What Makes Inference Possible?

As we discussed before, a system does not interact with the world as a whole, but only through an interface that determines what can be observed and what remains inaccessible. Information, therefore, is not a property of the world alone, but a relation established through interaction: only those structures that pass through the interface can become available to the system. This immediately constrains not only what can be known, but what can be distinguished. Before a system can reason about the world, it must first be able to distinguish it, and these distinctions are themselves induced by the observation interface.

Consider a sensor that samples a signal at a fixed rate. Any frequency components above the Nyquist limit are not directly observable; they are aliased into lower frequencies. From the perspective of the system, multiple distinct high-frequency signals produce exactly the same observed data, and no amount of additional observation can resolve this ambiguity because the distinction is not present at the interface. In this situation, it is meaningless for the system to assign probabilities to the true high-frequency components individually. The issue is not insufficient data, but the absence of a representable distinction. All such signals collapse into the same observable representation, and the only well-defined object is this equivalence class. What cannot be distinguished cannot be inferred.

What we call a hypothesis is nothing more than a structured distinction—a way of organizing possible observations into meaningful alternatives. If a distinction does not exist at the level of observation, then neither does the corresponding hypothesis. Bayesian inference is often described as a general principle of learning, in which a system maintains beliefs over hypotheses and updates them in light of data. Implicit in this formulation is a crucial assumption: that the hypothesis space is already given. In a physical system, however, this assumption does not hold. The system can only form hypotheses from distinctions that its interface makes available. Probability presupposes distinction, and distinction presupposes observability. We thus arrive at the second impossibility principle:

Impossibility II: Impossibility of Fully Internal Bayesian Inference

For any physical learning system, the support of the prior distribution is determined by the observation interface, not vice versa.

This statement sharpens the limitation of Bayesian inference in physical systems. A prior is not a freely specified distribution over an arbitrary hypothesis space; it cannot extend beyond the set of observable distinctions. States that do not exist at the interface are not assigned low probability—they are not part of the support at all. In this sense, inference is not merely uncertain about such states; it is undefined over them.

From this perspective, the Bayesian Brain Hypothesis and the Free Energy Principle should be understood as describing inference within an interface-induced hypothesis space, rather than over all possible latent causes. Their success reflects that probabilistic inference is effective within the space of accessible distinctions, not that the system represents the world without constraint. Latent variables are therefore not freely posited explanations, but structures constructed from observable regularities; what lies outside this space is not improbable—it is unrepresented. In this light, action does not merely reduce uncertainty; it reshapes the interface and thus the support of the prior, altering the domain over which inference can operate. Notably, the prior here should not be understood as uncertainty over the true state of the world, but as a distribution over the hypotheses available to the system, which are themselves constrained by the observation interface.

This also clarifies the relationship to the earlier result that semantics cannot arise without external constraints. Not all distinctions are meaningful, but more fundamentally, not all distinctions are even available. The interface determines what can exist as a hypothesis, while external constraints determine which of these become meaningful. Impossibility II restricts what can be represented; Impossibility I determines what is selected among those representations. A prior is therefore neither arbitrary nor purely internal: its support is fixed by observability, and its structure reflects selection.

Inference, therefore, does not begin with a prior. It begins with the interface.