What is the likelihood function?

What is the likelihood function?

• A. The probability of the parameter given data
• B. The probability of the observed data as a function of the parameter
• C. The prior distribution
• D. The posterior distribution

Answer: (B)

The likelihood function is the probability (or probability density) of the observed data considered as a function of the parameter, with the data fixed. It tells you how plausible different parameter values are given what you actually observed. This is why the statement describing the likelihood as the probability of the observed data as a function of the parameter is the best match. It differs from the probability of the parameter given the data (that’s the posterior after incorporating data), from the prior (the distribution of the parameter before seeing data), and from the posterior (the updated belief after observing data). In practice you treat the observed data as fixed and vary the parameter to see how the likelihood changes, often maximizing it to obtain the maximum likelihood estimate.

The likelihood function is the probability (or probability density) of the observed data considered as a function of the parameter, with the data fixed. It tells you how plausible different parameter values are given what you actually observed.

This is why the statement describing the likelihood as the probability of the observed data as a function of the parameter is the best match. It differs from the probability of the parameter given the data (that’s the posterior after incorporating data), from the prior (the distribution of the parameter before seeing data), and from the posterior (the updated belief after observing data). In practice you treat the observed data as fixed and vary the parameter to see how the likelihood changes, often maximizing it to obtain the maximum likelihood estimate.