I need to "construct a distribution" that I can run in a computer program, in a particular way that is not clearly specified.
Say I have the following function,
$$ f(x) = (4-2.1x^2 + \frac{y^4}{3})x^2 + xy + (-4 + 4x^2)y^2 $$
And I am going to explore the space $ x \in [-3,3], y \in [-2,2]$ in search of the global min.
Say I have $n$ samples, $x\in[-3,3],y\in[-2,2]$:
$$ (x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n) $$
These samples evaluate to values that are higher and lower. I throw away the bottom half of the values, and keep the top half.
In order to continue, I need to estimate the mutual information gain between the $y$'s and the $x's$, and then produce samples according to that distribution (eventually, I need to scale up to more dimensions). To do that, I need the conditional distribution, the joint distribution, etc.
Mutual Information Gain: $$ I[X;Y] = \int_Y \int_X p(x,y) log(\frac{p(x,y)}{p(x)p(y)})dxdy $$
How do I "estimate" the conditional distribution of all remaining $x$'s, given, say, all $y$'s, using the $\frac{n}{2}$ $x$'s remaining, relative to all $n$ $y$'s?