo (MCMC) algorithms for wide range of models commonly used in the social and behavioral sciences. It contains R functions to fit a number of regression models (linear regression, logit, ordinal probit, probit, Poisson regression, etc.), measurement models (item response theory and factor models), changepoint models (linear regression, binary probit, ordinal probit, Poisson, panel), and models for ecological inference. It also contains a generic Metropolis sampler that can be used to fit arbitrary models.
• The mcmc package consists of an R function for a random-walk Metropolis algorithm for a continuous random vector.
• The nimble package provides a general MCMC system that allows customizable MCMC for models written in the BUGS/JAGS model language. Users can choose samplers and write new samplers. Models and samplers are automatically compiled via generated C++. The package also supports other methods such as particle filtering or whatever users write in its algorithm language.

Bayesian packages for specific models or methods

• abc package implements several ABC algorithms for performing parameter estimation and model selection. Cross-validation tools are also available for measuring the accuracy of ABC estimates, and to calculate the misclassification probabilities of different models.
• abn is a package for modelling multivariate data using additive Bayesian networks. It provides routines to help determine optimal Bayesian network models for a given data set, where these models are used to identify statistical dependencies in messy, complex data.
• AdMit provides functions to perform the fitting of an adapative mixture of Student-t distributions to a target density through its kernel function. The mixture approximation can be used as the importance density in importance sampling or as the candidate density in the Metropolis-Hastings algorithm.
• The BaBooN package contains two variants of Bayesian Bootstrap Predictive Mean Matching to multiply impute missing data.
• bamlss provides an infrastructure for estimating probabilistic distributional regression models in a Bayesian framework. The distribution parameters may capture location, scale, shape, etc. and every parameter may depend on complex additive terms similar to a generalized additive model.
• The BART package provide flexible nonparametric modeling of covariates for continuous, binary, categorical and time-to-event outcomes.
• The bayesGARCH package provides a function which perform the Bayesian estimation of the GARCH(1,1) model with Student’s t innovations.
• bayesImageS is an R package for Bayesian image analysis using the hidden Potts model.
• bayesmeta is an R package to perform meta-analyses within the common random-effects model framework.
• BayesSummaryStatLM provides two functions: one function that computes summary statistics of data and one function that carries out the MCMC posterior sampling for Bayesian linear regression models where summary statistics are used as input.
• BayesTree implements BART (Bayesian Additive Regression Trees) by Chipman, George, and McCulloch (2006).
• bayesQR supports Bayesian quantile regression using the asymmetric Laplace distribution, both continuous as well as binary dependent variables.
• BayesFactor provides a suite of functions for computing various Bayes factors for simple designs, including contingency tables, one- and two-sample designs, one-way designs, general ANOVA designs, and linear regression.
• BayesVarSel calculate Bayes factors in linear models and then to provide a formal Bayesian answer to testing and variable selection problems.
• BayHaz contains a suite of R functions for Bayesian estimation of smooth hazard rates via Compound Poisson Process (CPP) priors.
• BAYSTAR provides functions for Bayesian estimation of threshold autoregressive models.
• bbemkr implements Bayesian bandwidth estimation for Nadaraya-Watson type multivariate kernel regression with Gaussian error.
• BCE contains function to estimates taxonomic compositions from biomarker data using a Bayesian approach.
• BCBCSF provides functions to predict the discrete response based on selected high dimensional features, such as gene expression data.
• bcp implements a Bayesian analysis of changepoint problem using Barry and Hartigan product partition model.
• BDgraph provides statistical tools for Bayesian structure learning in undirected graphical models for multivariate continuous, discrete, and mixed data.
• BLR provides R functions to fit parametric regression models using different types of shrinkage methods.
• The BMA package has functions for Bayesian model averaging for linear models, generalized linear models, and survival models. The complementary package ensembleBMA uses the BMA package to create probabilistic forecasts of ensembles using a mixture of normal distributions.
• bmixture provides statistical tools for Bayesian estimation for the finite mixture of distributions, mainly mixture of Gamma, Normal and t-distributions.
• BMS is Bayesian Model Averaging library for linear models with a wide choice of (customizable) priors. Built-in priors include coefficient priors (fixed, flexible and hyper-g priors), and 5 kinds of model priors.
• Bmix is a bare-bones implementation of sampling algorithms for a variety of Bayesian stick-breaking (marginally DP) mixture models, including particle learning and Gibbs sampling for static DP mixtures, particle learning for dynamic BAR stick-breaking, and DP mixture regression.
• bnlearn is a package for Bayesian network structure learning (via constraint-based, score-based and hybrid algorithms), parameter learning (via ML and Bayesian estimators) and inference.
• BNSP is a package for Bayeisan non- and semi-parametric model fitting. It handles Dirichlet process mixtures and spike-slab for multivariate (and univariate) response analysis, with nonparametric models for the means, the variances and the correlation matrix.
• BoomSpikeSlab provides functions to do spike and slab regression via the stochastic search variable selection algorithm. It handles probit, logit, poisson, and student T data.
• bqtl can be used to fit quantitative trait loci (QTL) models. This package allows Bayesian estimation of multi-gene models via Laplace approximations and provides tools for interval mapping of genetic loci. The package also contains graphical tools for QTL analysis.
• bridgesampling provides R functions for estimating marginal likelihoods, Bayes factors, posterior model probabilities, and normalizing constants in general, via different versions of bridge sampling (Meng and Wong, 1996).
• bsamGP provides functions to perform Bayesian inference using a spectral analysis of Gaussian process priors. Gaussian processes are represented with a Fourier series based on cosine basis functions. Currently the package includes parametric linear models, partial linear additive models with/without shape restrictions, generalized linear additive models with/without shape restrictions, and density estimation model.
• bspec performs Bayesian inference on the (discrete) power spectrum of time series.
• bspmma is a package for Bayesian semiparametric models for meta-analysis.
• bsts is a package for time series regression using dynamic linear models using MCMC.
• BVS is a package for Bayesian variant selection and Bayesian model uncertainty techniques for genetic association studies.
• coalescentMCMC provides a flexible framework for coalescent analyses in R.
• dclone provides low level functions for implementing maximum likelihood estimating procedures for complex models using data cloning and MCMC methods.
• deBInfer provides R functions for Bayesian parameter inference in differential equations using MCMC methods.
• dlm is a package for Bayesian (and likelihood) analysis of dynamic linear models. It includes the calculations of the Kalman filter and smoother, and the forward filtering backward sampling algorithm.
• EbayesThresh implements Bayesian estimation for thresholding methods. Although the original model is developed in the context of wavelets, this package is useful when researchers need to take advantage of possible sparsity in a parameter set.
• ebdbNet can be used to infer the adjacency matrix of a network from time course data using an empirical Bayes estimation procedure based on Dynamic Bayesian Networks.
• eigenmodel estimates the parameters of a model for symmetric relational data (e.g., the above-diagonal part of a square matrix), using a model-based eigenvalue decomposition and regression using MCMC methods.
• EntropyMCMC is an R package for MCMC simulation and convergence evaluation using entropy and Kullback-Leibler divergence estimation.
• evdbayes provides tools for Bayesian analysis of extreme value models.
• exactLoglinTest provides functions for log-linear models that compute Monte Carlo estimates of conditional P-values for goodness of fit tests.
• factorQR is a package to fit Bayesian quantile regression models that assume a factor structure for at least part of the design matrix.
• FME provides functions to help in fitting models to data, to perform Monte Carlo, sensitivity and identifiability analysis. It is intended to work with models be written as a set of differential equations that are solved either by an integration routine from deSolve, or a steady-state solver from rootSolve.
• The gbayes() function in Hmisc derives the posterior (and optionally) the predictive distribution when both the prior and the likelihood are Gaussian, and when the statistic of interest comes from a two-sample problem.
• ggmcmc is a tool for assessing and diagnosing convergence of Markov Chain Monte Carlo simulations, as well as for graphically display results from full MCMC analysis.
• gRain is a package for probability propagation in graphical independence networks, also known as Bayesian networks or probabilistic expert systems.
• The HI package has functions to implement a geometric approach to transdimensional MCMC methods and random direction multivariate Adaptive Rejection Metropolis Sampling.
• The hbsae package provides functions to compute small area estimates based on a basic area or unit-level model. The model is fit using restricted maximum likelihood, or in a hierarchical Bayesian way.
• iterLap performs an iterative Laplace approximation to build a global approximation of the posterior (using mixture distributions) and then uses importance sampling for simulation based inference.
• The function krige.bayes() in the geoR package performs Bayesian analysis of geostatistical data allowing specification of different levels of uncertainty in the model parameters. Ssee the Spatial view for more information.
• The lmm package contains R functions to fit linear mixed models using MCMC methods.
• matchingMarkets implements a structural model based on a Gibbs sampler to correct for the bias from endogenous matching (e.g. group formation or two-sided matching).
• MCMCglmm is package for fitting Generalised Linear Mixed Models using MCMC methods.
• The mcmcsamp() function in lme4 allows MCMC sampling for the linear mixed model and generalized linear mixed model.
• The mlogitBMA Provides a modified function bic.glm() of the BMA package that can be applied to multinomial logit (MNL) data.
• The MNP package fits multinomial probit models using MCMC methods.
• mombf performs model selection based on non-local priors, including MOM, eMOM and iMOM priors..
• NetworkChange is an R package for change point analysis in longitudinal network data. It implements a hidden Markovmultilinear tensor regression model. Model diagnostic tools using marginal likelihoods and WAIC are provided.
• openEBGM calculates Empirical Bayes Geometric Mean (EBGM) and quantile scores from the posterior distribution using the Gamma-Poisson Shrinker (GPS) model to find unusually large cell counts in large, sparse contingency tables.
• pacbpred perform estimation and prediction in high-dimensional additive models, using a sparse PAC-Bayesian point of view and a MCMC algorithm.
• predmixcor provides functions to predict the binary response based on high dimensional binary features modeled with Bayesian mixture models.
• prevalence provides functions for the estimation of true prevalence from apparent prevalence in a Bayesian framework. MCMC sampling is performed via JAGS/rjags.
• The pscl package provides R functions to fit item-response theory models using MCMC methods and to compute highest density regions for the Beta distribution and the inverse gamma distribution.
• The PAWL package implements parallel adaptive Metropolis-Hastings and sequential Monte Carlo samplers for sampling from multimodal target distributions.
• PReMiuM is a package for profile regression, which is a Dirichlet process Bayesian clustering where the response is linked non-parametrically to the covariate profile.
• revdbayes provides functions for the Bayesian analysis of extreme value models using direct random sampling from extreme value posterior distributions.
• The hitro.new() function in Runuran provides an MCMC sampler based on the Hit-and-Run algorithm in combination with the Ratio-of-Uniforms method.
• RSGHB can be used to estimate models using a hierarchical Bayesian framework and provides flexibility in allowing the user to specify the likelihood function directly instead of assuming predetermined model structures.
• rstiefel simulates random orthonormal matrices from linear and quadratic exponential family distributions on the Stiefel manifold using the Gibbs sampling method. The most general type of distribution covered is the matrix-variate Bingham-von Mises-Fisher distribution.
• RxCEcolInf fits the R x C inference model described in Greiner and Quinn (2009).
• SamplerCompare provides a framework for running sets of MCMC samplers on sets of distributions with a variety of tuning parameters, along with plotting functions to visualize the results of those simulations.
• SampleSizeMeans contains a set of R functions for calculating sample size requirements using three different Bayesian criteria in the context of designing an experiment to estimate a normal mean or the difference between two normal means.
• SampleSizeProportions contains a set of R functions for calculating sample size requirements using three different Bayesian criteria in the context of designing an experiment to estimate the difference between two binomial proportions.
• sbgcop estimates parameters of a Gaussian copula, treating the univariate marginal distributions as nuisance parameters as described in Hoff(2007). It also provides a semiparametric imputation procedure for missing multivariate data.
• SimpleTable provides a series of methods to conduct Bayesian inference and sensitivity analysis for causal effects from 2 x 2 and 2 x 2 x K tables.
• sna, an R package for social network analysis, contains functions to generate posterior samples from Butt’s Bayesian network accuracy model using Gibbs sampling.
• spBayes provides R functions that fit Gaussian spatial process models for univariate as well as multivariate point-referenced data using MCMC methods.
• spikeslab provides functions for prediction and variable selection using spike and slab regression.
• spikeSlabGAM implements Bayesian variable selection, model choice, and regularized estimation in (geo-)additive mixed models for Gaussian, binomial, and Poisson responses.
• spTimer fits, spatially predict and temporally forecast large amounts of space-time data using Bayesian Gaussian Process Models, Bayesian Auto-Regressive (AR) Models, and Bayesian Gaussian Predictive Processes based AR Models.
• ssgraph is for Bayesian inference in undirected graphical models using spike-and-slab priors for multivariate continuous, discrete, and mixed data.
• ssMousetrack estimates previously compiled state-space modeling for mouse-tracking experiments using the rstan package, which provides the R interface to the Stan C++ library for Bayesian estimation.
• stochvol provides efficient algorithms for fully Bayesian estimation of stochastic volatility (SV) models.
• The tgp package implements Bayesian treed Gaussian process models: a spatial modeling and regression package providing fully Bayesian MCMC posterior inference for models ranging from the simple linear model, to nonstationary treed Gaussian process, and others in between.
• vbmp is a package for variational Bayesian multinomial probit regression with Gaussian process priors. It estimates class membership posterior probability employing variational and sparse approximation to the full posterior. This software also incorporates feature weighting by means of Automatic Relevance Determination.
• The vcov.gam() function the mgcv package can extract a Bayesian posterior covariance matrix of the parameters from a fitted gam object.
• zic provides functions for an MCMC analysis of zero-inflated count models including stochastic search variable selection.

Post-estimation tools

• BayesPostEst allows to generate and plot postestimation quantities after estimating Bayesian regression models. Functionality includes predicted probabilities and first differences as well as model checks. The functions can be used with MCMC output generated by any Bayesian estimation tool including JAGS, BUGS, MCMCpack, and Stan.
• BayesValidate implements a software validation method for Bayesian softwares.
• MCMCvis performs key functions (visualizes, manipulates, and summarizes) for MCMC analysis. Functions support simple and straightforward subsetting of model parameters within the calls, and produce presentable and ‘publication-ready’ output. MCMC output may be derived from Bayesian model output fit with JAGS, Stan, or other MCMC samplers.
• The boa package provides functions for diagnostics, summarization, and visualization of MCMC sequences. It imports draws from BUGS format, or from plain matrices. boa provides the Gelman and Rubin, Geweke, Heidelberger and Welch, and Raftery and Lewis diagnostics, the Brooks and Gelman multivariate shrink factors.
• The coda (Convergence Diagnosis and Output Analysis) package is a suite of functions that can be used to summarize, plot, and and diagnose convergence from MCMC samples. coda also defines an mcmc object and related methods which are used by other packages. It can easily import MCMC output from WinBUGS, OpenBUGS, and JAGS, or from plain matrices. coda contains the Gelman and Rubin, Geweke, Heidelberger and Welch, and Raftery and Lewis diagnostics.
• plotMCMC extends coda by adding convenience functions to make it easier to create multipanel plots. The graphical parameters have sensible defaults and are easy to modify via top-level arguments.
• ramps implements Bayesian geostatistical analysis of Gaussian processes using a reparameterized and marginalized posterior sampling algorithm.

Packages for learning Bayesian statistics

• AtelieR is a GTK interface for teaching basic concepts in statistical inference, and doing elementary bayesian statistics (inference on proportions, multinomial counts, means and variances).
• The BaM package is an R package associated with Jeff Gill’s book, “Bayesian Methods: A Social and Behavioral Sciences Approach, Second Edition” (CRC Press, 2007).
• BayesDA provides R functions and datasets for “Bayesian Data Analysis, Second Edition” (CRC Press, 2003) by Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin.
• The Bolstad package contains a set of R functions and data sets for the book Introduction to Bayesian Statistics, by Bolstad, W.M. (2007).
• The LearnBayes package contains a collection of functions helpful in learning the basic tenets of Bayesian statistical inference. It contains functions for summarizing basic one and two parameter posterior distributions and predictive distributions and MCMC algorithms for summarizing posterior distributions defined by the user. It also contains functions for regression models, hierarchical models, Bayesian tests, and illustrations of Gibbs sampling.

Packages that link R to other sampling engines

• bayesmix is an R package to fit Bayesian mixture models using JAGS .
• BayesX provides functionality for exploring and visualizing estimation results obtained with the software package BayesX .
• Boom provides a C++ library for Bayesian modeling, with an emphasis on Markov chain Monte Carlo.
• BRugs provides an R interface to OpenBUGS . It works under Windows and Linux. BRugs used to be available from CRAN, now it is located at the CRANextras repository.
• brms implements Bayesian multilevel models in R using Stan . A wide range of distributions and link functions are supported, allowing users to fit linear, robust linear, binomial, Pois- son, survival, response times, ordinal, quantile, zero-inflated, hurdle, and even non-linear models all in a multilevel context.
• There are two packages that can be used to interface R with WinBUGS . R2WinBUGS provides a set of functions to call WinBUGS on a Windows system and a Linux system.
• There are three packages that provide R interface with Just Another Gibbs Sampler (JAGS) : rjags, R2jags, and runjags.
• All of these BUGS engines use graphical models for model specification. As such, the gR task view may be of interest.
• rstan provides R functions to parse, compile, test, estimate, and analyze Stan models by accessing the header-only Stan library provided by the `StanHeaders’ package. The Stan project develops a probabilistic programming language that implements full Bayesian statistical inference via MCMC and (optionally penalized) maximum likelihood estimation via optimization.
• pcFactorStan provides convenience functions and pre-programmed Stan models related to the paired comparison factor model. Its purpose is to make fitting paired comparison data using Stan easy.
• R2BayesX provides an R interface to estimate structured additive regression (STAR) models with ‘BayesX’.

The Bayesian Inference Task View is written by Jong Hee Park (Seoul National University, South Korea), Andrew D. Martin (University of Michigan, Ann Arbor, MI, USA), and Kevin M. Quinn (UC Berkeley, Berkeley, CA, USA). Please email the task view maintainer with suggestions.

## How to implement the Bayesian average algorithm for a binary rating system

I have a system where people can up vote or down vote an item and I want to display the results of that as a 5 star rating.

I have been trying use the Bayesian Rating algorithm explained here and here with no success.

For example: I have three items (A, B and C) in my database:

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A = 500 UP and 500 down votes B = 0 UP and 1000 down votes C = 0 UP and 1000 down votes

How do i calculate the Bayesian average rating for each item so that it has a score on a scale of 1 to 5?

This blog post, How Not To Sort By Average Rating, describes exactly your situation, and how to solve it using a Wilson Score confidence interval. Reddit used this to good effect.

AvgRating = Sum of up votes for all items * 5 / Sum of all votes

CurRating = Sum of up votes on current item * 5/ Number of votes on current item

So plugging in your numbers evaluating the weight for A.

AvgRating = 0 (Remember do not include numbers for the item you are evaluating in this calculation)

CurRating = 500 * 5 / 1000 = 2.5

Total Votes = 2000 + 1000 = 3000

((1000 * 0) + (1000 * 2.5)) * 5 / 3000 = 4.166

I forgot to add, do NOT include any items in any calculation or sum above that have no votes or it will throw the weights off.

EDIT – Simplified Solution:

I should note that there is a simplified solution to the problem that can be performed. I only demonstrated longhand form for comprehension. The compressed algorithm looks like:

SET = Anything not related to the current evaluation target where votes is greater than zero.

TARGET = The element you are currently trying to evaluate

Again plugging in with your numbers evaluating ‘A’ for clarification and proof:

## 1.9. Naive Bayes¶

Naive Bayes methods are a set of supervised learning algorithms based on applying Bayes’ theorem with the “naive” assumption of conditional independence between every pair of features given the value of the class variable. Bayes’ theorem states the following relationship, given class variable $$y$$ and dependent feature vector $$x_1$$ through $$x_n$$ , :

Using the naive conditional independence assumption that

for all $$i$$ , this relationship is simplified to

Since $$P(x_1, \dots, x_n)$$ is constant given the input, we can use the following classification rule:

and we can use Maximum A Posteriori (MAP) estimation to estimate $$P(y)$$ and $$P(x_i \mid y)$$ ; the former is then the relative frequency of class $$y$$ in the training set.

The different naive Bayes classifiers differ mainly by the assumptions they make regarding the distribution of $$P(x_i \mid y)$$ .

In spite of their apparently over-simplified assumptions, naive Bayes classifiers have worked quite well in many real-world situations, famously document classification and spam filtering. They require a small amount of training data to estimate the necessary parameters. (For theoretical reasons why naive Bayes works well, and on which types of data it does, see the references below.)

Naive Bayes learners and classifiers can be extremely fast compared to more sophisticated methods. The decoupling of the class conditional feature distributions means that each distribution can be independently estimated as a one dimensional distribution. This in turn helps to alleviate problems stemming from the curse of dimensionality.

On the flip side, although naive Bayes is known as a decent classifier, it is known to be a bad estimator, so the probability outputs from predict_proba are not to be taken too seriously.

## 1.9.1. Gaussian Naive Bayes¶

GaussianNB implements the Gaussian Naive Bayes algorithm for classification. The likelihood of the features is assumed to be Gaussian:

The parameters $$\sigma_y$$ and $$\mu_y$$ are estimated using maximum likelihood.

## 1.9.2. Multinomial Naive Bayes¶

MultinomialNB implements the naive Bayes algorithm for multinomially distributed data, and is one of the two classic naive Bayes variants used in text classification (where the data are typically represented as word vector counts, although tf-idf vectors are also known to work well in practice). The distribution is parametrized by vectors $$\theta_y = (\theta_,\ldots,\theta_)$$ for each class $$y$$ , where $$n$$ is the number of features (in text classification, the size of the vocabulary) and $$\theta_$$ is the probability $$P(x_i \mid y)$$ of feature $$i$$ appearing in a sample belonging to class $$y$$ .

The parameters $$\theta_y$$ is estimated by a smoothed version of maximum likelihood, i.e. relative frequency counting:

where $$N_ = \sum_ x_i$$ is the number of times feature $$i$$ appears in a sample of class $$y$$ in the training set $$T$$ , and $$N_ = \sum_^ N_$$ is the total count of all features for class $$y$$ .

The smoothing priors $$\alpha \ge 0$$ accounts for features not present in the learning samples and prevents zero probabilities in further computations. Setting $$\alpha = 1$$ is called Laplace smoothing, while $$\alpha is called Lidstone smoothing. ## 1.9.3. Complement Naive Bayes¶ ComplementNB implements the complement naive Bayes (CNB) algorithm. CNB is an adaptation of the standard multinomial naive Bayes (MNB) algorithm that is particularly suited for imbalanced data sets. Specifically, CNB uses statistics from the complement of each class to compute the model’s weights. The inventors of CNB show empirically that the parameter estimates for CNB are more stable than those for MNB. Further, CNB regularly outperforms MNB (often by a considerable margin) on text classification tasks. The procedure for calculating the weights is as follows: where the summations are over all documents \(j$$ not in class $$c$$ , $$d_$$ is either the count or tf-idf value of term $$i$$ in document $$j$$ , $$\alpha_i$$ is a smoothing hyperparameter like that found in MNB, and $$\alpha = \sum_ \alpha_i$$ . The second normalization addresses the tendency for longer documents to dominate parameter estimates in MNB. The classification rule is:

i.e., a document is assigned to the class that is the poorest complement match.

Rennie, J. D., Shih, L., Teevan, J., & Karger, D. R. (2003). Tackling the poor assumptions of naive bayes text classifiers. In ICML (Vol. 3, pp. 616-623).

## 1.9.4. Bernoulli Naive Bayes¶

BernoulliNB implements the naive Bayes training and classification algorithms for data that is distributed according to multivariate Bernoulli distributions; i.e., there may be multiple features but each one is assumed to be a binary-valued (Bernoulli, boolean) variable. Therefore, this class requires samples to be represented as binary-valued feature vectors; if handed any other kind of data, a BernoulliNB instance may binarize its input (depending on the binarize parameter).

The decision rule for Bernoulli naive Bayes is based on

which differs from multinomial NB’s rule in that it explicitly penalizes the non-occurrence of a feature $$i$$ that is an indicator for class $$y$$ , where the multinomial variant would simply ignore a non-occurring feature.

In the case of text classification, word occurrence vectors (rather than word count vectors) may be used to train and use this classifier. BernoulliNB might perform better on some datasets, especially those with shorter documents. It is advisable to evaluate both models, if time permits.

C.D. Manning, P. Raghavan and H. Schütze (2008). Introduction to Information Retrieval. Cambridge University Press, pp. 234-265.

A. McCallum and K. Nigam (1998). A comparison of event models for Naive Bayes text classification. Proc. AAAI/ICML-98 Workshop on Learning for Text Categorization, pp. 41-48.

V. Metsis, I. Androutsopoulos and G. Paliouras (2006). Spam filtering with Naive Bayes – Which Naive Bayes? 3rd Conf. on Email and Anti-Spam (CEAS).

## 1.9.5. Categorical Naive Bayes¶

CategoricalNB implements the categorical naive Bayes algorithm for categorically distributed data. It assumes that each feature, which is described by the index $$i$$ , has its own categorical distribution.

For each feature $$i$$ in the training set $$X$$ , CategoricalNB estimates a categorical distribution for each feature i of X conditioned on the class y. The index set of the samples is defined as $$J = \< 1, \dots, m \>$$ , with $$m$$ as the number of samples.

The probability of category $$t$$ in feature $$i$$ given class $$c$$ is estimated as:

where $$N_ = |\ = t, y_j = c\>|$$ is the number of times category $$t$$ appears in the samples $$x_$$ , which belong to class $$c$$ , $$N_ = |\< j \in J\mid y_j = c\>|$$ is the number of samples with class c, $$\alpha$$ is a smoothing parameter and $$n_i$$ is the number of available categories of feature $$i$$ .

CategoricalNB assumes that the sample matrix $$X$$ is encoded (for instance with the help of OrdinalEncoder ) such that all categories for each feature $$i$$ are represented with numbers $$0, . n_i – 1$$ where $$n_i$$ is the number of available categories of feature $$i$$ .

## 1.9.6. Out-of-core naive Bayes model fitting¶

Naive Bayes models can be used to tackle large scale classification problems for which the full training set might not fit in memory. To handle this case, MultinomialNB , BernoulliNB , and GaussianNB expose a partial_fit method that can be used incrementally as done with other classifiers as demonstrated in Out-of-core classification of text documents . All naive Bayes classifiers support sample weighting.

Contrary to the fit method, the first call to partial_fit needs to be passed the list of all the expected class labels.

For an overview of available strategies in scikit-learn, see also the out-of-core learning documentation.

The partial_fit method call of naive Bayes models introduces some computational overhead. It is recommended to use data chunk sizes that are as large as possible, that is as the available RAM allows.

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Binary.com имеет все необходимые для ведения деятельности международные лицензии, подтверждающие его надежность и прозрачность. Центральные офисы находятся на Британских островах и Мальте. Депозиты клиентов размещают только в самых надежных банках, имеющих высокие рейтинги. Об уже имеющемся неудачном опыте RegenMarketsGroup в этом сегменте представители предпочитают помалкивать.

На данный момент у Binary.com есть большой набор инструментов для торговли, в развитие программного обеспечения вкладывается немало средств. Котировки берутся напрямую с межбанка, а не поставляются через посредников. Такой подход гарантирует их точность.

## Торговая платформа Binary.com

У брокера — своя торговая платформа, разработанная профессиональными программистами. Ее можно назвать инновационной, учитывая целый ряд особенностей.

Есть 5 видов контрактов, которые предлагаются к покупке. При совершении сделки трейдер может настроить согласно своим предпочтениям любой из параметров. Увеличить или уменьшить можно размер ставки, а также выставить желаемую прибыль. Стоимость лота при этом посчитается сама.

Некоторое неудобство состоит в том, что нет доступа к графикам и индикаторам. Чтобы эту возможность получить, нужно скачать дополнительную программу, ссылка на нее есть в личном кабинете.

Брокер предлагает попробовать инструмент Random, его волатильность одинакова в любое время, не зависит от выхода новостей, торговля доступна в любой день. Индексы позиционируются как полезное новшество для новичков, поскольку они более стабильны ввиду их автоматической генерации, не подверженной резким всплескам, как на реальных рынках.

К преимуществам платформы можно отнести:

• протокол безопасности;
• обширный выбор сроков заключения контрактов;
• быстрота исполнения всех действий, несмотря на большую нагрузку: Binary.com проводит более 1 млн. транзакций в день;
• впечатляющий ассортимент торговых инструментов: более 25 валютных пар, акции, различные индексы, сырьевые товары, Random.

Binary.com — популярный международный брокер

Прежде чем остановить свой выбор на Binary.com, учтите преимущества и недостатки сотрудничества с ним.

• низкая комиссия;
• небольшой первоначальный депозит — всего 5 долларов США, евро или фунтов;
• оперативный ввод и вывод денег;
• отмечен международными наградами;
• наличие демо-счета;
• ставка — от 1 доллара США, евро или фунта.
• сложная для восприятия торговая платформа, в которой новичкам легко запутаться;
• небольшой объем обучающих материалов, представленных на сайте.

Стать клиентом Binary.com очень просто. Регистрация не сложная, введенный пароль нужно подтвердить через e-mail.

### Обучение работе с бинарными опционами

При изучении материалов, предоставляемых брокером, вы почерпнете базовые знания о бинарных опционах, особенностях торговли. Все это доступно в любое время в личном кабинете.

На сайте есть информация о торговой площадке, типах контрактов, активах, финансовых рынках, есть немало полезных советов, они пригодятся и начинающим, и более опытным трейдерам.

Регулярно проводятся вебинары, рассматривающие отдельные аспекты торговли опционами, стратегии более подробно.

В целом, обучение качественное, хотя и немного усложненное.

Демо-счет

Доступен всем клиентам без ограничений. Открыть его можно без внесения реальных средств. Регистрируясь на сайте, у вас сразу же спросят, какой вид счета вы желаете иметь.

Демо помогает разобраться с непривычным терминалом без ущерба для депозита, протестировать новую стратегию. Его полезно изучить перед тем, как открывать реальный счет, ведь, возможно, площадка вам совсем не понравится.

Трейдеру предоставляется 10 тыс. долларов, чтобы оценить преимущества Binary.com. Вывести их, к сожалению, нельзя, но тренироваться можно сколько угодно.

### Ввод и вывод денег Binary.com

Еще один плюс, который смело можно засчитать брокеру — пополнение депозита в разных валютах: фунты, евро, доллары США. Начать торговлю можно, перечислив 5 любых указанных денежных единиц. Максимальный перевод — 150 тыс.

Поддерживается Union Pay, платежные карты, популярные системы QIWI, Яндекс.Деньги, Webmoney и др. Деньги перечисляются практически моментально, не считая банковского перевода, который обрабатывается до 3 дней.

Вывести деньги представляется возможным только после прохождения процедуры верификации аккаунта. Для этого трейдер предъявляет представителю брокера сканированные документы, подтверждающие личность.

Для вывода клиент указывает в личном кабинете сумму, которую он хочет получить. Заявки по правилам обрабатываются до 3 банковских дней, скорость выводв – в зависимости от выбранного способа, хотя отзывы пользователей отмечают, что нередко случаются длительные задержки или вообще операция не проходит.

Итоги

Binary.com можно считать настоящим новатором в сфере бинарных опционов. Необычная торговая платформа, имеющая большой набор инструментов и полезных опций, предлагающая несколько способов торговли, несомненно, заслуживает того, чтобы трейдер изучил ее возможности.

Отметим также большой стаж работы компании, ее вклад в разработку современного софта, завоевавшего несколько международных наград. Она гордится своеобразным рекордом — уже открыто около 1 млн. счетов по всему миру.

Трудности, которые могут отпугнуть новичка — сложность учебных материалов, большое количество цифр и обилие функций торговой площадки.

Ознакомиться со всеми брокерами и выбрать надежного брокера бинарных опционов вы можете в Рейтинге брокеров бинарных опционов на нашем сайте.

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