In a previous article, we saw why training with MSE forces a model to report only the conditional mean, and how Gaussian NLL lets it learn an honest per-step uncertainty σ as well. That was the one-step story. This post picks up where that left off. Everything we built in Part I — the Gaussian NLL, the two-term loss, the proof that the model learns both the conditional mean and the conditional variance — was about one-step-ahead prediction: given a context window of T observed values, predict the distribution of the very next value x_{T+1}.

At inference time, in any real application, this is rarely enough. You want to know what happens over the next 10, 100, or 1000 steps. You want a forecast, not a single prediction. And if your model is probabilistic, you want that forecast to carry honest uncertainty all the way to the end of the horizon. This post is about how to do this correctly.

One could think immediately of the naive approach, that is applying the model repeatedly, feeding its predictions back as inputs is secretly a deterministic operation that discards the uncertainty the model spent so much effort learning. Fixing this requires thinking carefully about what a joint distribution over future values actually is, and how to approximate it with Monte Carlo sampling. The fix is barely more code than the naive version, but it gives a fundamentally different and fundamentally better forecast.

Everything here is built and checked in the companion notebook: the two rollout approaches, applied to the same trained model, on the same signal, with coverage and PIT diagnostics (all explained in this post).