Calculates the log-likelihood of \(\mathbf{X}\) following a mutivariate normal distribution.
mvnll(X, mu, Sigma, neg = TRUE)
| X | Numeric matrix. Values of the \(k\)-dimensional random variable \(\mathbf{X}\). |
|---|---|
| mu | Numeric vector. Location parameter mean vector \(\boldsymbol{\mu}\) of length \(k\). |
| Sigma | Numeric matrix. \(k \times k\) variance-covariance matrix \(\boldsymbol{\Sigma}\). |
| neg | Logical.
If |
The natural log of the likelihood function for the multivariate normal (or multivariate Gaussian, or joint normal) distribution is given by $$ \ln \mathcal{L} \left( \boldsymbol{\mu}, \boldsymbol{\Sigma} \mid \mathbf{X} \right) = \mathcal{l} \left( \boldsymbol{\mu}, \boldsymbol{\Sigma} \mid \mathbf{X} \right) \\ \mathcal{l} \left( \boldsymbol{\mu}, \boldsymbol{\Sigma} \mid \mathbf{X} \right) = - \frac{1}{2} \left[ \ln \left( | \boldsymbol{\Sigma} | \right) + \left( \mathbf{x} - \boldsymbol{\mu} \right)^{\prime} \boldsymbol{\Sigma}^{-1} \left( \mathbf{x} - \boldsymbol{\mu} \right) + k \ln \left( 2 \pi \right) \right] %(\#eq:dist-mvnL) $$ with \(k\)-dimensional random vector \(\mathbf{X} \in \boldsymbol{\mu} + \textrm{span}\left(\boldsymbol{\Sigma}\right) \subseteq \mathbf{R}^k\), \(\boldsymbol{\mu}\) is the location parameter mean \(\left( \boldsymbol{\mu} \in \mathbf{R}^k \right)\), and \(\boldsymbol{\Sigma}\) is the variance-covariance matrix \(\left( \boldsymbol{\Sigma} \in \mathbf{R}^{k \times k} \right)\).
The negative log-likelihood is given by $$ - \mathcal{l} \left( \boldsymbol{\mu}, \boldsymbol{\Sigma} \mid \mathbf{X} \right) = \frac{1}{2} \left[ \ln \left( | \boldsymbol{\Sigma} | \right) + \left( \mathbf{x} - \boldsymbol{\mu} \right)^{\prime} \boldsymbol{\Sigma}^{-1} \left( \mathbf{x} - \boldsymbol{\mu} \right) + k \ln \left( 2 \pi \right) \right] . %(\#eq:dist-mvnnegll) $$
Wikipedia: Multivariate Normal Distribution