A popular method for fine-tuning a model is to find the optimal parameters, $\hat{\theta}$, by maximizing the total conditional log-likelihood over a dataset $\mathcal{D}$ of prompt-response pairs. This approach, equivalent to minimizing the negative log-likelihood loss, seeks parameters that make the observed outputs $\mathbf{y}$ most probable given the inputs $\mathbf{x}$. In some cases, the prompt $\mathbf{x}$ is decomposed into an instruction $\mathbf{c}$ and a user input $\mathbf{z}$, such that $\mathbf{x} = (\mathbf{c}, \mathbf{z})$. The formal expression is: $\hat{\theta} = \arg \max_{\theta} \sum_{(\mathbf{x},\mathbf{y}) \in \mathcal{D}} \log \mathrm{Pr}_{\theta}(\mathbf{y}|\mathbf{x}) = \arg \max_{\theta} \sum_{(\mathbf{x},\mathbf{y}) \in \mathcal{D}} \log \mathrm{Pr}_{\theta}(\mathbf{y}|\mathbf{c}, \mathbf{z})$ where $\mathrm{Pr}_{\theta}(\cdot|\cdot)$ is the probability predicted by an LLM with the parameters $\theta$.