a) no - complex enough model such as a CNN will be able to transcribe very well

b) yes - can be influenced by human emotions

c) no* - complex enough model will be able to transcribe very well

d) yes - can be influenced by other external confounding factors not accounted for in the model such as going viral on tiktok

a)

$$ P(y = 1 | x) = 1/2 \\ = \frac{1}{1+exp(-w^T x + b)} = \frac{1}{2}\\ = 1+exp(-(w^Tx+b))=2\\ = exp⁡(−(w^Tx+b))=1\\ −(w^Tx+b) = 0\\ w^Tx+b = 0 $$

b)

$$ P(y = 1 | x) = 3/4 \\ = \frac{1}{1+exp(-w^T x + b)} = \frac{3}{4}\\ = 1+exp(-(w^Tx+b))= \frac{4}{3}\\ = exp⁡(−(w^Tx+b))=\frac{1}{3}\\ −(w^Tx+b) = ln(3)\\ w^Tx+b = -ln(3) $$

c)

$$ P(y = 1 | x) = 1/4 \\ = \frac{1}{1+exp(-w^T x + b)} = \frac{1}{4}\\ = 1+exp(-(w^Tx+b))= 4\\ = exp⁡(−(w^Tx+b))=3\\ −(w^Tx+b) = ln(3)\\ w^Tx+b = -ln(3) $$

$$ % Probability density function for a spherical Gaussian p(x|y) = \frac{1}{(2\pi\sigma^2)^{\frac{d}{2}}} \exp\left(-\frac{||x - \mu_y||^2}{2\sigma^2}\right)\\ \text{Bayes' theorem: }\\ P(y = 1|x) = \frac{p(x|y = 1)P(y = 1)}{p(x|y = 1)P(y = 1) + p(x|y = 2)P(y = 2)}\\ \text{Substituting the Gaussian distributions and priors into Bayes':}\\ P(y = 1|x) = \frac{\exp\left(-\frac{||x - \mu_1||^2}{2\sigma^2}\right)}{\exp\left(-\frac{||x - \mu_1||^2}{2\sigma^2}\right) + \exp\left(-\frac{||x - \mu_2||^2}{2\sigma^2}\right)}\\

% Simplifying by taking the difference of the exponents\\ P(y = 1|x) = \frac{1}{1 + \exp\left(-\left(\frac{||x - \mu_1||^2 - ||x - \mu_2||^2}{2\sigma^2}\right)\right)}\\

% Denote Delta mu and c \Delta\mu = \mu_2 - \mu_1, c = \frac{||\mu_2||^2 - ||\mu_1||^2}{2\sigma^2}\\

% Final form of the posterior probability using logistic function

P(y = 1|x) = \frac{1}{1 + \exp\left(-\left(x^T\frac{\Delta\mu}{\sigma^2} - c\right)\right)} $$

The equation for the likelihood of y = 1 given x has the form of the logistic regression function where the inputs are transformed linearly according to the mean vectors of the Gaussian distributions for each class and the variance. The squashing function, in this case, the logistic sigmoid function, ensures that the output is a valid probability between 0 and 1.