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Could you explain for me what is "likelihood principle (from R.A. Fisher)" without using any (or too serious) mathematical language? What will be the difference between "it is likely" and "it is probable" if we follow this principle?

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marked as duplicate by whuber Jul 31 '13 at 15:08

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Let's say the data comes from a distribution with a parameter $\theta$ whose value we don't know. You can think of the likelihood function ${L(\theta)}$, very roughly, as "the probability of observing the data given this value of $\theta$".

From this comes the maximum likelihood principle: when we want to estimate $\theta$ from some data, we solve for the value that maximises $L(\theta)$. This value is the one that is "most likely" to have generated the data we've observed. This principle is the basis of a huge variety of statistical techniques, including many of the most-used ones such as linear regression.

This description is technically incorrect, since the likelihood is not actually a probability. That's why it's called a likelihood function, not a probability (density) function. But as long as you're not trying to prove theorems or derive new methods, this should suffice.

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