Suppose that we are training a nave Bayes classifier and a logistic regression classifier: f f :

Suppose that we are training a naïve Bayes classifier and a logistic regression classifier: $f$ : $X\to Y$, which maps a $d$-dimensional real-valued feature vector $X\in R^d$ to a binary class label $Y\in \{0,1\}$. In the naïve Bayes classifier, we assume that all $X_i$ where $i=1,\dots ,n$ are conditionally independent given the class label $Y$ and the class prior $P\left(Y\right)$ follow the Bernoulli distribution with $P(Y=1)=\theta$. Now, prove the equivalence of logistic regression and naïve Bayes under these two assumptions.

a. For each $X_i$, we assume it is drawn from the Gaussian distribution $P(X_i\mid Y=k)\sim$ $N({\mu }_{ik},{\sigma }_{ik})$ where $k=0,1$. We also assume that ${\sigma }_{i0}={\sigma }_{i1}={\sigma }_i$.

b. For each $X_i$, we assume it is drawn from the Bernoulli distribution $P(X_i=1\mid Y=k)=p_k$ where $k=0,1$.