The probability distribution sums to 1 because it accounts for all possible data outcomes under a fixed model, satisfying the axioms of probability. The likelihood curve does not integrate to 1 because it measures relative support for different parameter values given fixed data, and is not constrained to be a probability distribution over the parameter space.
Conditions: Probability view: parameter fixed, data varies over the entire sample space.; Likelihood view: data fixed, parameter varies over its domain.
The probability distribution sums to 1 because it accounts for all possible data outcomes under a fixed model, satisfying the axioms of probability. The likelihood curve does not integrate to 1 because it measures relative support for different parameter values given fixed data, and is not constrained to be a probability distribution over the parameter space.
Conditions: Probability view: parameter fixed, data varies over the entire sample space.; Likelihood view: data fixed, parameter varies over its domain.
Probability fixes the parameter and varies the data, resulting in a normalized distribution where the probabilities sum to 1. Likelihood fixes the data and varies the parameter, resulting in a non-normalized curve whose integral does not equal 1.
Conditions: Both views are based on the same conditional expression P(x∣θ).; Probability view: parameter is fixed, data varies.; Likelihood view: data is fixed, parameter varies.
Probability fixes the parameter and varies the data, resulting in a normalized distribution where the probabilities sum to 1. Likelihood fixes the data and varies the parameter, resulting in a non-normalized curve whose integral does not equal 1.
Conditions: Both views are based on the same conditional expression P(x∣θ).; Probability view: parameter is fixed, data varies.; Likelihood view: data is fixed, parameter varies.
The maximum likelihood estimate is defined as the parameter value that corresponds to the peak (maximum) of the likelihood curve. It is the value of the parameter that gives the highest likelihood for the observed data.
Conditions: A likelihood function L(p) over candidate parameter values is available.; The data are fixed.
The maximum likelihood estimate is defined as the parameter value that corresponds to the peak (maximum) of the likelihood curve. It is the value of the parameter that gives the highest likelihood for the observed data.
Conditions: A likelihood function L(p) over candidate parameter values is available.; The data are fixed.