However, one numerical problem here is that we are multiply together many numbers less than one.
To solve the problem, we can use log\loglog to convert multiplication into addition.
logp(D)=∑ilogp(xi)+logpθ(yi∣xi)=∑ilogpθ(yi∣xi)+ const \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 }logp(D)=∑ilogp(xi)+logpθ(yi∣xi)=∑ilogpθ(yi∣xi)+ const
θ⋆←argmaxθ∑ilogpθ(yi∣xi)\theta^{\star} \leftarrow \arg \max _{\theta} \sum_{i} \log p_{\theta}\left(y_{i} \mid x_{i}\right)θ⋆←argmaxθ∑ilogpθ(yi∣xi)