---
# System prepended metadata

title: 'Notation:'

---

Notation:
$x$: prompt (includes few shot examples)
$z$: cot part of completion
$y$: answer part of completion
$c = (y,z)$: completion, cot + answer together 
$s(y, c)$: indicator of correctness of answer included in completion $c$
$p()$: original model
$p_{new}()$ probability under new model after gradient update

Here is one algorithm, using importance weighting:

1. we sample many completions $c_i\sim p$ from our original model
2. we evaluete $\log p(c_i\vert x)$
3. We update the model to obtain $p_{new}$
4. We evaluate $\log p_{new}(c_i\vert x)$ for each of our original samples.
5. We estimate the difference in correctness using the following formula:

\begin{align}
&\mathbb{E}_{c\sim p_{new}(c\vert x)} s(y, c) - \mathbb{E}_{c\sim p(c\vert x)} s(y, c) = \\
&\mathbb{E}_{c\sim p(c\vert x)} \frac{p_{new}(c)}{p(c)}s(y, c) - \mathbb{E}_{c\sim p(c\vert x)} s(y, c) =\\
&\mathbb{E}_{c\sim p(c\vert x)} \left( \frac{p_{new}(c\vert x)}{p(c\vert x)} - 1\right)s(y, c) \approx \\
&\frac{1}{N} \sum_{i=1}^N \left( \frac{p_{new}(c_i\vert x)}{p(c_i\vert x)} - 1\right)s(y, c_i)
\end{align}

If $s$ returns a $0$ when the answer is incorrect, the estimator may have a very high variance, as it ignores all the unsuccessful prompts. So it's perhaps better for $s$ to be $-1$ when things are incorrect, and then transform the result back to a probability of correctness.

In theory, you should be able to calculate $\log p$ and $\log p_{new}$ of any string, because this log probability is used in the gradient descent part. to extract $p(c\vert x)$ one can do $\log p(x, c) - \log p(x)$.