A likelihood function L(a) is the probability or probability density for the occurrence of a sample configuration x_1, , x_n given that the probability density f(x;a). In frequentist inference, a likelihood function (often simply the likelihood) is a function of the parameters of a statistical model, given specific observed data. Likelihood functions play a key role in frequentist inference, especially methods of estimating a parameter from a set of statistics.‎Marginal likelihood · ‎Frequentist inference · ‎Prosecutor's fallacy. The aim is to estimate $\theta \in \mathbb{R}^k$ which is a vector of unknown parameters. The likelihood function is defined as the joint density $L(\data{X};\theta).


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- Likelihood & LogLikelihood | STAT

In such a situation, the likelihood function factors into a product of individual likelihood functions. The logarithm of this product is a sum of individual logarithms, and the derivative of a sum of terms is often easier to compute than the derivative of a product.

Consider a coin toss. My answer will be similar to Example 1 on Wikipedia. What is the probability of likelihood function two heads in a row.

Likelihood Function -- from Wolfram MathWorld

A typical statistical question is: The 10 data points and possible Gaussian distributions from which the data were drawn. Calculating the Maximum Likelihood Likelihood function Now that we have an intuitive understanding of what maximum likelihood estimation is we can move on to learning how to calculate the parameter values.

The values that we find are likelihood function the maximum likelihood estimates MLE.

Suppose we have three data likelihood function this time and we assume that they have been generated from a process that is adequately described by a Gaussian distribution.

These points are 9, 9. What we want to calculate is the total probability of observing all of the data, i.

Likelihood function - Wikipedia

To do this we would likelihood function to calculate some conditional probabilities, which can get very difficult. The assumption is that each data point is generated independently of the others. This assumption makes the maths much easier. If the events i.

Likelihood function probability density of observing a single data point x, that is generated from a Gaussian distribution is given by: In our example the total joint probability density of observing the three data points is given by: All we have to do is find the derivative of the function, set the derivative function to zero and then rearrange the equation to make the parameter of interest the likelihood function of the equation.

If you would like a more detailed explanation then just let me know in the comments. The log likelihood The above expression for the total probability is actually quite a pain to differentiate, so it is almost always simplified by taking the natural logarithm of the expression.

This is absolutely fine because the natural logarithm is a monotonically increasing function. A probability density function is a function of x, your data point, and it will tell you how likely it is that certain data points appear.

A likelihood function, on the other hand, takes the data set as a given, and represents the likeliness of different parameters for your distribution.

Likelihood Function: Overview / Simple Definition

The reality, though, is actually quite different. But for the likelihood function, likelihood function compare two different parameter points. The area under their curves does not have to add up to 1.