Editing Likelihood function (section)

====Example: the gamma distribution====
The [[gamma distribution]] is an exponential family with two parameters, <math display="inline">\alpha</math> and <math display="inline">\beta</math>. The likelihood function is

<math display="block">\mathcal{L} (\alpha, \beta \mid x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}.</math>

Finding the maximum likelihood estimate of <math display="inline">\beta</math> for a single observed value <math display="inline">x</math> looks rather daunting. Its logarithm is much simpler to work with:

<math display="block">\log \mathcal{L}(\alpha,\beta \mid x) = \alpha \log \beta - \log \Gamma(\alpha) + (\alpha-1) \log x  - \beta x. \, </math>

To maximize the log-likelihood, we first take the [[partial derivative]] with respect to <math display="inline">\beta</math>:

<math display="block">\frac{\partial \log \mathcal{L}(\alpha,\beta \mid x)}{\partial \beta} = \frac{\alpha}{\beta} - x.</math>

If there are a number of independent observations <math display="inline">x_1, \ldots, x_n</math>, then the joint log-likelihood will be the sum of individual log-likelihoods, and the derivative of this sum will be a sum of derivatives of each individual log-likelihood:

<math display="block">
\begin{align}
& \frac{\partial \log \mathcal{L}(\alpha,\beta \mid x_1, \ldots, x_n)}{\partial \beta} \\
&= \frac{\partial \log \mathcal{L}(\alpha,\beta \mid x_1)}{\partial \beta} + \cdots + \frac{\partial \log \mathcal{L}(\alpha,\beta \mid x_n)}{\partial \beta} \\
&= \frac{n \alpha} \beta - \sum_{i=1}^n x_i.
\end{align}
</math>

To complete the maximization procedure for the joint log-likelihood, the equation is set to zero and solved for <math display="inline">\beta</math>:

<math display="block">\widehat\beta = \frac{\alpha}{\bar{x}}.</math>

Here <math display="inline">\widehat\beta</math> denotes the maximum-likelihood estimate, and <math display="inline">\textstyle \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i</math> is the [[sample mean]] of the observations.