# MAP in high dimensions

I’m currently trying to find the maximum of my posterior distribution but I’m having trouble finding the correct maximum since I have a lot of local maxima.

The sampled chains can be seen in this image:

I have three RVs: amplitude, l & m with shape=3

This is how I currently try to find the maximum:
For each RV combination (amplitude[i],l[i],m[i]) I calculate a KDE in 3D. The maximum of this KDE is then used to find the best values for (amplitude[i],l[i],m[i]).

The problem I’m having is that, as far as I know, in order to find the global maximum I should calculate the KDE in the higher dimensional space where my axes are given by all RVs i=1,2,3. This leads to 9 axes. If I evaluate this KDE on a grid with just 10 points per axis I already get an 10^9 sized array. With a finer gridding this results in a memory overflow…
This also scales very badly with more parameters shape=N \rightarrow 10^N

Is there any other way to find the global maximum?

I can’t answer your question directly, but I can note that this is part of the reason MAPs are generally discouraged. The amount of mass near the MAP shrinks rapidly as dimensional increases.

@cluhmann Is there a any good alternative for MAP, that I could use instead?

The answer to that question requires further information about why you are trying to find the MAP? If you were to actually get it, what were you planning on doing with it?

The answer to that question requires further information about why you are trying to find the MAP? If you were to actually get it, what were you planning on doing with it?

Let me quickly explain what I’m trying to do:
I have a 2d image with three gaussian/normal distribution in it, which looks like this. For every distribution in the image I want to estimate 2 postitional paramters called l&m, which correspond to the x- and y-axes, and an Amplitude A.

My observed variables are called visibilities and are given by the Fourier transform of this image multiplied by a sparse matrix, which simulates a measurement. Each pixel contains one visibility.
Fourier transform (real and imaginary part):

Multiplied with sparse matrix:

Every non zero pixel is a measured visibility V_i.
I have a forward model, which generates three normal distributions with parameters \tilde{A},\tilde{l},\tilde{m} and simulates the measurement process, called \tilde{V}(\tilde{A},\tilde{l},\tilde{m}).

Now the MAP part:
I calculate the likelihood as the product of normal distributions with mean \tilde{V_i} and observed value V: \mathcal{L}=\prod_i^N \mathcal{N}(V_i \vert \tilde{V_i}, \sigma_i).
I use some priors on the amplitude range of my three distributions as well as priors on \tilde{l},\tilde{m} and \sigma.
My goal is to fit the parameters \tilde{A},\tilde{l},\tilde{m} for the three distributions. So I’m trying to find the parameters for the global maximum of the posterior function.

The (short) Bayesian answer is that the values reflected in the MAP (if you could find it) should not be trusted as (particularly) more true/correct than other values. Or, to put it another way, all parameter values are credible to some degree. The posterior tells you how credible each potential parameter value (or vector of values) is.