However, one numerical problem here is that we are multiply together many numbers less than one.

To solve the problem, we can use $\log$ to convert multiplication into addition.

$\log p(\mathcal{D})=\sum_{i} \log p\left(x_{i}\right)+\log p_{\theta}\left(y_{i} \mid x_{i}\right) =\sum_{i} \log p_{\theta}\left(y_{i} \mid x_{i}\right)+\text { const }$

$\theta^{\star} \leftarrow \arg \max _{\theta} \sum_{i} \log p_{\theta}\left(y_{i} \mid x_{i}\right)$