Package 'PDEnaiveBayes'

Title: Plausible Naive Bayes Classifier Using PDE
Description: A nonparametric, multicore-capable plausible naive Bayes classifier based on the Pareto density estimation (PDE) featuring a plausible approach to a pitfall in the Bayesian theorem covering low evidence cases. Stier, Q., Hoffmann, J., and Thrun, M.C.: "Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naive Bayes" (2026), Machine Learning and Knowledge Extraction (MAKE), <DOI:10.3390/make8010013>.
Authors: Michael Thrun [aut, cph, cre] (ORCID: <https://orcid.org/0000-0001-9542-5543>), Quirin Stier [aut, rev] (ORCID: <https://orcid.org/0000-0002-7896-4737>), Tim Robin Neldner [ctr, ctb]
Maintainer: Michael Thrun <[email protected]>
License: GPL-3
Version: 0.2.9
Built: 2026-06-03 06:56:39 UTC
Source: https://github.com/mthrun/pdebayes

Help Index

Plausible Naive Bayes Classifier Using PDE

Description

A nonparametric, multicore-capable plausible naive Bayes classifier based on the Pareto density estimation (PDE) featuring a plausible approach to a pitfall in the Bayesian theorem covering low evidence cases. Stier, Q., Hoffmann, J., and Thrun, M.C.: "Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naive Bayes" (2026), Machine Learning and Knowledge Extraction (MAKE), <DOI:10.3390/make8010013>.

Details

Pareto Density Estimated naive Bayes Classifier Index of help topics: 

ApplyBayesTheorem4Likelihoods
                        ApplyBayesTheorem4Likelihoods
defineOrEstimateDistribution
                        defineOrEstimateDistribution
fitParameters           fitParameters
GetLikelihoods          GetLikelihoods
getPriors               getPriors
Hepta                   Hepta introduced in [Ultsch, 2003]
PDEnaiveBayes-package   Plausible Naive Bayes Classifier Using PDE
PlotBayesianDecision2D
                        PlotBayesianDecision2D
PlotLikelihoodFuns      PlotLikelihoodFuns
PlotLikelihoods         PlotLikelihoods
PlotNaiveBayes          PlotNaiveBayes
PlotPosteriors          PlotPosteriors
Predict_naiveBayes      Predict_naiveBayes
predict.PDEbayes        predict.PDEbayes
Train_naiveBayes        Train_naiveBayes
Train_naiveBayes_multicore
                        Train_naiveBayes_multicore

(PDENB) of [Stier et al., 2026].

Author(s)

Michal Thrun

Maintainer: Michael Thrun <[email protected]>

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

[Thrun et al., 2020] Thrun, M. C., Gehlert, T., & Ultsch, A.: Analyzing the Fine Structure of Distributions, PloS one, Vol. 15(10), pp. e0238835, doi 10.1371/journal.pone.0238835 2020.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi 10.1016/j.dib.2020.105501, 2020.

[Ultsch et al., 2015] Ultsch, A., Thrun, M. C., Hansen-Goos, O., & L?tsch, J.: Identification of Molecular Fingerprints in Human Heat Pain Thresholds by Use of an Interactive Mixture Model R Toolbox (AdaptGauss), International journal of molecular sciences, Vol. 16(10), pp. 25897-25911, doi 10.3390/ijms161025897, 2015.

Examples

if(requireNamespace("FCPS")){
V=FCPS::ClusterChallenge("Hepta",1000)
Data=V$Hepta
Cls=V$Cls
ind=1:length(Cls)
indtrain=sample(ind,800)
indtest=setdiff(ind,indtrain)
#parametric
#model=Train_naiveBayes(Data[indtrain,],Cls[indtrain],Gaussian=TRUE)
#ClsTrain=model$ClsTrain
#table(Cls[indtrain],ClsTrain)

#res=Predict_naiveBayes(Data[indtest,], Model = model)
#table(Cls[indtest],res$ClsTest)

#PDEbayes
model=Train_naiveBayes(Data[indtrain,],Cls[indtrain],Gaussian=FALSE)
ClsTrain=model$ClsTrain
table(Cls[indtrain],ClsTrain)

res=Predict_naiveBayes(Data[indtest,], Model = model)
table(Cls[indtest],res$ClsTest)
}

ApplyBayesTheorem4Likelihoods

Description

Calculates the posteriors, for given likelihoods and priors using the Bayes Theorem

Usage

ApplyBayesTheorem4Likelihoods(Likelihoods,Priors,threshold=.Machine$double.eps*1000)

Arguments

Likelihoods

List of d numeric matrices, one per feature, each matrix with 1:k columns containing the distribution of class 1:k.
Priors

[1:k] Numeric vector with prior probability for each class.
threshold

(Optional: Default=0.00001).

Value

Posteriors

[1:n, 1:d] Numeric matrix with posterior probability according to the bayes theorem.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

if(requireNamespace("FCPS")){
  data(Hepta)
  Data=Hepta$Data
  Cls=Hepta$Cls
  #parametric
  #V=Train_naiveBayes(Data,Cls,Gaussian=TRUE)
  #ClsTrain=V$ClsTrain
  #table(Cls,ClsTrain)
  
  #non-parametric
  V=Train_naiveBayes(Data,Cls,Gaussian=FALSE)
  ClsTrain=V$ClsTrain
  table(Cls,ClsTrain)
}

defineOrEstimateDistribution

Description

The function estimates the distribution of values within a features that belong to a specific class, i.e., the conditional probability of the likelihood

Usage

defineOrEstimateDistribution(Feature,ClassInd,Gaussian=FALSE,ParetoRadius=NULL,
InternalPlotIt=FALSE,SD_Threshold=0.001,...)

Arguments

Feature

[1:n] Numeric Vector
ClassInd

Integer Vector with class indices
Gaussian

(Optional: Default=TRUE). Assume gaussian distribution.
ParetoRadius

Optional [1:d] numerical vector for pareto radii computed priorly, see ParetoRadius
InternalPlotIt

Optional: Default=FALSE). Create plot if set to TRUE.
SD_Threshold

Optional: Default=0.001.
...

Robust: Optional: Default=FALSE, TRUE: robust estimation of mean and std in case of Gaussian=TRUE

Type: (Optional: Default=2, 1=original PDE, 2= improved PDE

na.rm: (Optional: Default=TRUE). Remove na.

Value

Kernels

[1:m] Numeric vector with kernels (x-values) of a 1D pdf.
PDF

[1:m] Numeric vector with the distribution values of a 1D pdf.
Theta

Numeric vector with parameters of gaussian of mean and standard deviation - NULL if no gaussian used.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

if(requireNamespace("FCPS")){
data(Hepta)
Data=Hepta$Data
Cls=Hepta$Cls
Priors=getPriors(Cls)
}

fitParameters

Description

Fit Gaussian parameters.

Usage

fitParameters(Feature,ClassInd,Robust=FALSE,na.rm=TRUE,SD_Threshold=0.0001)

Arguments

Feature

[1:n] Numeric Vector
ClassInd

Integer Vector with class indices
Robust

(Optional: Default=FALSE). Robust computation if set to TRUE.
na.rm

(Optional: Default=TRUE). Remove na.
SD_Threshold

(Optional: Default=0.00001).

Value

Parameters

[1:2] Numeric vector with Mean and Std.

Author(s)

Michael Thrun

Examples

if(requireNamespace("FCPS")){
data(Hepta)
Data=Hepta$Data
Cls=Hepta$Cls
Priors=getPriors(Cls)
}

GetLikelihoods

Description

Yields the likelihoods per feauture and class as values of distribution either defined by Gaussian or estimated form the data using pareto density estimation.

Usage

GetLikelihoods(Data,Cls,...)

Arguments

Data

[1:n,1:d] matrix of training data. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
Cls

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.
...

Further arguements for defineOrEstimateDistribution Robust=TRUE: robustly estimated gaussians na.rm=TRUE: remove NaNs Threshold: threshold for which the standard deviation cannot be smaller (defaul 0.0001)

Details

Due to pareto density estimation per class and feature, usually the number of rows in each element of c_Kernels_list and ListOfLikelihoods varies and does not equal the number of rows of data n.

Value

c_Kernels_list

List of d numeric matrices, one per feature, each matrix with 1:k columns containing the kernels of class 1:k
ListOfLikelihoods

List of d numeric matrices, one per feature, each matrix with 1:k columns containing distribution values (likelihood) of class 1:k

Thetas

If Gaussian=TRUE: List of d numeric matrices, one per feauture, each matrix with 1:k rows containing the mean in the first column and the standard deviation in teh seconf columd of class 1:k Otherwise: NULL
ParetoRadiusPerFeauture

Numeric vector with estimated pareto radius per feature.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

if(requireNamespace("FCPS")){
data(Hepta)
Data=Hepta$Data
Cls=Hepta$Cls
Priors=getPriors(Cls)
}

getPriors

Description

Get a prior via class proportions.

Usage

getPriors(Cls)

Arguments

Cls

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.

Value

Priors

[1:k] Numeric vector with prior probability for each class.

Author(s)

Michael Thrun

Examples

if(requireNamespace("FCPS")){
data(Hepta)
Data=Hepta$Data
Cls=Hepta$Cls
Priors=getPriors(Cls)
}

Hepta introduced in [Ultsch, 2003]

Description

Clearly defined clusters, different variances. Detailed description of dataset and its clustering challenge is provided in [Thrun/Ultsch, 2020].

Usage

data("Hepta")

Details

Size 212, Dimensions 3, stored in Hepta$Data

Classes 7, stored in Hepta$Cls

References

[Ultsch, 2003] Ultsch, A.: Maps for the visualization of high-dimensional data spaces, Proc. Workshop on Self organizing Maps (WSOM), pp. 225-230, Kyushu, Japan, 2003.

[Thrun/Ultsch, 2020] Thrun, M. C., & Ultsch, A.: Clustering Benchmark Datasets Exploiting the Fundamental Clustering Problems, Data in Brief, Vol. 30(C), pp. 105501, doi:10.1016/j.dib.2020.105501, 2020.

Examples

data(Hepta)
str(Hepta)

PlotBayesianDecision2D

Description

Plots estimation of decision boundary in a 2D slice of the data using the posteriors

Usage

PlotBayesianDecision2D(X, Y, Posteriors, Class = 1, NoBins,
CellColorsOrPallette, Showpoints = TRUE, xlim, ylim, xlab, ylab, main,
PlotIt = TRUE)

Arguments

X

Numeric vector with point coordinates of first dimension of data selection.
Y

Numeric vector with point coordinates of second dimension of data selection.
Posteriors

[1:n, 1:Class] matrix of posteriors.
Class

Optional,Integer defining which class to look at.
NoBins

Optional,Number of bins for class posteriori.
CellColorsOrPallette

Optional, Either a function defining the color palette of a character vector or character vector of length NoBins stating colors.
Showpoints

Optional, TRUE, points are displayed.
xlim

Optional,Numeric vector of length 2 stating limits of x axis.
ylim

Optional,Numeric vector of length 2 stating limits of y axis.
xlab

Optional,Character stating name of x axis.
ylab

Optional,Character stating name of y axis.
main

Optional, Character name of title
PlotIt

Optional, TRUE: prints GGPLOT2 object, FALSE: not shown plot.

Details

Boundaries are assumed to be zero for plotting.

Value

List of:

Mapping

List containing a map for colors, kernels and bin number.
GGobj

ggplot2 object containing 2D visualization of Posteriori.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

Data = as.matrix(iris[,1:4])
Cls = as.numeric(iris[,5])

TrainIdx = c(17, 73, 46, 29, 68, 35, 131, 62, 132, 127, 71, 72, 
144, 99, 93, 13, 38, 21, 102, 53, 36, 111, 114, 96, 57, 74, 145, 
86, 3, 16, 52, 59, 140, 40, 122, 109, 6, 91, 79, 15, 108, 139, 
37, 76, 20, 115, 66, 28, 100, 117, 44, 78, 80, 150, 146, 142, 
9, 90, 45, 58, 134, 11, 87, 125, 141, 118, 136, 48, 124, 47, 
8, 27, 33, 92, 130, 54, 65, 104, 23, 98, 129, 123, 34, 128, 135, 
51, 64, 5, 94, 83, 42, 116, 101, 43, 7, 12, 82, 1, 84, 138, 2, 
56, 4, 106, 120)

TestIdx = c(60, 10, 75, 70, 81, 18, 97, 95, 67, 22, 55, 143, 
88, 24, 105, 26, 119, 31, 107, 63, 41, 61, 32, 147, 89, 14, 121, 
19, 113, 49, 126, 112, 25, 77, 137, 103, 50, 30, 149, 110, 39, 
69, 148, 85, 133)

TrainX = Data[TrainIdx, ]
TestX  = Data[TestIdx, ]
TrainY = Cls[TrainIdx]
TestY  = Cls[TestIdx]

VPDENB = Train_naiveBayes(Data = TrainX, Cls = TrainY, Plausible = FALSE)

PlotBayesianDecision2D(X = TrainX[, 1], Y = TrainX[, 2],
Posteriors = VPDENB$Posteriors, Class = 1)

PlotLikelihoodFuns

Description

Plots the class-conditional Likelihoods per feature, given the generating likelihood functions.

Usage

PlotLikelihoodFuns(LikelihoodFuns,Data,PlausibleLikelihoodFuns=NULL,
Epsilon=NULL,PlausibleCenters=NULL,PlotCutOff=4,xlim)

Arguments

LikelihoodFuns

List with Likelihoods generating functions
Data

Numeric matrix with data.
PlausibleLikelihoodFuns

List with plausible Likelihoods.
Epsilon

Numeric scalar defining epsilon fo plausible likelihoods.
PlausibleCenters

Numeric vector [1:k] plausible centers used to compute plausible likelihoods.
PlotCutOff

scalar defining the how many feature starting from 1 should be plotted or numerical vector defining the index of features to be plotted in second case should not be too many otherwise plot yields an error.
xlim

Numeric vector of length 2 stating limits of x axis.

Value

No return value.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

Data = as.matrix(iris[,1:4])
Cls = as.numeric(iris[,5])

TrainIdx = c(17, 73, 46, 29, 68, 35, 131, 62, 132, 127, 71, 72, 
144, 99, 93, 13, 38, 21, 102, 53, 36, 111, 114, 96, 57, 74, 145, 
86, 3, 16, 52, 59, 140, 40, 122, 109, 6, 91, 79, 15, 108, 139, 
37, 76, 20, 115, 66, 28, 100, 117, 44, 78, 80, 150, 146, 142, 
9, 90, 45, 58, 134, 11, 87, 125, 141, 118, 136, 48, 124, 47, 
8, 27, 33, 92, 130, 54, 65, 104, 23, 98, 129, 123, 34, 128, 135, 
51, 64, 5, 94, 83, 42, 116, 101, 43, 7, 12, 82, 1, 84, 138, 2, 
56, 4, 106, 120)

TestIdx = c(60, 10, 75, 70, 81, 18, 97, 95, 67, 22, 55, 143, 
88, 24, 105, 26, 119, 31, 107, 63, 41, 61, 32, 147, 89, 14, 121, 
19, 113, 49, 126, 112, 25, 77, 137, 103, 50, 30, 149, 110, 39, 
69, 148, 85, 133)

TrainX = Data[TrainIdx, ]
TestX  = Data[TestIdx, ]
TrainY = Cls[TrainIdx]
TestY  = Cls[TestIdx]

VPDENB = Train_naiveBayes(Data = TrainX, Cls = TrainY, Plausible = FALSE)

PlotLikelihoodFuns(LikelihoodFuns = VPDENB$Model$PDFs_funs, Data = TrainX)

PlotLikelihoods

Description

Plots the Likelihoods per feature.

Usage

PlotLikelihoods(Likelihoods, Data, PlausibleLikelihoods=NULL,Epsilon=NULL,
PlausibleCenters=NULL,PlotCutOff=4,xlim)

Arguments

Likelihoods

List with Likelihoods.
Data

Numeric matrix with data.
PlausibleLikelihoods

List with plausible Likelihoods.
Epsilon

Numeric scalar defining epsilon fo plausible likelihoods.
PlausibleCenters

Numeric vector [1:k] plausible centers used to compute plausible likelihoods.
PlotCutOff

scalar defining the how many feature starting from 1 should be plotted or numerical vector defining the index of features to be plotted in second case should not be too many otherwise plot yields an error.
xlim

Numeric vector of length 2 stating limits of x axis.

Details

Boundaries are assumed to be zero for plotting.

Value

No return value.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

Data = as.matrix(iris[,1:4])
Cls = as.numeric(iris[,5])

TrainIdx = c(17, 73, 46, 29, 68, 35, 131, 62, 132, 127, 71, 72, 
144, 99, 93, 13, 38, 21, 102, 53, 36, 111, 114, 96, 57, 74, 145, 
86, 3, 16, 52, 59, 140, 40, 122, 109, 6, 91, 79, 15, 108, 139, 
37, 76, 20, 115, 66, 28, 100, 117, 44, 78, 80, 150, 146, 142, 
9, 90, 45, 58, 134, 11, 87, 125, 141, 118, 136, 48, 124, 47, 
8, 27, 33, 92, 130, 54, 65, 104, 23, 98, 129, 123, 34, 128, 135, 
51, 64, 5, 94, 83, 42, 116, 101, 43, 7, 12, 82, 1, 84, 138, 2, 
56, 4, 106, 120)

TestIdx = c(60, 10, 75, 70, 81, 18, 97, 95, 67, 22, 55, 143, 
88, 24, 105, 26, 119, 31, 107, 63, 41, 61, 32, 147, 89, 14, 121, 
19, 113, 49, 126, 112, 25, 77, 137, 103, 50, 30, 149, 110, 39, 
69, 148, 85, 133)

TrainX = Data[TrainIdx, ]
TestX  = Data[TestIdx, ]
TrainY = Cls[TrainIdx]
TestY  = Cls[TestIdx]

VPDENB = Train_naiveBayes(Data = TrainX, Cls = TrainY, Plausible = FALSE)

PlotLikelihoods(Likelihoods = VPDENB$Model$ListOfLikelihoods, Data = TrainX)

PlotNaiveBayes

Description

Visualize the class-conditional distributions of the Pareto Density estimated naive Bayes model (PDENB) [Stier et al., 2026].

Usage

PlotNaiveBayes(Model, FeatureNames, ClassNames, DatasetName = "Data",
nrows = 1, FeatureOrder, NumFeaturesPerRow = 4, Colors,
IndividualFigures = FALSE)

Arguments

Model

List with elements Priors,c_2List_Train.
FeatureNames

Character vector of names with a name for each feature contained in the data used to create the naive bayes model.
ClassNames

Character vector of class names to present in the legend of the plots.
DatasetName

Character title for each plot.
nrows

Number of rows inside one plot.
FeatureOrder

Numeric vector representing the order of the features to be displayed.
NumFeaturesPerRow

Maximum number of features to be displayed in one plot.
Colors

Character vector of color names. The length of the vector must be the same as the number of classes within the data modeled by the naive Bayes classifier.
IndividualFigures

Optional boolean: If set to TRUE, it returns a list of the individual figures for customization.

Details

Boundaries are assumed to be zero for plotting.

Value

Cls

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.
Posteriors

[1:n, 1:l] Numeric matrices with posterior probabilities.
DataLikelihoodsPerClass

list of length d, each element is a matrix [1:n,1:k] of interpolated class likelihoods per feature d

Author(s)

Quirin Stier

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

Data = as.matrix(iris[,1:4])
Cls = as.numeric(iris[,5])
DatasetName = "Iris"

TrainIdx = c(17, 73, 46, 29, 68, 35, 131, 62, 132, 127, 71, 72, 
144, 99, 93, 13, 38, 21, 102, 53, 36, 111, 114, 96, 57, 74, 145, 
86, 3, 16, 52, 59, 140, 40, 122, 109, 6, 91, 79, 15, 108, 139, 
37, 76, 20, 115, 66, 28, 100, 117, 44, 78, 80, 150, 146, 142, 
9, 90, 45, 58, 134, 11, 87, 125, 141, 118, 136, 48, 124, 47, 
8, 27, 33, 92, 130, 54, 65, 104, 23, 98, 129, 123, 34, 128, 135, 
51, 64, 5, 94, 83, 42, 116, 101, 43, 7, 12, 82, 1, 84, 138, 2, 
56, 4, 106, 120)

TestIdx = c(60, 10, 75, 70, 81, 18, 97, 95, 67, 22, 55, 143, 
88, 24, 105, 26, 119, 31, 107, 63, 41, 61, 32, 147, 89, 14, 121, 
19, 113, 49, 126, 112, 25, 77, 137, 103, 50, 30, 149, 110, 39, 
69, 148, 85, 133)

TrainX = Data[TrainIdx, ]
TestX  = Data[TestIdx, ]
TrainY = Cls[TrainIdx]
TestY  = Cls[TestIdx]

VPDENB = Train_naiveBayes(Data = TrainX, Cls = TrainY, Plausible = FALSE)

FeatureNames = colnames(Data)

PlotNaiveBayes(Model = VPDENB$Model, FeatureNames = FeatureNames)

PlotPosteriors

Description

Plots posteriors either using a panel of plots based on PlotBayesianDecision2D or in 1D as a line plot [Stier et al., 2026].

Usage

PlotPosteriors(Data, Posteriors, Class = 1, CellColorsOrPallette,
Showpoints = TRUE)

Arguments

Data

Either numeric matrix [1:n, 1:d] with data or one column of data.
Posteriors

[1:n, 1:Class] matrix of posteriors.
Class

Integer defining which class to look at if numeric matrix is given, for column of data all posteriors are overlayed in line plot.
CellColorsOrPallette

Either a function defining the color palette of a character vector or character vector of length NoBins stating colors.
Showpoints

TRUE, points are displayed.

Details

Plotting posteriors in one directions only often does not give any insight. The default option using PlotBayesianDecision2D os often more useful.

Value

GGobj

ggplot2 object containing 2D visualization of Posteriori.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

Examples

Data = as.matrix(iris[,1:4])
Cls = as.numeric(iris[,5])

TrainIdx = c(17, 73, 46, 29, 68, 35, 131, 62, 132, 127, 71, 72, 
144, 99, 93, 13, 38, 21, 102, 53, 36, 111, 114, 96, 57, 74, 145, 
86, 3, 16, 52, 59, 140, 40, 122, 109, 6, 91, 79, 15, 108, 139, 
37, 76, 20, 115, 66, 28, 100, 117, 44, 78, 80, 150, 146, 142, 
9, 90, 45, 58, 134, 11, 87, 125, 141, 118, 136, 48, 124, 47, 
8, 27, 33, 92, 130, 54, 65, 104, 23, 98, 129, 123, 34, 128, 135, 
51, 64, 5, 94, 83, 42, 116, 101, 43, 7, 12, 82, 1, 84, 138, 2, 
56, 4, 106, 120)

TestIdx = c(60, 10, 75, 70, 81, 18, 97, 95, 67, 22, 55, 143, 
88, 24, 105, 26, 119, 31, 107, 63, 41, 61, 32, 147, 89, 14, 121, 
19, 113, 49, 126, 112, 25, 77, 137, 103, 50, 30, 149, 110, 39, 
69, 148, 85, 133)

TrainX = Data[TrainIdx, ]
TestX  = Data[TestIdx, ]
TrainY = Cls[TrainIdx]
TestY  = Cls[TestIdx]

VPDENB = Train_naiveBayes(Data = TrainX, Cls = TrainY, Plausible = FALSE)
#default option
PlotPosteriors(Data = TrainX, Posteriors = VPDENB$Posteriors, Class = 1)

# alternative option
PlotPosteriors(Data = TrainX[,3], Posteriors = VPDENB$Posteriors)

Predict_naiveBayes

Description

Predict classification with naive Bayes model [Stier et al., 2026].

Usage

Predict_naiveBayes(Data, Model, ...)

Arguments

Data

[1:n,1:d] matrix of test data. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
Model

Optional, list with elements Priors,c_2List_Train,Thetas, alternative set arguements seperatly

...

Priors: Optional, if Model missing, then [1:k] Numeric vector with prior probability for each class.

c_2List_Train: Optional, if Model missing, then c_2List_Train is the output of GetLikelihoods: a list of two three elements of Kernels, Likelihoods per feature and class, optional Thetas or PlausibleCenters depending on parameter setting

Thetas: Optional, if Model missing, then If c_2List_Train is missing, alternatively the parameters mean and standard deviation of the gaussian distributions per class and feaures.

PlotIt: Optional: Default=FALSE, TRUE: Plots Likelihoods

PlotCutOff: Optional: Scalar indicating how many features (starting from 1) should be plotted, or a numerical vector specifying the indices of the features to plot. Note: In the second case, avoid selecting too many features, as this may cause the plot to fail

Details

The function is implemented in a way so that one can combine training and test data although it is intended to be applied on test data only.

Value

Cls

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.
Posteriors

[1:n, 1:l] Numeric matrices with posterior probabilities.
DataLikelihoodsPerClass

list of length d, each element is a matrix [1:n,1:k] of interpolated class likelihoods per feature d

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

See Also

Train_naiveBayes

Examples

if(requireNamespace("FCPS")){
V=FCPS::ClusterChallenge("Hepta",1000)
Data=V$Hepta
Cls=V$Cls
ind=1:length(Cls)
indtrain=sample(ind,800)
indtest=setdiff(ind,indtrain)

#PDEbayes
model=Train_naiveBayes(Data[indtrain,],Cls[indtrain],Gaussian=FALSE)
ClsTrain=model$ClsTrain
table(Cls[indtrain],ClsTrain)

res=Predict_naiveBayes(Data[indtest,], Model = model)
table(Cls[indtest],res$ClsTest)
}

predict.PDEbayes

Description

Predict a classification with the Pareto Density estimated naive Bayes model [Stier et al., 2026] . (PDENB).

Usage

predict.PDEbayes(object, newdata, type = c("class", "response","prob"), ...)

Arguments

object

Model obtained from training routine in PDEnaiveBayes package.
newdata

[1:n,1:d] matrix of test data. It consists of n cases of d-dimensional data points. Every case has d attributes, variables or features.
type

Optional parameter.
...

Gaussian: Optional: Default=TRUE). Assume gaussian distribution. Plausible: (Optional: TRUE: uses plausble bayesian theorem, FALSE non-plausible bayesian theorem Type: (Optional: default=1, 1 = original PDE, 2 = R native density estimation Threshold: Threshold for which the standard deviation cannot be smaller (default =1e-12) PlotIt: Optional: Default=FALSE, TRUE: Plots Likelihoods PlotCutOff: Optional: Scalar indicating how many features (starting from 1) should be plotted, or a numerical vector specifying the indices of the features to plot. Note: In the second case, avoid selecting too many features, as this may cause the plot to fail ParetoRadiusPerFeauture: Optional [1:d] numerical vector for pareto radii computed priorly, see ParetoRadius or {ParetoRadius_fast} cl: Optional: a cluster object, created by parallel, if given and ParetoRadiusPerFeauture missing, then ParetoRadiusPerFeauture is compputed multicore otherwise single core Robust: Optional: Default=FALSE, TRUE: robust estimation of mean and std in case of Gaussian=TRUE

Details

The function is implemented in a way so that one can combine training and test data although it is intended to be applied on test data only.

Value

Cls

Numeric vector with predicted class associated with newdata.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

See Also

Train_naiveBayes

Examples

if(requireNamespace("FCPS")){
V=FCPS::ClusterChallenge("Hepta",1000)
Data=V$Hepta
Cls=V$Cls
ind=1:length(Cls)
indtrain=sample(ind,800)
indtest=setdiff(ind,indtrain)

model=Train_naiveBayes(Data[indtrain,],Cls[indtrain],Gaussian=FALSE)
ClsTrain=model$ClsTrain
table(Cls[indtrain],ClsTrain)

ClsTest=predict.PDEbayes(object = model, newdata = Data[indtest,])
table(Cls[indtest],ClsTest)
}

Train_naiveBayes

Description

Trains a Pareto Density estimated naive Bayes model (PDENB) of [Stier et al., 2026].

Usage

Train_naiveBayes(Data,Cls,Predict=TRUE,Priors,...)

Arguments

Data

[1:n,1:d] matrix of training data. It consists of n cases of d-dimensional data points. Every case hasd attributes, variables or features.

Cls

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.

Predict

Optional, boolean to decide extent of output. In case of TRUE, yields ClsTrain and Posteriors, else it yields only Model and Thetas. Note: Only if Predict is set to TRUE, parameter EvalPlausible can be set true!

Priors

Optional, [1:k] numerical vector defining the prior probabilities of the k classes. If missing, estimated from Cls.

...

Gaussian: Optional: Default=TRUE). Assume gaussian distribution.

Plausible: (Optional: TRUE: uses plausble bayesian theorem, FALSE non-plausible bayesian theorem.

Type: (Optional: default=1, 1 = original PDE, 2 = R native density estimation.

Threshold: Threshold for which the standard deviation cannot be smaller (default =1e-12).

PlotIt: Optional: Default=FALSE, TRUE: Plots Likelihoods.

PlotCutOff: Optional: Scalar indicating how many features (starting from 1) should be plotted, or a numerical vector specifying the indices of the features to plot. Note: In the second case, avoid selecting too many features, as this may cause the plot to fail.

ParetoRadiusPerFeauture: Optional [1:d] numerical vector for pareto radii computed priorly, see ParetoRadius or {ParetoRadius_fast}

cl: Optional: a cluster object, created by parallel, if given and ParetoRadiusPerFeauture missing, then ParetoRadiusPerFeauture is compputed multicore otherwise single core

Robust: Optional: Default=FALSE, TRUE: robust estimation of mean and std in case of Gaussian=TRUE.

GlobalPR: Optional: Default=TRUE, FALSE: estimation of pareto radius for each class individually.

Details

Precomputation of ParetoRadiusPerFeauture can be usefull to make cross-validation faster although it should be only done on the training data.

If Plausible is not given, both options are evalauted using shannon information.

c_Kernels_list and ListOfLikelihoods have d elements each storing a matrix [1:m,1:k], usually m!=n. In contrast to DataLikelihoodsPerClass in which by interpolation the matrix are of size [1:n,1:k]

Value

Model

List of model parameters and results.
c_Kernels_list

List of matrices, where each matrix represent the kernels of one feature for all classes.
ListOfLikelihoods

List of matrices, where each matrix represent the likelihood of one feature for all classes.
PDFs_funs

Nested list of depth 1, where the first index assigns the feature index and the second index assigns the class. The elements are functions for the density estimation for each feature and each class.
ParetoRadiusPerFeauture

Numeric vector which stores the pareto radius for each feature.
Theta

Parameters mean and standard deviation of the Gaussian distributions per class and feaures.
Priors

Numeric vector which stores the prior probability of each class to appear.
PlausibleCenters

[1:k, 1:f] Numeric matrix which stores the centers for each feature and each class, where the row index assigns features and the column index assigns classes.
ClsTrain

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.
Posteriors

[1:n, 1:k] Numeric matrices with posterior probabilities.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

See Also

Predict_naiveBayes

Examples

if(requireNamespace("FCPS")){
data(Hepta)
Data=Hepta$Data
Cls=Hepta$Cls

#non-parametric
V=Train_naiveBayes(Data,Cls,Gaussian=FALSE)
ClsTrain=V$ClsTrain
table(Cls,ClsTrain)
}

Train_naiveBayes_multicore

Description

Trains a Pareto Density estimated naive Bayes model (PDENB) with multicore parallelity

Usage

Train_naiveBayes_multicore(cl=NULL,Data,Cls,Predict=FALSE,Priors,UseMemshare=FALSE,...)

Arguments

cl

Object instance of package parallel.
Data

[1:n,1:d] matrix of training data. It consists of n cases of d-dimensional data points. Every case hasd attributes, variables or features.

Cls

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.

Predict

Optional, boolean to decide extent of output. In case of TRUE, yields ClsTrain and Posteriors, else it yields only Model and Thetas. Note: Only if Predict is set to TRUE, parameter EvalPlausible can be set true!

Priors

Optional, [1:k] numerical vector defining the prior probabilities of the k classes. If missing, estimated from Cls.

UseMemshare

Optional boolean. If set to TRUE, then package functionality from Memshare is used, else classic library parallel is used.

...

Gaussian: Optional: Default=TRUE). Assume gaussian distribution.

Plausible: (Optional: TRUE: uses plausble bayesian theorem, FALSE non-plausible bayesian theorem

Type: (Optional: default=1, 1 = original PDE, 2 = R native density estimation

Threshold: Threshold for which the standard deviation cannot be smaller (default =1e-12)

PlotIt: Optional: Default=FALSE, TRUE: Plots Likelihoods

PlotCutOff: Optional: Scalar indicating how many features (starting from 1) should be plotted, or a numerical vector specifying the indices of the features to plot. Note: In the second case, avoid selecting too many features, as this may cause the plot to fail

ParetoRadiusPerFeauture: Optional [1:d] numerical vector for pareto radii computed priorly, see ParetoRadius or {ParetoRadius_fast}

cl: Optional: a cluster object, created by parallel, if given and ParetoRadiusPerFeauture missing, then ParetoRadiusPerFeauture is compputed multicore otherwise single core

Robust: Optional: Default=FALSE, TRUE: robust estimation of mean and std in case of Gaussian=TRUE

GlobalPR: Optional: Default=TRUE, FALSE: estimation of pareto radius for each class individually.

Details

Precomputation of ParetoRadiusPerFeauture can be usefull to make cross-validation faster although it should be only done on the training data.

If Plausible is not given, both options are evalauted using shannon information.

c_Kernels_list and ListOfLikelihoods have d elements each storing a matrix [1:m,1:k], usually m!=n. In contrast to DataLikelihoodsPerClass in which by interpolation the matrix are of size [1:n,1:k]

Value

Model

List of model parameters and results.
c_Kernels_list

List of matrices, where each matrix represent the kernels of one feature for all classes.
ListOfLikelihoods

List of matrices, where each matrix represent the likelihood of one feature for all classes.
PDFs_funs

Nested list of depth 1, where the first index assigns the feature index and the second index assigns the class. The elements are functions for the density estimation for each feature and each class.
ParetoRadiusPerFeauture

Numeric vector which stores the pareto radius for each feature.
Theta

Parameters mean and standard deviation of the Gaussian distributions per class and feaures.
Priors

Numeric vector which stores the prior probability of each class to appear.
PlausibleCenters

[1:k, 1:f] Numeric matrix which stores the centers for each feature and each class, where the row index assigns features and the column index assigns classes.
ClsTrain

[1:n] numerical vector with n numbers defining the classification. It has k unique numbers representing the arbitrary labels of the classification.
Posteriors

[1:n, 1:k] Numeric matrices with posterior probabilities.

Author(s)

Michael Thrun

References

[Stier et al., 2026] Stier, Q.,Hoffmann, J. & Thrun, M. C.: Classifying with the Fine Structure of Distributions: Leveraging Distributional Information for Robust and Plausible Naïve Bayes, Machine Learning and Knowledge Extraction (MAKE), Vol. 8(1), 13, doi 10.3390/make8010013, MDPI, 2026.

See Also

Predict_naiveBayes

Examples

if(requireNamespace("FCPS")){
data(Hepta)
Data=Hepta$Data
Cls=Hepta$Cls

#non-parametric
V=Train_naiveBayes_multicore(cl=NULL,Data=Data,Cls=Cls,Gaussian=FALSE,Predict=TRUE)
ClsTrain=V$ClsTrain
table(Cls,ClsTrain)
}