Perspective

Homogenization without the corrector

Two fields face the same problem. There is a hidden variable moving faster than the one you care about, and you would like a single effective description that does not mention it. Both fields take the same first step. They take opposite second steps, and the second step is where all the content is.

The shared first step

Suppose the state has a slow part you model and a fast part you do not observe. Volatility that clusters, a regime that switches, a latent scale that drifts. The obvious move is to average the fast part over its stationary law and keep the average.

In homogenization this is called the fast mixing limit, or the averaged limit, and the resulting coefficients are the effective coefficients. In split conformal prediction the same move is called pooling the residuals. You take the errors your model made across every past state, put them in one pile, and read a quantile off the pile. Pooling residuals over states is averaging the fast variable over its invariant measure. The two operations are the same operation.

Where they part

Homogenization theory regards the averaged limit as the beginning of the analysis. The standard adjective for it, in the literature, is naive. What follows is the interesting part: a solvability condition that says when the averaging is legitimate, a corrector that recovers the leading behaviour the average destroyed, and a rate that says how large the error is in terms of the separation between the two time scales.

Split conformal prediction regards the same limit as the end of the analysis, and attaches a coverage certificate to it.

The certificate is not wrong. It is the claim that the interval covers the truth with the stated frequency on average over inputs, and that claim is true, finite sample, without distributional assumptions. The question this page asks is what the second step would have found.

What stopping costs

Here is a pair of data generating processes that split conformal prediction cannot tell apart.

World A. The forecast error is drawn from the same heavy tailed law every period, a Student t with three degrees of freedom. Large errors happen and they are bad luck. Nothing about today made today dangerous, and one interval width is correct forever.

World B. Each period has its own volatility, drawn at random, and given that volatility the error is Gaussian. Large errors happen on high volatility periods. If you knew the volatility you would know in advance that the period was dangerous, and the correct interval is narrow when it is calm and wide when it is not.

These are not similar. They are marginally identical, because the Student t is exactly a Gaussian scale mixture. Pile up a long run of errors from each world and you have drawn twice from the same distribution. Split conformal only ever looks at that pile, so the calibration scores, the quantile, the intervals and the coverage agree in law. No output of the method differs between the worlds.

Three degrees of freedom is a realistic tail index for daily financial returns, so this is not a constructed pathology. It is the standard disagreement about return series, stated twice. Are the tails fat, or is the volatility stochastic? Every practitioner has a view. Conformal prediction returns the same interval either way and certifies it either way, which is precisely why a guarantee can be true and uninformative at the same time.

The numbers above are reproduced by two short scripts, one checking the closed form against numerical integration and Monte Carlo, the other checking that the two worlds are statistically indistinguishable to the method.

The hypothesis nobody states

Residual pooling has an adequacy condition. It is legitimate when the state dependence in the error law mixes away over the calibration window, and misleading when it does not. Conformal prediction never states this condition, because the coverage guarantee holds either way. That is the appeal, and it is also why the condition goes unexamined.

Homogenization is the theory of exactly that condition. It names the separation parameter, gives the solvability condition under which the averaged limit is the right leading term, and quantifies the error in terms of that parameter. Whether the information discarded by pooling admits a clean expansion in the same parameter is open, and it is worth asking, because in the fast mean reverting models the corrections to the conditional density are already known in closed form.

The honest trade

None of this makes conformal prediction a mistake, and the asymmetry that saves it is real. Homogenization needs a model. You cannot compute a corrector without writing down what the fast variable is doing. Conformal prediction needs nothing, which is why it travels so well.

So the trade is clear once it is stated. The corrector is what you sacrifice in order not to specify anything. In some settings that is a good trade, and where the base model's error law really is state invariant the corrector is zero and pooling is optimal. The objection is not to the trade. It is to making it without pricing it, and then reading the certificate as though it were evidence that nothing was given up.

Where the modelling lives

The corrector is the part a certificate cannot see. It is also the part that improves a proper score, which is why conformalizing a forecast leaves the log score exactly where it was. If you want the sharpness back there is only one place to get it, and it is upstream, in the conditional model of scale and shape.

Averaging the fast variable is a good first step. Every field that has faced this problem takes it. The difference is whether you treat the average as the answer or as the leading term of something you intend to correct.