# Most probable inputs assignment in Gaussian process

Given

1. A Guassian process $$w(\mu^*,\Sigma^*)$$
2. $$n$$ observations of the output
3. And $$n$$ potential inputs.

But the assignment of the inputs to the observations is unknown. The goal is to find a inputs-to-outputs matching that leads to the highest probability.

Formally, the objective is to find a permutation of the given input values $$X'_{i}\in Per\{x'_{1},...,x'_{n}\}$$ (where $$Per$$ is the permutation group includes all possible permutations) that maximize the probability of observing the given vector of output values $$Y'=[y'_{1}...y'_{n}]$$ at the corresponding input. Or:

$$argmax_{i}\{Pr(Y'|X'_{i},w)\}~~/or/~~argmax_{i}\{\mathcal N(\mu^*_{X'_{i}},\Sigma^*_{X'_{i}})_{Y'}\}$$

Is there any solution for the above problem better than brute-force?