Finding Global Minima in Likelihood Functions

In summary: Other than that, there are many global optimizers that can be used but for a problem like this, you can also consider using Bayesian optimization methods. These methods can provide a posterior distribution instead of just a point estimate, which can be helpful in some cases. However, if the function is computationally expensive, you may want to consider other options such as MCMC sampling optimization. Keep in mind that the number of parameters may increase in the future, so it's important to choose an optimization method that can handle higher dimensions.
  • #1
tworitdash
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I have a likelihood function that has one global minima, but a lot of local ones too. I attach a figure with the likelihood function in 2D (it has two parameters). I have added a 3D view and a surface view of the likelihood function. I know there are many global optimizers that can be used to obtain the location of the global minimum point in the likelihood function. However, I want to know what basic optimizer principles that I can use (that I can also derive and implement myself) for a problem like this. If you see the 3D view, you may find many local minima. I am also open to suggestions that involve Bayesian type of optimization where I will get a posterior and not just a point estimate. I am open to that as well. I have tried MCMC type sampling optimization, however, they are computationally expensive. The number of parameters may increase later.
 

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  • #2
Is it literally just this function you want to optimize?

You already did it, by drawing a graph. More formally if that's unsatisfying, for low dimensions and fast evaluation functions you can just evaluate the function at every point on a fine grid and pick the point with the best value. If you want a little extra precision you can run any optimizer from there to find the local extremum near that point.
 
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