Hi, thanks for the question :). This is essentially the approximation to the expectation by sampling. If we have E_p[f(X)] = int_X p(X) * f(X) dX Then we can just approximate this by sampling a couple of variates from p, evaluate f at these points and then take the mean over the evaluations. Or mathematically E_p[f(X)] \approx 1/L sum_i f(x_i) with x_i ~ p I think this should also relate to your question: In the notation with the expectation operator E the distribution appears in the subscript E_p. When equally defining it by an integral it appears in the multiplication, but once we approximate it by sampling it only appears as the distribution we sample from. We never evaluate the probability of those samples. Maybe what you are referring to is a weighted mean. This can be seen as an approximation to the expectation by importance sampling: en.m.wikipedia.org/wiki/Importance_sampling I hope that answered your question 😊 let me know if sth is unclear.
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ffs finally a tutorial with black background!
Appreciate it 😊
I really like the white on black.
at around 6:00-6:30 shouldn't the mean be weighted by the probability?
Hi,
thanks for the question :).
This is essentially the approximation to the expectation by sampling.
If we have
E_p[f(X)] = int_X p(X) * f(X) dX
Then we can just approximate this by sampling a couple of variates from p, evaluate f at these points and then take the mean over the evaluations.
Or mathematically
E_p[f(X)] \approx 1/L sum_i f(x_i)
with
x_i ~ p
I think this should also relate to your question: In the notation with the expectation operator E the distribution appears in the subscript E_p. When equally defining it by an integral it appears in the multiplication, but once we approximate it by sampling it only appears as the distribution we sample from. We never evaluate the probability of those samples.
Maybe what you are referring to is a weighted mean. This can be seen as an approximation to the expectation by importance sampling: en.m.wikipedia.org/wiki/Importance_sampling
I hope that answered your question 😊 let me know if sth is unclear.