**Definition: Likelihood function** (see Casella & Berger, 2002): Let $f(\boldsymbol x \mid \boldsymbol \theta)$ denote the joint probability density function (PDF) of the sample $\boldsymbol X = (X_1,\ldots,X_n)^{\mathsf T}$, where $\boldsymbol \theta \in \Theta$ is some set of parameters and $\Theta$ is the parameter space. We define the _likelihood function_ $\mathcal L \colon \Theta \to [0, \infty)$ by $\mathcal L(\boldsymbol \theta \mid \boldsymbol x) = f(\boldsymbol x \mid \boldsymbol \theta)$ for some realisation $\boldsymbol x = (x_1,\ldots,x_n)^{\mathsf T}$ of $\boldsymbol X$. The _log-likelihood function_ $\ell\colon\Theta\to\mathbb R$ is defined by $\ell(\boldsymbol \theta \mid \boldsymbol x) = \log\mathcal L(\boldsymbol\theta \mid \boldsymbol x)$.The _maximum likelihood estimate_ (MLE) $\hat{\boldsymbol\theta}$ is the parameter $\boldsymbol\theta$ that maximises the likelihood function, $\hat{\boldsymbol{\theta}} = argmax_{\boldsymbol{\theta} \in \Theta} \mathcal{L}(\boldsymbol{\theta} \mid \boldsymbol x) = argmax_{\boldsymbol\theta \in \Theta} \ell(\boldsymbol\theta \mid \boldsymbol x)$.
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