MLE Argmax

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$\left\{\begin{matrix} \theta^{MLE*} = \arg\max_{\theta} P(D|\theta) \\ \theta^{MAP*} = \arg\max_{\theta} P(\theta|D) \end{matrix}\right.$

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$\begin{align*} \theta^{MLE*} &= \arg\max_{\theta} P(D|\theta) \\ &= \arg\max_{\theta} \log{P(D|\theta)} \\ &= \arg\min_{\theta} -\log{P(D|\theta)} \\ &= \arg\min_{\theta} \frac{1}{2\sigma^2}\sum_{i=1}^N(y_i-f(x_i, \theta))^2 - C_1\\ &= \arg\min_{\theta} \sum_{i=1}^N(y_i-f(x_i, \theta))^2 \\ &= \arg\min_{\theta} \mathcal{L}(\theta) \\ \end{align*}$

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$\begin{aligned}\hat{\theta}_{MLE}(x)&=\arg\max_{\theta}f(x|\theta) \\\hat{\theta}_{MAP}(x)&=\arg\max_{\theta}\frac{f(x|\theta)g(\theta)}{\int_{\theta}f(x|\theta')g(\theta')d\theta'} \\&=\arg\max_{\theta}f(x|\theta)g(\theta)\end{aligned}$

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$\begin{align*} \theta^{MAP*} &= \arg\max_{\theta} P(\theta|D) \\ &= \arg\max_{\theta} P(\theta) P(D|\theta) \\ &= \arg\max_{\theta} P(\theta) P(D|\theta) \\ &= \arg\max_{\theta} (\log{P(\theta)} + \log{P(D|\theta)}) \\ &= \arg\min_{\theta} (\frac{1}{2\sigma_2^2}\sum_{i=1}^M \|\theta_i\|^2 + \frac{1}{2\sigma_1^2} \sum_{i=1}^N\| y_i - f(x_i, \theta) \|^2 ) \\ &= \arg\min_{\theta} (\frac{\sigma_1^2}{\sigma_2^2} \sum_{i=1}^M \|\theta_i\|^2 + \sum_{i=1}^N\| y_i - f(x_i, \theta) \|^2 ) \\ \end{align*}$