6. Gaussian Mixture
Let µ0, µ1 ∈ R d , and let Σ0, Σ1 be two d × d positive definite matrices (i.e. symmetric with positive eigenvalues).
We now introduce the two following pdf over R d :
These pdf correspond to the multivariate Gaussian distribution of mean µ0 and covariance Σ0, denoted Nd(µ0, Σ0), and the multivariate Gaussian distribution of mean µ1 and covariance Σ1, denoted Nd(µ1, Σ1).
We now toss a balanced coin Y, and draw a random variable X in R d , following this process : if the coin lands on tails (Y = 0) we draw X rom Nd(µ0, Σ0), and if the coin lands on heads (Y = 1) we draw X from Nd(µ1, Σ1).
(a) Calculate P(Y = 0|X = x), the probability that the coin landed on tails given X = x ∈ R d , as a function of µ0, µ1, Σ0, Σ1, and x. Show all the steps of the derivation.
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