Your task:

Recall that Naive Bayes assumes all xi$$s are conditionally independent given y$.$ However, in this assignment, you will be building a not$-$so$-$Naive Bayes model where now xi$$s are correlated. In particular, we will only consider two features x$1$ and x$2,$ where x$1$ and x$2$ are not conditionally independent given y$.$ We assume f$($x$1,$x$2|$y $=$ k$)$ is a bi$-$variate Gaussian

$1$

distribution N$(\backslash$mu $1$k$,\backslash$mu $2$k$,\backslash$sigma $1,\backslash$sigma $2,\backslash$rho $),$ where $\backslash$mu $1$k and $\backslash$mu $2$k are means of x$1$ and x$2$ given y $=$ k$,\backslash$sigma $1$ and $\backslash$sigma $2$ are standard deviations of x$1$ and x$2,$ and $\backslash$rho is the correlation between x$1$ and x$2.$ Notethat$\backslash$mu $1$k and$\backslash$mu $2$k dependonthevaluekofy,but$\backslash$sigma $1,\backslash$sigma $2,$and$\backslash$rho donotdependony. Also recall that the density of a bi$-$variate Gaussian distribution, given $(\backslash$mu $1$k$,\backslash$mu $2$k$,\backslash$sigma $1,\backslash$sigma $2,\backslash$rho $),$ is:

$\backslash$sigma $2($x$1\backslash$mu $1$k$)2+\backslash$sigma $12($x$2\backslash$mu $2$k$)22\backslash$rho $\backslash$sigma $1\backslash$sigma $2($x$1\backslash$mu $1$k$)($x$2\backslash$mu $2$k$)2(1\backslash$rho $2)\backslash$sigma $2\backslash$sigma $2$

$12$

f$($x$1,$x$2|$y $=$ k$)$P$($y $=$ k$)$ P$($y $=$ k$|$x$1,$x$2)=$ Pi f$($x$1,$x$2|$y $=$ i$)$P$($y $=$ i$)$

Build a not$-$so$-$Naive$-$Bayes model that uses x$1$ and x$2$ as features to predict y$.$ Use train data to build your model and make predictions for the test set. When making predictions use the optimal Bayes classifier idea, i$.$e$.,$ if P$($y $=1|$x$1,$x$2)>=0\mathrm{.}5,$ classify the test instance as $1,$ else classify the test instance as $0.$ Finally, report the accuracy of your model in the TEST set.

Function Parameters:

train: This is the training data set used to train your model. Particularly, use train set to determine the necessary parameters of the model $\backslash$mu $1$k$,\backslash$mu $2$k$,\backslash$sigma $1,\backslash$sigma $2,\backslash$rho and P$($y $=$ i$).$ DO NOT use the training set to make predictions and compute the accuracy of your model as you should do it using the test set. train should be a pandas dataframe, that has three columns x$1,$ x$2,$ y$.$ Please note that you should use the entire train data to build your model, in other words, you DON$$T need to further split the training data to create another test set. You can find a sample training set in HW$5$traindata.csv$.$

test: This is the test data set used to make predictions and test the performance of your model. Please note that test data set should not be used during the training of the model and only be used to make predictions and compute the accuracy. test is also a pandas dataframe, that has three columns x$1,$ x$2,$ y$.$ A sample test data set can be found in HW$5$testdata.csv

Output: Your function should return class predictions for the test set and overall accu$-$ racy of your model in the test set. The type of predictions should be a numpy array and the type of the reported accuracy should be a numpy float.