Machine Learning , Bayesian decision classification, GMM

Implement Bayesian decision classification using Gaussian Mixture Models (GMM) to classify new data points into one of two classes based on 2D synthetic data.

1-Load synthetic 2D data for two classes (X1 and X2) from a given Excel file. 2- Load new observations to be classified into one of the two classes from the same Excel file. 3- Implement GMM for both classes (X1 and X2). 4- Calculate the prior probabilities for both classes. 5- Compute the likelihood probabilities using GMM for the new observations. 6- Apply Bayes' Theorem to calculate the posterior probabilities. 7- Use the decision rule to classify new observations based on the higher posterior probability. 8- Visualize the results, including posterior probabilities and decision boundaries, for different GMM component numbers.

\begin{tabular}{l|r|r|r|r|r|r|} 1 & X1_data_1 & X1_data_2 & \multicolumn{1}{|c|}{ X2_data_1 } & X2_data_2 & New_Observations_1 & New_Observations_2 \\ \cline { 2 - 8 } & 102.4835708 & 49.56276989 & 181.9596821 & 72.17190913 & 135.6301453 & 54.56924752 \\ \hline 3 & 103.2384427 & 54.81624329 & 185.932116 & 74.0813578 & 96.20130463 & 75.26528317 \\ \hline 4 & 98.82923313 & 49.25959393 & 172.4541941 & 66.36781924 & 179.419111 & 79.99011857 \\ \hline 5 & 107.8960641 & 52.4268417 & 182.8209643 & 71.98988443 & 199.6300521 & 64.4401097 \\ \hline 6 & 97.65262807 & 51.71572551 & 182.8210324 & 84.9215649 & 174.5886157 & 78.06680055 \\ \hline 7 & 97.68291154 & 48.5272332 & 183.1268961 & 74.39802681 & 183.461213 & 53.65718356 \\ \hline 8 & 101.2098114 & 43.94967662 & 185.2252829 & 72.52282747 & 139.5598549 & 49.52057953 \\ \hline 9 & 91.37541084 & 48.22189071 & 178.2731992 & 72.93947515 & 194.945613 & 66.21611221 \\ \hline 10 & 94.9358444 & 50.99373732 & 175.767062 & 69.08280548 & 115.1507086 & 68.50952395 \\ \hline 11 & 95.45987962 & 45.53390356 & 177.3415544 & 70.31709718 & 157.9941065 & 57.53569513 \end{tabular}