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I have a large number of samples (say $N$) from a multimodal joint probability distribution, for example:

a     b     c     d     e     f 
0.01  5.37  2.87  3.22  1.93  5.38
0.07  5.24  2.99  3.56  2.02  6.01
...

I want to find the values $(a,b,c,d,e,f)$ at which the joint probability density function attains its maximum value.

How can this be best achieved?

rhombidodecahedron
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  • Let's tumble that multimodal distribution and view it as an optimisation problem where we want to find the global minimum of function with multiple local minima. What would be the solution there? Compulsory link to [No free lunch in search and optimization](https://en.wikipedia.org/wiki/No_free_lunch_in_search_and_optimization). Short answer: It really depends on your particular problem. (+1 because this seemingly innocuous question describes an absolutely universal problem.) – usεr11852 Mar 10 '18 at 13:09
  • "It really depends on your particular problem" can you give some examples? – rhombidodecahedron Mar 10 '18 at 13:58
  • Sure here we are: https://stats.stackexchange.com/questions/185631/ – usεr11852 Mar 10 '18 at 14:43
  • @usεr11852 if I am not mistaken, these seem like sampling methods, but I already have the samples – rhombidodecahedron Mar 10 '18 at 17:42
  • Yes, that is what says on the tin, but the container of the tin is an estimation method; we do not sample for the sake of sampling. What we care (here) is the estimate that attains a maximum value of a particular cost function. Standard techniques like kernel density estimation in multi-dimensional setting (even 3D smoothing is a computational pain - this is a manifestation of the curse of dimensionality). So we cannot smooth to estimate "a density" so what can we do? We can only "sample around", i.e. back to that link. :D – usεr11852 Mar 10 '18 at 18:01
  • See also here: https://stats.stackexchange.com/questions/33625 same story. – usεr11852 Mar 10 '18 at 18:02

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