Visualising (unnormalised) posterior functions

[MichaelClerx]

The theory of samplers usually starts from the idea that we want to calculate some function over an unknown distribution, but we don't know the distribution exactly, so we draw a bunch of samples and estimate the function over them instead.

But a lot of the time in applications we're never calculating this function, we're just drawing (some 2d projections of) the resulting posterior.

The way we do that is a bit odd: We (1) throw away the actual (unnormalised) function values, e.g. the y-coordinates, (2) we use the density of samples in each region (x-coordinate) as a proxy for the function value.

Why not just (1) save all points we evaluate (xs and ys) and then (2) draw the function

Essentially this means using the (accepted and rejected) points in parameter space as a grid to draw our function with, instead of using a regular or uniformly sampled grid over the prior