# Understanding maximum likelihood estimation

I am told

the method of maximum likelihood says we should use the model that assigns the greatest probability to the data we have observed; formally, the maximum likelihood estimator is found by solving
$$\hat{\theta}= arg_{\theta}max\{p(x|\theta)\}$$
where $$p(x|\theta)$$ is called the likelihood function.

Am I correct reading $$p(x|\theta)$$ in English as "the probability of the data given the parameters"? I am confused because we at first seem to be told that MLE is about the probability of the parameters given the data.

• Thank you, In English "Likelihood" is similar to "probability" Jan 4 at 19:27
• Thank you. How would you read $p(x|\theta)$ in English? Jan 4 at 19:32

The parameters are not random variables but fixed unknowns (at least in the likelihood approach to inference). It is thus incorrect to talk of a probability distribution on the parameters. The MLE is the value of the parameters that makes the actual observations the most likely to have occurred $$p(x|\hat\theta) = \max_\theta p(x|\theta)$$ The quantity $$p(x|θ)$$ thus reads as
1. the probability of observing $$x$$ when the parameter value is equal to $$θ$$ in a discrete setting and
2. the density of $$x$$ when the parameter value is equal to $$θ$$ in a continuous setting.
(I would even avoid given since this could be interpreted as a conditional probability or density, which does not make sense if $$\theta$$ is not a random variable, i.e., outside a Bayesian framework.)