Short Note — Any Two (all three covered for completeness):

(a) Naive Bayes:
Bayes classifier with strong independence assumption.
$P(C_k|x) \propto P(C_k)\prod_i P(x_i|C_k)$. Types: Gaussian (continuous $x_i\sim N(\mu_{k,i},\sigma^2)$), Multinomial (counts/TF for text), Bernoulli (binary). Training: Estimate priors $P(C_k)=N_k/N$, likelihoods via MLE/MAP with Laplace smoothing $\alpha=1$ to avoid zero counts. Prediction: $\hat C =\arg\max_k P(C_k)\prod P(x_i|C_k)$ (log sum for stability). Pros: Fast $O(n d)$, works well on high-dim sparse text, needs little data. Cons: independence violated often, poor probability calibration. Use for spam, sentiment, document classification.

(b) Logistic Regression:
Discriminative linear classifier modeling $P(y=1|x)=\sigma(w^T x+b)$, $\sigma(z)=1/(1+e^{-z})$. Decision boundary linear. Loss = binary cross-entropy $J=-\frac1m\sum[y\log\hat y+(1-y)\log(1-\hat y)]$ (MLE). Gradient $
abla J = X^T(\hat y - y)/m$, solved via GD/SGD/Newton. Regularize$L1/L2$. Multi-class via softmax$P(y=k|x)=\exp(w_k^T x)/\sum \exp(w_j^T x)$. Interpretable coefficients (odds ratio), outputs probabilities, good baseline. Example:$w^T x=0$ → 0.5 threshold.

(c) Back Propagation:
Algorithm to train MLPs by efficient gradient via chain rule. Forward: $a^{(l)}=f(W^{(l)}a^{(l-1)}+b^{(l)})$, compute loss. Backward: $\delta^{(L)}=
abla L \odot f'(z^{(L)})$,$\delta^{(l)}=W^{(l+1)T}\delta^{(l+1)}\odot f'(z^{(l)})$, grads$\partial L/\partial W^{(l)}=\delta^{(l)}a^{(l-1)T}$. Update$W:=W-\eta\partial L/\partial W$. Enables deep learning,$O(W)$complexity vs naive$O(W^2)$. Needs differentiable activations, suffers vanishing gradient (solved by ReLU, residual, batchnorm).

For exam attempt any two → choose (a) and (b) for completeness, or (b)+(c) as requested.