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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.

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

To quote the originator of the notion of likelihood, R.A. Fisher :

I suggest that we may speak without confusion of the likelihood of one value of p being thrice the likelihood of another (…) likelihood is not here used loosely as a synonym of probability, but simply to express the relative frequencies with which such values of the hypothetical quantity p would in fact yield the observed sample.

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    $\begingroup$ So is the vertical line symbol ambiguous? Sometimes meaning "given" and sometimes meaning "when" ? As a programmer I feel like I can't get maths to compile! $\endgroup$
    – Kirsten
    Jan 5 at 8:25

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