Some notes on Elizabeth Donoway's work, which proposes EDL (Excess Description Length), an information-theoretic measure of the predictive information the model's parameters has learned from the training data. Her work formalizes and answers the question of "What is elicitation vs. learning".
Given a finetuning dataset, let $\mathcal{D}$ be our train set with sample count $n$, and let $\mathcal{D}_{test}$ be the test set. The train and test sets are disjoint: $\mathcal{D}_{test} \cap \mathcal{D} = \emptyset$.
Pick a number of training steps $T$. Partition the training data into disjoint sets $\mathcal{D} = \mathcal{D}_1 \sqcup \cdots \sqcup \mathcal{D}_T$.
Given initial model parameters $\theta_0$, define MDL (Minimum Description Length) as: $$\sum_{t =1,...,T} \sum_{(x,y) \in \mathcal{D}_t}l((x,y),\theta_t),$$
where $\theta_t = \text{update}(\theta_{t-1}, \mathcal{D}_{t-1})$ and $l((x,y),\theta)$ is the loss of the model parameterized by $\theta$ evaluated on data point $(x,y)$. Each datapoint $(x,y)$ is composed of a prompt $x$ and completion $y$, each with token count $|x|$ and $|y|$ respectively.
We denote $C$ to be the number of completion tokens in the train set defined as $$C=\sum_{t=1,...,T}\sum_{(x,y) \in \mathcal{D}_t} |y|.$$
We denote $C_{test}$ to be the number of completion tokens in the test set defined as $$C_{test}=\sum_{(x,y) \in \mathcal{D}_{test}} |y|.$$
EDL (excess description length) is calculated as
$$\text{MDL} - \frac{C}{C_{test}}\sum_{(x,y) \in \mathcal{D}_{test}}l((x,y),\theta_T).$$
One measure of elicitability is then the derivative with respect to dataset size of the EDL per train completion token,
$$\frac{\partial}{\partial n}\bigg(\frac{EDL}{C}\bigg),$$
where a more negative value indicates a more elicitable model. That is to say, a model that doesn't learn more from more examples.