Maximum Likelihood Estimation
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- tags
- Statistics, Machine Learning,
- source
- (Goodfellow, Bengio, and Courville 2016; Wasserman 2004)
The maximum likelihood estimator (MLE) denoted by \(\hat{\theta}_n\), is the value which maximizes the likelihood (or log-likelihood) function defined as
\begin{align*} \mathscr{L}_n(\theta) &= \prod_i f(\xvec_i;\theta) &\quad \triangleright \textbf{Likelihood Function} \\ l_n(\theta) &= \log \mathscr{L}_n(\theta) = \sum_i \log f(\xvec_i; \theta) &\quad \triangleright \textbf{Log-Likelihood Function} \end{align*}
where instances \(\xvec_1, \xvec_2, \ldots, \xvec_n\) are IID with PDF \(f(\xvec; \theta)\). We can take the log of the product because the function will have the same stationary points and will have less numerical issues with under/overflow.
The MLE estimator \(\thetavec_{\text{MLE}}\) is one which maximizes the likelihood function:
\[\thetavec_{\text{MLE}} = \text{argmax}_\thetavec \sum_{i} \log p_{model}(\xvec_i).\]
This can be viewed as minimizing the the dissimilarity between the empirical distribution \(\hat{p}_data\) and the model distribution \(p_{model}\) by minimizing the KL Divergence. Because we only have control over the model distribution, this is the same as minimizing
\[-\expected_{X\sim\hat{p}(data)} [\log p_{model}(X)]\]
This corresponds exactly with minimizing the Cross-Entropy.