To begin, load the Titanic dataset from Seaborn and focus on the relevant variables. The target variable for this analysis is 'survived', which indicates if a passenger survived $(1)$ or died $(0).$ The other variables will serve as input features.

In a Markdown cell, identify the categorical and numerical features by examining the possible values each feature can assume. Remember, there are four categorical and four numerical features.

Next, partition the data into training and testing sets, ensuring to stratify based on the target variable to maintain the distribution. Extract $20\%$ of the data for testing. The resulting DataFrames should be named df$\_$train and df$\_$test. Print the dimensions of these datasets.

Using the training data, calculate the probability of each target label, P$($yk$).$

For the categorical features in the training data, calculate and show P$($xj$$yk$),$ the probability of each feature given the target label.

Identify and display the means and standard deviations of the continuous features. The probability of observing any value for these features is assumed to follow the Normal Distribution:

P$($xj$$yk$)=$N$($xj$,$xj$)$

where $$j and $$j are calculated from each feature j and target label yk$.$

Construct a prediction function that calculates:

P$($yk$$x$($i$))=$argmaxk in $\{0,1\}[$P$($yk$)\backslash$times $$jfP$($xj$($i$)$yk$)]$

where yk$=0$ means Died and yk$=1$ means Survived. This function should calculate the above expression for k$=0$ and k$=1,$ and return both predictions for each instance considered.

def calculate$\_$class$\_$predictions$($data$,$ return$\_$normed$\_$probs$=$ False$)$:

predicted$\_$probs$\_$per$\_$instance $=$ np$.$empty$($shape$=(0,2))$

## Construct function here

predicted$\_$probs$\_$per$\_$instance $=$ np$.$vstack$([$predicted$\_$probs$\_$per$\_$instance, np$.$array$($prob$\_$died, prob$\_$survived$)])$

return$($predicted$\_$probs$\_$per$\_$instance$)$

Using your function, calculate the predictions from the training data and then compute the confusion matrix and accuracy score.

Repeat the prediction using the test data and calculate the confusion matrix and accuracy score for this set as well.