# On the "Causality" Step in Policy Gradient Derivations: A Pedagogical Reconciliation of Full Return and Reward-to-Go

arXiv: https://arxiv.org/abs/2604.04686 · HTML: https://mlsanity.com/paper/2604.04686/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Deep RL algorithms and representations, one of 15 directions in Deep reinforcement learning methods.
- Builds on 0 of its 4 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Placed in

- Deep reinforcement learning methods / [Deep RL algorithms and representations](https://mlsanity.com/deeprl/deep-rl-algorithms-and-representations/)

## Abstract (arXiv)

In introductory presentations of policy gradients, one often derives the REINFORCE estimator using the full trajectory return and then states, by ``causality,'' that the full return may be replaced by the reward-to-go. Although this statement is correct, it is frequently presented at a level of rigor that leaves unclear where the past-reward terms disappear. This short paper isolates that step and gives a mathematically explicit derivation based on prefix trajectory distributions and the score-function identity. The resulting account does not change the estimator. Its contribution is conceptual: instead of presenting reward-to-go as a post hoc unbiased replacement for full return, it shows that reward-to-go arises directly once the objective is decomposed over prefix trajectories. In this formulation, the usual causality argument is recovered as a corollary of the derivation rather than as an additional heuristic principle.