I am new to the machine learning area. Then I am studying a paper that considers the problem of logistic regression in Bayesian. In general, in regression or logistic regression when they assume the prior or likelihood is Gaussian why in the formula of Gaussian $p(x)= \frac{1}{\sqrt(2\pi \sigma)}\mathrm{exp}(-\frac{(x-\mu)^2}{2\sigma^2})$ instead of $x$ they put $y_i-\theta^tx_i$?

I am new to the machine learning area. Then I am studying a paper that considers the problem of logistic regression in Bayesian. In general, in regression or logistic regression when they assume the prior or likelihood is Gaussian why in the formula of Gaussian $p(x)= \frac{1}{\sqrt{2\pi \sigma^2}}\mathrm{exp}(-\frac{(x-\mu)^2}{2\sigma^2})$ instead of $x$ they put $y_i-\theta^tx_i$?

I am new to the machine learning area. Then I am studying a paper that considers the problem of logistic regression in Bayesian. Then I do not know in the $\pi$ formula in the following which one is prior?

Define the vector $Y=(y_1,\cdots,y_n)\in \{0,1\}^n$ and let $X$ be the $n\times d$ matrix with $x_i$ as $i^{th}$-row. We choose the prior $\pi_0$ to be a Gaussian distribution with zero mean and covariance matrix proportional to the inverse of the sample covariance matrix $\sum_X=\frac{1}{n} X^TX$. Plugging in the formula for the prior and likelihood, we find that the posterior density is given by $\pi(\theta)=\pi(\theta|X,Y)\propto \mathrm{exp}\left\{ Y^TX\theta - \sum_{i=1}^n \log(1+e^{\theta^Tx_i})-\alpha||\sum_X^{1/2}\theta||_2^2\right\}$.

And in general, in regression or logistic regression when they assume the prior or likelihood is Gaussian why in the formula of Gaussian $p(x)= \frac{1}{\sqrt(2\pi \sigma)}\mathrm{exp}(-\frac{(x-\mu)^2}{2\sigma^2})$ instead of $x$ they put $y_i-\theta^tx_i$?

I am new to the machine learning area. Then I am studying a paper that considers the problem of logistic regression in Bayesian. In general, in regression or logistic regression when they assume the prior or likelihood is Gaussian why in the formula of Gaussian $p(x)= \frac{1}{\sqrt(2\pi \sigma)}\mathrm{exp}(-\frac{(x-\mu)^2}{2\sigma^2})$ instead of $x$ they put $y_i-\theta^tx_i$?

I am new to the machine learning area. Then I am studying a paper that considers the problem of logistic regression in Bayesian. Then I do not know in the $\pi$ formula in the following which one is prior?

Define the vector $Y=(y_1,\cdots,y_n)\in \{0,1\}^n$ and let $X$ be the $n\times d$ matrix with $x_i$ as $i^{th}$-row. We choose the prior $\pi_0$ to be a Gaussian distribution with zero mean and covariance matrix proportional to the inverse of the sample covariance matrix $\sum_X=\frac{1}{n} X^TX$. Plugging in the formula for the prior and likelihood, we find that the posterior density is given by $\pi(\theta)=\pi(\theta|X,Y)\propto \mathrm{exp}\left\{ Y^TX\theta - \sum_{i=1}^n \log(1+e^{\theta^Tx_i})-\alpha||\sum_X^{1/2}\theta||_2^2\right\}$.

And in general, in regression or logistic regression when they assume the prior or likelihood is Guassian why in the formula of Guassian $p(x)= \frac{1}{\sqrt(2\pi \sigma)}\mathrm{exp}(-\frac{(x-\mu)^2}{2\sigma^2})$ in stead of $x$ they put $y_i-\theta^tx_i$?

I am new to the machine learning area. Then I am studying a paper that considers the problem of logistic regression in Bayesian. Then I do not know in the $\pi$ formula in the following which one is prior?

Define the vector $Y=(y_1,\cdots,y_n)\in \{0,1\}^n$ and let $X$ be the $n\times d$ matrix with $x_i$ as $i^{th}$-row. We choose the prior $\pi_0$ to be a Gaussian distribution with zero mean and covariance matrix proportional to the inverse of the sample covariance matrix $\sum_X=\frac{1}{n} X^TX$. Plugging in the formula for the prior and likelihood, we find that the posterior density is given by $\pi(\theta)=\pi(\theta|X,Y)\propto \mathrm{exp}\left\{ Y^TX\theta - \sum_{i=1}^n \log(1+e^{\theta^Tx_i})-\alpha||\sum_X^{1/2}\theta||_2^2\right\}$.

And in general, in regression or logistic regression when they assume the prior or likelihood is Gaussian why in the formula of Gaussian $p(x)= \frac{1}{\sqrt(2\pi \sigma)}\mathrm{exp}(-\frac{(x-\mu)^2}{2\sigma^2})$ instead of $x$ they put $y_i-\theta^tx_i$?