Let θ1 and θ2 be unknown parameters for the component densities and respectively. Assume that θ1 and θ2 are initially statistically independent, so that (a) Show that after one sample x1 from the...

Let θ₁
and θ₂
be unknown parameters for the component densities

and

respectively. Assume that θ₁
and θ₂
are initially statistically independent, so that

(a) Show that after one sample x₁
from the mixture density is observed,

can no longer be factored as

If

(b) What does this imply in general about the statistical dependence of parameters in unsupervised learning?