In statistics , a likelihood function (often simply the likelihood ) is a function of the parameters of a statistical model , defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values. Likelihood functions play a key role in statistical inference , especially methods of estimating a parameter from a set of statistics . In non-technical parlance, "likelihood" is usually a synonym for " probability " but in statistical usage, a clear technical distinction is made. One may ask "If I were to flip a fair coin 100 times, what is the probability of it landing heads-up every time?" or "Given that I have flipped a coin 100 times and it has landed heads-up 100 times, what is the likelihood that the coin is fair?" but it would be improper to switch "likelihood" and "probability" in the two sentences. If a probability distribution depends on a parameter, one may on one hand consider—for a given value of the parameter—the probability (density) of the different outcomes, and on the other hand consider—for a given outcome—the probability (density) this outcome has occurred for different values of the parameter. The first approach interprets the probability distribution as a function of the outcome, given a fixed parameter value, while the second interprets it as a function of the parameter, given a fixed outcome. In the latter case the function is called the "likelihood function" of the parameter, and indicates how likely a parameter value is in light of the observed outcome.

In statistics , a likelihood function (often simply the likelihood ) is a function of the parameters of a statistical model , defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values. Likelihood functions play a key role in statistical inference , especially methods of estimating a parameter from a set of statistics . In non-technical parlance, "likelihood" is usually a synonym for " probability " but in statistical usage, a clear technical distinction is made. One may ask "If I were to flip a fair coin 100 times, what is the probability of it landing heads-up every time?" or "Given that I have flipped a coin 100 times and it has landed heads-up 100 times, what is the likelihood that the coin is fair?" but it would be improper to switch "likelihood" and "probability" in the two sentences. If a probability distribution depends on a parameter, one may on one hand consider—for a given value of the parameter—the probability (density) of the different outcomes, and on the other hand consider—for a given outcome—the probability (density) this outcome has occurred for different values of the parameter. The first approach interprets the probability distribution as a function of the outcome, given a fixed parameter value, while the second interprets it as a function of the parameter, given a fixed outcome. In the latter case the function is called the "likelihood function" of the parameter, and indicates how likely a parameter value is in light of the observed outcome.

In statistics, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values. Likelihood functions play a key role in statistical inference, especially methods of estimating a parameter from a set of statistics. In non-technical parlance, "likelihood" is usually a synonym for "probability" but in statistical usage, a clear technical distinction is made. One may ask "If I were to flip a fair coin 100 times, what is the probability of it landing heads-up every time?" or "Given that I have flipped a coin 100 times and it has landed heads-up 100 times, what is the likelihood that the coin is fair?" but it would be improper to switch "likelihood" and "probability" in the two sentences. If a probability distribution depends on a parameter, one may on one hand consider—for a given value of the parameter—the probability (density) of the different outcomes, and on the other hand consider—for a given outcome—the probability (density) this outcome has occurred for different values of the parameter. The first approach interprets the probability distribution as a function of the outcome, given a fixed parameter value, while the second interprets it as a function of the parameter, given a fixed outcome. In the latter case the function is called the "likelihood function" of the parameter, and indicates how likely a parameter value is in light of the observed outcome.