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As far as I understand (strict) Frequentists treat hypothesis (model parameters) as fixed and don't allow to assign probabilities to a range of model parameters. That is the reason why they compute confidence intervals through variation in data, not variation in model parameters.

Maximum-likelihood estimation is considered to be frequentist approach. But if you search for best fit parameter value that maximizes likelihood function then you are automatically assigning probabilities to other parameter values (they are not improbable anymore, just less likely than best fit).

In other words, you cannot take a derivative of likelihood function $\dfrac{dP(E|H)}{dH} = 0$ because H is fixed in P(E|H) if you are frequentist. So MLE is really just MAP $\dfrac{dP(H|E)}{dH} = 0$ with implicit flat priors.

To sum up, in MLE one has to treat model parameters as a variable with different likelihoods given experimental data. Then doesn't it become Bayesian (with implicit flat prior) and stops being frequentist?

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    $\begingroup$ " But if you search for best fit parameter value that maximizes likelihood function then you are automatically assigning probabilities to other parameter values" is not true. I am trying to find numerical values for the parameters that maximize the likelihood function. There are no probabilities for the parameters involved. The MLE is the parameter values(s) that maximize the probability of seeing the data we actually saw; the statement is about the probability of observing the data given the parameter values, not the other way around. $\endgroup$ – jbowman May 21 '18 at 4:13
  • $\begingroup$ @jbowman why not write that as an answer? $\endgroup$ – Juho Kokkala May 22 '18 at 6:02
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Copied and extended from comments -

"But if you search for best fit parameter value that maximizes likelihood function then you are automatically assigning probabilities to other parameter values" is not true. We are trying to find numerical values for the parameters that maximize the likelihood function. There are no probabilities for the parameters involved. The MLE is the parameter values(s) that maximize the probability of seeing the data we actually saw; the statement is about the probability of observing the data given the parameter values, not the other way around.

Since the function we are trying to optimize has the data as fixed values and varies the parameters, it's no longer a probability function, for which the parameters are fixed and the (unobserved) data varies. Hence the name "likelihood".

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