**Segmentation based on Mixture and Bayesian Models applied to Prostate Cancer patients**
\*\*This is a Double-blind review, please do not include authors information in this version \*\*
Received Date: \*date
Accepted Date: \*date
Published Date: \*date
**HIGHLIGHTS**
- Segmentation of data using probabilistic models.
- Machine learning with the EM algorithm.
- Application of Mixed Models to Prostate Cancers.
- Gaussian Mixture Model and other algorithms.
**ABSTRACT**
*The growth in the amount of data in companies puts analysts in difficulties when extracting hidden knowledge from data. Several models have emerged that focus on the notion of distances while ignoring the notion of conditional probability density. This research study focuses on segmentation using mixture models and Bayesian networks for medical data mining. As enterprise data becomes large, there is a way to apply Data Ming methods to make sense of it using classification methods. We designed different models with different architectures and then applied these models to the medical database. The algorithms were implemented for the real data. The objective is to classify individuals according to the conditional probability density of random variables, in addition to identifying causalities between traits from tests of conditional independence and a correlation measure, both based on χ 2. After a quick illustration on several models (decision tree, SVM, K-means, Bayes), we applied our method to data from an epidemiological study (done at the University of Kinshasa university clinics) case-control of prostate cancer. Thus, we found after interpretation of the results followed by discussion that our model allows us to classify a new individual with an accuracy of 96% contrary to other non-probabilistic models which gave the result at 78%.*
***Keywords:** Classification, Mixture models, Gaussian mixture, Bayesian networks, Bayes classifier.*
# INTRODUCTION
In recent years, an increasing amount of data has been generated by companies, whether it is medical data, such as patient records seen in a hospital and their pathologies, banking data, such as credit card transactions, or industrial data, such as measurements from sensors on a production line, or any other type of data imaginable.
In each case, the analyst seeks to extract hidden information from the data. But no matter how big the data is, it will not allow him to do so. This gave rise to the idea of classifying the data, so that the most similar data could be grouped together in a single class. This technique is called automatic classification (SERHAL, H., ABDALLAH, N., MARION, J.M., & al., 2022 ; PREUX P., 2009).
Automatic data classification is the algorithmic categorization of objects. It consists of assigning a class or category to each object (or individual) to be classified, based on statistical data.
For a long time, this mathematical technique was based on the notion of distance, i.e. two elements were closer according to the distance separating them, as presented by *Ph. PREUX* in (PREUX P., 2009) and *CÔTÉ, Mélina and LAMARCHE, Benoît* in (CÔTÉ, Mélina and LAMARCHE, Benoît, 2022). Then, the problem was to know which metric to choose for a good ranking. Moreover, the choice of the number of classes was very crucial and sometimes depended on one analyst to another. Another problem was the sensitivity to atypical data, i.e. data that deviated a little from the others. The other problem was how to define the probability of an individual belonging to one class of data and not to another.
To do this, a question was raised, namely: *How to classify individuals according to the conditional probability density of the random variables?*
In an attempt to address some of the above problems, probabilistic models have emerged.
Thus, our study will focus on the probabilistic approach (OU, Guiliang, HE, Yulin, and HUANG, Joshua Zhexue , 2021) which has largely corrected the problem posed above. This approach is the use of the notion of conditional probability density as a Gaussian, or a mixture of Gaussians in classification. This time, it is the statistical parameters such as the mean or the variance that allow us to say that two elements are very close according to their proximity to their mean.
The above methods will be applied to prostate cancer, which is the leading cancer in men over 50 in industrialised countries. As diagnostic practices and treatment options have continued to evolve, making it possible to detect small tumours and guide targeted treatments with the aim of minimising the morbidity of the therapy (N Makni, 2013; [N Makni](https://scholar.google.fr/citations?user=HRxrLYEAAAAJ&hl=fr&oi=sra), [P Puech](https://scholar.google.fr/citations?user=Mp5IkGkAAAAJ&hl=fr&oi=sra), [O Colot](https://scholar.google.fr/citations?user=0KLh9r8AAAAJ&hl=fr&oi=sra), S Mordon, [N Betrouni](https://scholar.google.fr/citations?user=YcQ9x6MAAAAJ&hl=fr&oi=sra), 2011). The set of methods mentioned above will be crucial for the automated processing of medical data, with the aim of assisting and guiding the practitioner in the diagnostic decision making and therapeutic gesture, for the specific treatments of prostate cancer.
# METHODOLOGY
In the framework of our research we have opted for algorithms based on the mixture model (BIERNACKI, Christophe, CELEUX, Gilles, ECHENIM, Anwuli, et al., 2016) and Bayesian networks (OU, Guiliang, HE, Yulin, and HUANG, Joshua Zhexue , 2021). The motivation and detailed overview of the tasks and the proposed approach are described here. Today, it remains to be seen that prostate cancer has serious effects on men already beyond the age of 40, even leading to death.
Our approach addresses methods that help to understand and extract knowledge from masses of data on prostate cancer (Medjaher, Kamal, Amine M., & Noureddine Z., 2008), which can enable a province, a country to understand the harms of this disease based on other factors affecting it and their probabilities on the evolution of the said disease
# Segmentation by Mixture Model
The mixture model, by definition, is a statistical model used to parametrically estimate the distribution (density function) of random variables by modelling them as a sum of several other simple distributions as presented in (BIERNACKI, Christophe, CELEUX, Gilles, ECHENIM, Anwuli, et al., 2016; Nikseresht, Afshin, 2008)
This definition means that it is assumed that the whole population is represented by a probability distribution which is a mixture of C probability distributions associated with the classes. The main objective is to identify the C distributions by estimating their parameters. This identification consists of assuming that the observed data are realisations of a random vector X of unknown distribution P. The objective is to reconstruct P from its realisations.
The probability density \(f(x,\Theta)\ \)at a point is unknown. The principle of the model is to decompose this density into a sum of K components \(f_{k}\)( \(x\), \(\theta_{k}\)) (k =1, . . ., c) corresponding to the C classes whose parameters will be estimated \(\theta_{k}\)(k =1, . . . , c) from a sample X.
\[f(x,\Theta) = \sum_{k = 1}^{C}\pi_{k}f_{k}(x,\theta_{k})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\]
Where \(\pi_{k}\)is the a priori probability of component k. It does check the probability conditions such that \(\sum_{k = 1}^{C}{\pi_{k} = 1}\), and 0 ≤ \(\pi_{\text{k\ }}\)≤1. Θ and \(\theta_{k}\)are respectively the parameters of the model \(f(x,\Theta)\ \)and \(f_{k}\)( \(x\), \(\theta_{k}\)).
If X is a continuous distribution, i.e. we have an infinite number of components, then (1) is written
\(f(x,\Theta)\ = \int_{1}^{\infty}\pi_{k}f_{k}(x,\theta_{k})\) (2)
The probability densities \(f_{k}\)( \(x\), \(\theta_{k}\)) can be a distribution (parameter) from the large family of statistical distributions such as (Gaussian, Student).
But in the following we will detail the Gaussian mixture because it has some advantages compared to other distributions. In particular, in practice the specification of the components of a mixture is not always guided by the singularity of the data to be modelled. Gaussian mixtures, which assume conditional populations distributed according to a normal distribution, are of great interest because of their flexibility, their ability to approximate a wide variety of densities, their mathematically simple use and the generality of the normal distribution as shown by the central limit theorem.
# Gaussian Mixture Model
In order to describe Gaussian mixing, we first define the density function of the Gaussian distribution. We then introduce the notion of Gaussian mixing.
***Gaussian distribution***
The Gaussian distribution, also known as the normal distribution, is the best known of the probabilistic distributions. It has been widely used to model the distribution of continuous random variables (KIM, JaeKwang and SHAO, Jun, 2021; J Griffon, 2019).
> In the case of a simple random variable X, the density function of the Gaussian distribution can be written as follows:
> \(f\ (x,\mu,\sigma^{2}) = \frac{1}{\sigma\sqrt{2\pi}}e^{- \frac{1}{2\sigma^{2}}{(x - \mu)}^{2}}\) (3)
Where \(\mu\)is the mean \(\sigma^{2}\)and is the variance. If the distribution is multidimensional, the density function of the multivariate Gaussian distribution takes the following form:
\(f\)( \(x\), \(\mu_{k}{,\Sigma}_{k}\))=\(\frac{1}{{(2\pi)}^{\frac{d}{2}}\left| \Sigma_{k} \right|^{\frac{1}{2}}}e^{- \frac{1}{2}\left( x - \mu_{k} \right)^{t}\Sigma_{k}^{- 1}(x - \mu_{k})}\) (4)
with\(\ \Sigma_{k}\)the variance-covariance matrix, \(\mu_{k}\)the matrix of means, *d* the dimension of the Space of individuals.
Gaussian mixing has long been used in statistical learning. It can model any numerical data set, but with arbitrary precision.
The Gaussian mixture model is a convex combination of several Gaussian components. It is particularly used in cases where the data under study cannot be modelled by a simple Gaussian. In other words, if the data structure is naturally composed of several groups, it is necessary to represent them by a Gaussian mixture model rather than a simple Gaussian distribution (Nikseresht, Afshin, 2008).
To formalize this, simply replace each \(f_{k}\)( \(x\), \(\theta_{k}\)) by the density function of the Gaussian distribution. The density function of the Gaussian distribution is then written as follows:
\(f(x,\Theta)\ = \sum_{k = 1}^{C}\pi_{k}\mathcal{N}(x,\mu_{k}{,\Sigma}_{k})\) (5)
Where \(\mathcal{N}\)( \(x\), \(\mu_{k}{,\Sigma}_{k}\)) is the k component of the Gaussian mixture.
In this context, we define the set of weights as π = { \(\pi_{k}\)}, the set of means by µ= { \(\mu_{k}\)}, and the k-th class covariance matrices we denote Σ = { \(\Sigma_{k}\)}. We also define Θ = { \(\theta_{k}\)} and \(\theta_{k}\)= { \(\pi_{k}\), \(\mu_{k}{,\Sigma}_{k}\)}.
***Figure 1**: Density mixtures*
Here is a representation of the Gaussian components and the continuous curve shows the mixing density of these two Gaussians.
After a brief overview of Gaussian mixing, we now want to explain how the parameters of the Gaussian mixing model can be estimated, using the most well-known technique in this field: the EM (Estimation-Maximization) algorithm.
**Estimation of the parameters of mixing models**
Several estimation methods have already been developed in inferential statistics to estimate the parameters of the probability distribution of a sample, including point estimation, confidence interval estimation, etc., but the best known is the maximum likelihood method which is defined as follows:
Let the sample X= { \(x_{1}\). . ., \(x_{n}\)} with \(x_{1}\). . ., \(x_{n}\)independent realizations of a random vector X, the likelihood law of the data with respect to the model of parameter Θ is written :
\(L(\Theta)\ = \prod_{i = 1}^{n}{f(x_{i},\Theta)}\) (6)
*Θ= {* \(\theta_{k}\)*} and* \(\theta_{k}\)*= {* \(\pi_{k}\)*,* \(\mu_{k}{,\Sigma}_{k}\)*}.*
It is easier to maximize the log-likelihood instead of the likelihood function itself. If Θ maximises \(\ln\)(L( Θ)), then it also maximises L( Θ). This is due to the monotonicity of the log-likelihood function. The log-likelihood function is written:
\(ln\ L(\Theta) = \sum_{i = 1}^{n}\ln\ f(x_{i},\Theta)\) (7)
> This method is centralised by the EM algorithm in the case of Gaussian mixtures:
>
> ***EM algorithm***
Proposed by *DEMPSTER* in (DEMPSTER A., LAIRD N., RUBIN D, 1977; Biernacki C., Celeux G., & Govaert G., 2003) allows structural learning to evaluate the parameters and the score of all the networks in this neighbourhood, and to choose the best one for the next iteration in order to achieve a good ranking.
It is represented as follows:
> ***Input:*** C the number of components in the mixture, X a set of observations,
>
> \(\Theta^{(0)}\)= \[ \(\pi_{1}^{(0)}\), . . ., \(\pi_{c}^{(0)}\), \(\mu_{1}^{(0)}\), . . . ,\(\mu_{c}^{(0)}\), \(\Sigma_{1}^{(0)}\), . . . , \(\Sigma_{c}^{(0)}\)\] an initial value of the model parameters.
***Output:*** A (local) maximum \(\Theta^{*}\)of the likelihood law
1. > S ← 0 ;
2. > **Initialization of** the model \(\Theta^{(0)}\);
3. > **Do**
4. > (Step **-E: Estimation)** A posteriori probability calculation :
5. > **For i= 1.... N**
6. > **For k= 1.... C**
7.
> \(\gamma\)( \(z_{\text{ik}}\)) =\(\frac{\pi_{k}\mathcal{N(}x_{i},\mu_{k}{,\Sigma}_{k})}{\sum_{j = 1}^{C}{\pi_{j}\mathcal{N(}x_{i},\mu_{j}{,\Sigma}_{j})}}\)
8. > **Next**
9. > **Next**
10. > (Stage **- M: Maximisation)**
11. > **For k= 1.... C**
12. > Estimation of the a priori probability \(\pi_{k}\)of the k-th class :
> \(\pi_{k}^{(s + 1)}\)=\(\frac{\ 1}{N}\sum_{i = 1}^{N}{\gamma(z_{\text{ik}})}\)
>
> Estimation of the average \(\mu_{k}\)of class k
>
> \(\mu_{k}^{(s + 1)}\)=\(\frac{{\sum_{i = 1}^{N}{\gamma(z_{\text{ik}})}\text{.x}}_{i}}{\sum_{i = 1}^{N}{\gamma(z_{\text{ik}})}}\)
>
> Estimation of the \(\Sigma_{k}\)covariance matrix of class k
>
> \(\Sigma_{k}^{(s + 1)}\)=\(\frac{\sum_{i = 1}^{N}{\gamma\left( z_{\text{ik}} \right)}.(x_{i} - \mu_{k}^{\left( s + 1 \right)}){(x_{i} - \mu_{k}^{\left( s + 1 \right)})}^{t}}{\sum_{i = 1}^{N}{\gamma(z_{\text{ik}})}}\)
13. > **Next**
14. > **S** ← s+1
15. > **Up to** convergence
The convergence of the algorithm is achieved in one of two cases: either the difference in the likelihood function between two consecutive steps is negligible, or the new estimated parameters do not change with respect to the previous step \[9\].
Each individual is assigned to the class to which he or she is most likely to belong and this is determined by \(\gamma\)( \(z_{\text{ik}}\)).
The EM algorithm surely converges to a local optimum likelihood. This convergence depends directly on the initialization phase of the algorithm. A bad initialization could lead to a bad model, while a good initialization will rather lead to a convergence towards a global likelihood optimum.
To overcome the problems associated with the EM algorithm, a few variants have been proposed, including the SEM, CEM algorithm, which will be developed in our next articles.
In the next section, we present how the Bayesian classifier can be used to predict the class that a new individual may occupy.
# Bayesian Classifier
Bayesian Networks (BNs) have many advantages (probabilistic modelling of uncertainty, possibility of reasoning in both the diagnostic and symptom-diagnostic directions, etc.) which make them preferred tools in the diagnostic framework, particularly for medical diagnosis, where they were used as early as the 1980s in (KAPPEN H., WIEGERINCK W., TER BRAAK E., 2010) for a presentation of some applications of BNs in the medical field). Setting up a RB to model such a problem is fairly straightforward when the problem is simple (few variables, sufficient data and/or availability of an expert to learn the probabilities).
Thus, the naive Bayes classification (Philippe Leray, Olivier François, 2012), which has been used for a long time in statistical pattern recognition, can be considered as a very simple Bayesian network with all discrete variables, with the hypothesis that all symptoms are conditionally independent of the diagnosis. But then a classic question arises in the Machine Learning community: how to discretize continuous variables? This naive RB can benefit from the contributions of the "Bayesian networks" community to circumvent this difficulty, by making the assumption that the conditional probability density (CPD) is a Gaussian (mixed naive RB) (FRANÇOIS, Olivier, 2006; LAURITZEN S., 2016; OU, Guiliang, HE, Yulin, and HUANG, 2021), or a mixture of Gaussians (Nikseresht, Afshin, 2008).
***Bayesian Classifier***
> Given k classes and X a feature vector of size M, we choose the class \(C_{k}\)that maximises P( \(C_{k}\)/X), the probability that X belongs to \(C_{k}\): P ( \(\frac{C_{K}}{X)} = \max_{i = 1\ldots.k}P(\frac{C_{i}}{X)}\)with :
>
> \(P(\frac{C_{i}}{X)} = \frac{P(\frac{X}{C_{i})}P(C_{i})}{P(X)}\ \); (8)
>
> P(X)= \(\sum_{i = 1}^{k}{P\left( C_{i} \right)P(\frac{X}{C_{i})}}\)and
>
> P(Ci ) = \(\frac{N_{i}}{N}\)is the probability of observing the class \(C_{k}\)given the example set N.
>
> Or P(Ci )=1/k.
>
> \(P(\frac{X}{C_{i})}:\ \)The likelihood of the event "observe data x" if it is of class with examples of size N. This term is more difficult or impossible to estimate than the previous one. In the absence of other information, we use the "naive Bayes hypothesis".
# *Estimate of*\(\mathbf{P}\mathbf{(}\frac{\mathbf{X}}{\mathbf{C}_{\mathbf{i}}\mathbf{)}}\)
> When random variables are derived from a sequence of random events, their probability density takes the form of the normal distribution, N( \(\mu\), \(\sigma\)). This is proved by the central limit theorem. This is a frequent case in nature.
>
> The parameters of N( \(\mu\), \(\sigma\)) are the first and second moments of the examples. Thus, we can estimate them for any number of examples. We can even estimate the moments when there are no bounds (Xmax - Xmin) or when X is a continuous variable.
>
> In this case, p(x) is a "density" and we need a parametric function for p ( ).
In most cases, the density function for p(x) can be used.
\(p\left( x \right) = N\left( x;\mu,\sigma^{2} \right) = \frac{1}{\sqrt{2\pi}\sigma}e^{- \frac{{(x - \mu)}^{2}}{2\sigma^{2}}}\) (9)
> The basic "e" is: e = 2.718281828....
>
> The term \(\frac{1}{\sqrt{2\pi}\sigma}\)is used to normalise the function so that its area is 1.
# *Estimation of a Vector of Random Variables*
> For a vector of properties D
>
> \(\overrightarrow{\mu} \equiv E\left\{ \overrightarrow{X} \right\} = \frac{1}{M}\sum_{m = 1}^{M}\overrightarrow{X_{m}} = \begin{pmatrix}
> \mu_{1} \\
> \mu_{2} \\
> \ldots. \\
> \mu_{D} \\
> \end{pmatrix} = \begin{pmatrix}
> E\{ X_{1}\} \\
> E\{ X_{2}\} \\
> \ldots. \\
> E\{ X_{D}\} \\
> \end{pmatrix}\) (10)
>
> For D dimensions, the covariance between the variables xi and xj is estimated from M observations\(\left\{ \overrightarrow{X}m \right\}\)
\({\sigma_{\text{ij}}}^{2}\)=E{( \(X_{i} - E\left\{ X_{i} \right\}\))( \(X_{j} - E\left\{ X_{j} \right\}\))}
> \({\sigma_{\text{ij}}}^{2}\)= \(\frac{1}{M}\sum_{m = 1}^{M}{\left( X_{\text{im}} - \mu_{i} \right)(}\left( X_{\text{jm}} - \mu_{j} \right)\)And again, to avoid bias, we can use : \({\sigma_{\text{ij}}}^{2}\)=\(\frac{1}{M - 1}\sum_{m = 1}^{M}{\left( X_{\text{im}} - \mu_{i} \right)(}\left( X_{\text{jm}} - \mu_{j} \right)\)
>
> These coefficients make up a variance covariance matrix C\(C_{x} \equiv E\{\lbrack\overrightarrow{X} - \overrightarrow{\mu}\rbrack{\lbrack\overrightarrow{X} - \overrightarrow{\mu}\rbrack}^{T}\}\)
\[C_{x} \equiv \begin{pmatrix}
{\sigma_{11}}^{2} & {\sigma_{11}}^{2}\ \ldots.. & {\sigma_{1D}}^{2} \\
\ldots. & \ \ \ \ \ \ \ \ \ \ \ \ \ldots. & \ldots\ldots \\
{\sigma_{D1}}^{2} & {\sigma_{D2}}^{2}\ldots\ldots. & {\sigma_{\text{DD}}}^{2} \\
\end{pmatrix}\]
As we have just demonstrated with formula (10), it is also possible that the normal distribution takes the form (11) in the case of a vector of properties, \(\overrightarrow{X}\)the normal distribution :
> p(x)=N(x; \(\mu\), \(\sigma^{2}\))=\(\frac{1}{\left( 2\pi \right)^{\frac{D}{2}}\ det\left( C \right)^{\frac{1}{2}}}e^{- \frac{1}{2}\left( \overrightarrow{X} - \overrightarrow{\mu} \right){C_{x}}^{- 1}\left( \overrightarrow{X} - \overrightarrow{\mu} \right)}\) (11)
>
> The term \(\left( 2\pi \right)^{\frac{D}{2}}\det\left( C \right)^{\frac{1}{2}}\)is a normalisation factor.
>
> According to the law of large numbers, we have :
>
> \(P(\frac{X}{C_{i})} = \frac{1}{{(2\pi)}^{\frac{D}{2}}\sqrt{\det\text{Ψ}_{i}}}e^{- \frac{1}{2}\left( x - \mu_{i} \right){\text{Ψ}_{i}}^{- 1}\left( x - \mu_{i} \right)}\) (12)
>
> With
>
> \(\text{Ψ}_{i} = \frac{1}{N}\sum_{n = 1}^{N}{\left( x_{n} - \mu_{i} \right)\left( x_{n} - \mu_{i} \right)^{T}}\ \)the variance-covariance matrix
>
> Certain assumptions must be taken into account and play a very important role, namely
>
> \- Normal distribution within each class C i
>
> \- The variables are correlated with each other
# *Risk Assessment*
> Not all decisions have the same impact and it all depends on the field of application as the cases differ:
>
> \- Lending to a high risk patient versus not lending to a low risk patient
>
> \- Medical diagnosis: possible impacts of not detecting a serious disease
>
> \- Intrusion detection
>
> \- Targeting a customer who is not interested in the product
>
> \- Assignment of a consumer to another class of consumers who do not have the same characteristics.
>
> All these questions lead us to evaluate the risk of misallocation of individuals. This risk can be quantified by the loss function: L ( \(\alpha_{i}\); C j)
>
> Performing an action when the true class is Cj. Thus, the expected risk of an action \(\alpha_{i}\)is evaluated as follows:
>
> \(R(\alpha_{i}/x) = \sum_{k = 1}^{K}{\mathcal{L}\left( \alpha_{i},c_{k} \right)P\left( \frac{c_{k}}{x} \right)}\) (13)
>
> And the action that minimises the risk is :
\[\alpha_{i} = {\arg\min}_{\forall\alpha}R(\frac{\alpha}{x})\]
> To change the loss function that modifies the risk, one must change the cost associated with a false negative to the cost of the false positive.
>
> Based on the confusion matrix (Guiliang, HE, Yulin, and HUANG, Joshua Zhexue, 2021; FRANÇOIS O., 2006; Philippe L., Olivier F., 2012).
>
> **Table 1:** Confusion matrix for risk assessment
> We have : \(\mathcal{L}\left( \alpha = 1,C = 0 \right) = \lambda_{\text{FP}}\)
>
> \(\mathcal{L}\left( \alpha = 0,C = 1 \right) = \lambda_{\text{FN}}\)
>
> This brings us to the zero-one loss function of a matrix of class k :
>
> L ( \(\alpha_{i}\), \(c_{j}\))=\(\left\{ \begin{matrix}
> 0\ \text{si}\alpha_{i} = c_{j} \\
> 1\ \text{si}\alpha_{i} \neq c_{j} \\
> \end{matrix} \right.\ \)
>
> The corresponding risk is then :
>
> R (\(\alpha_{i}/x) = \ \sum_{k = 1}^{K}{\mathcal{L}\left( \alpha_{i},c_{k} \right)P\left( \frac{c_{k}}{x} \right)}\)
>
> \=\(\sum_{k \neq i}^{K}{P\left( \frac{c_{k}}{x} \right)}\)
>
> \=1-\(P\left( \frac{c_{i}}{x} \right)\)
# RESULTS AND INTERPRETATIONS
In our application the objective is to classify people living with prostate cancer and to determine the level of intervention and follow-up, the correlation between the different variables that are taken into account; as well as to predict the class to be assigned to a new individual.
Based on the Dataset we designed, we ran several machine learning algorithms developed in Python to achieve the interpretation of the results.
#
#
# Data Source
The research sample was drawn from the database of the University Clinics of the University of Kinshasa.
The data collection period was from January 2017 to December 2021. The first step was to collect data by selecting all patients diagnosed with prostate disease. To determine the classification of disease types, compiled on the basis of a system of categories and grouped into a disease according to predetermined criteria known as the International Statistical Classification of Diseases and Related Health Problems, Tenth Revision (ICD10, 2014) .
# Displaying the prostate cancer dataset, coding variables and applying models
**This is the presentation of the data that constitute our Dataset on which the different models will be applied in order to make the objective analyses and prove the advantage of our model.**
**Figure 2. DataSet containing Prostate Cancer data**
**This is a table showing the situation of the evolution of prostate cancer patients. On the basis of these data our methodology is applied.**
**Figure 3**: Importing libraries
**Figure 4**: Coding of categorical variables
**Figure 5**: Analysis of the dataset with some statistical parameters
**Figure 6**: Data separation for training
**Figure 7**: Separating training and test data
**Figure 8:** Nominalisation
**Figure 9**: Modelling with the combination of classifiers (SVM, Decision tree and K-Means)
# Validation of the model with the Roc Curve
After training with the non-probabilistic models on the data we found the results to be 78% satisfactory. This is shown in Figure 3 by the rocky curve. By seeing this, we can say that our models have learned well and can classify with good accuracy a new individual in its class (prostate cancer or not).
**Figure 10. The Roc curve**
**Modelling with the Mixture and Bayes models**
**Figure 11:** Application of mixture model and Bayes classifier
We obtained the same result, the accuracy of the Bayesian model is 96%, with this the classification of a new individual is done with a high accuracy, our chosen classifier is very efficient.
**CONCLUSIONS AND DISCUSSIONS**
In this paper, we have proposed a new method inspired by the work on data segmentation for machine learning using probabilistic models.
The objective was to classify individuals according to the conditional probability density of random variables, in addition to identifying causalities between traits using tests of conditional independence and a correlation measure, both based on χ 2. After a quick illustration on several models (decision tree, SVM, K-means, Bayes), we applied our method to data from an epidemiological case-control study of prostate cancer. We found that our model allows us to classify a new individual with an accuracy of 96% contrary to other non-probabilistic models.
The results obtained show that the methods developed are sufficiently accurate, rapid and robust to be used in a clinical context. These tools prove their ability to offer a gain in turnaround time and reproducibility of diagnostic and therapeutic decisions in the prostate.
# REFERENCES
SERHAL, Hassan, ABDALLAH, Nassib, MARION, Jean-Marie, et al. (2022). "Overview on prediction, detection, and classification of atrial fibrillation usingwavelets and AI on ECG". *Computers in Biology and Medicine*, p. 105168.
PREUX P., (2009). "Data mining "*Lecture notes available at* , *Université Lille 3*.
CÔTÉ, Mélina and LAMARCHE, Benoît. (2022). "Artificial intelligence in nutrition research: perspectives on current and future applications". *AppliedPhysiology, Nutrition, and Metabolism*, vol. 47, no. 1, pp. 1-8.
RENOOIJ S., (2001). "Probability Elicitation for BeliefNetworks: Issues to Consider",*Knowledge Engineering Review*, *vol. 16, no. 3*, pp. 255-269.
BOUNEBACHE, Said Karim, QUANTIN, Catherine, BENZENINE, Éric, et al. (2018). " Bibliographic Review of Database Linking Methods: Applications and Perspectives in the Case of Public Health Data". *Journal de la société française de statistique, vol. 159, no. 3, pp.* 79-123.
B.A. Turlach. (1993). "Bandwidth selection in kernel density estimation: A review. *CORE and Institut de Statistique, pages 23-493*.
BIERNACKI, Christophe, CELEUX, Gilles, ECHENIM, Anwuli, et al. (2016). "The MIXMOD mixture analysis software for classification and discriminant analysis*". MODULAD Journal, no. 35*, pp. 25-44.
Nikseresht, Afshin. (2008). "Estimation of probabilistic mixture models: a proposal for distributed and decentralized operation". *Diss. University of Nantes*.
KIM, JaeKwang and SHAO, Jun. (2021). "Statistical Methods for Processing Incomplete Data". *Chapman and Hall/CRC*.
DEMPSTER A., LAIRD N., RUBIN D. (1977). "Maximum LikelihoodfromIncompete Data Via the EM Algorithm", *Journal of the Royal Statistical Society, vol. B 39*, pp. 1-38*.*
C. Biernacki, G. Celeux, and G. Govaert. (2003). "Choosingstarting values for the EM algorithm for getting the highestlikelihood in multivariategaussian mixture models. ComputationalStatistics and Data Analysis", 41 :561-575.
KAPPEN H., WIEGERINCK W., TER BRAAK E. (2010). "Decision support for medical diagnosis", *MEIJ J., Ed., Dealing with the data flood. Mining data, text and multimedia, The Hague: STT/ Bewetong (Studycentre for Technology Trends, 65)*.
LAURITZEN S. (2016). "Propagation of Probabilistics,Means and Variances in Mixed Graphical Association Models", *Journal of the American Statistical Association, vol. 87*, pp. 1098-1108.
Philippe Leray, Olivier François. (2012). "Réseaux Bayésiens de Méthodologie de Classification et d'Illustration en Diagnostic Médical", *INSA Rouen/ PSI, FRE CNRS 2645 BP 08 - Av. de l'Université 76801St-Etienne du Rouvray Cédex*.
Medjaher, Kamal, Amine Mechraoui, & Noureddine Zerhouni (2008). "Diagnosis and prognosis of failures by Bayesian networks.*" 4th Journées Francophones sur les Réseaux Bayésiens, JFRB'2008. Vol. 1*.
FRANÇOIS, Olivier. (2006). "From Bayesian network structure identification to pattern recognition from complete or incomplete information". *PhD thesis. INSA of Rouen.*
OU, Guiliang, HE, Yulin, and HUANG, Joshua Zhexue. (2021). "A compressed naive hidden Bayesian classifier". *In: 2021 International Joint Conference on Neural Networks (IJCNN). IEEE*. p. 1-7.
World Health Organization. (2014). "ICD10: International Statistical Classification of Diseases and Related Health Problems: Tenth Revision, 2nd edition".
Nasr Makni. (2010). "Identification, diagnostic support and planning methods using multi-modality imaging for focal prostate cancer therapies". PhD thesis, University of Sciences and Technology of Lille - Lille I
[N Makni](https://scholar.google.fr/citations?user=HRxrLYEAAAAJ&hl=fr&oi=sra), [P Puech](https://scholar.google.fr/citations?user=Mp5IkGkAAAAJ&hl=fr&oi=sra), [O Colot](https://scholar.google.fr/citations?user=0KLh9r8AAAAJ&hl=fr&oi=sra), S Mordon, [N Betrouni](https://scholar.google.fr/citations?user=YcQ9x6MAAAAJ&hl=fr&oi=sra). (2011). [Hybrid approach combining Markov fields and statistical shape model for prostate contour extraction in MRI](https://www.sciencedirect.com/science/article/pii/S1959031811000923), IRBM – Elsevie.
Griffon, J., (2019). [Ultrasound multi-parametric modelling and characterisation for in vivo assessment of tumour heterogeneity and its influence on therapeutic progression](https://theses.hal.science/tel-02946781/), theses.hal.science.