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Maximum likelihood estimation considers the likelihood of data given parameters. In a non-parametric model like k-means clustering, can MLE still be used?

I know a Gaussian mixture model can use MLE to make a clustering algorithm, but it's parametric so I'm not interested in this case.

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The premise of maximum likelihood estimation is that there is a probability distribution parameterized by variables. The goal is to find the variables that optimize the metric, which is the joint probability.

If you have a non-parametric model with no distribution, the metric is not a likelihood. In the case of k-means, it is some distance metric between your centers and data. So the terminology here is you are not doing a MLE, but minimization of a loss function.

The computational techniques to find optimal values are similar (eg. gradient descent algorithms) but MLE strictly refers to maximizing the defined joint probability.

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