# Uncertainty, evidence and inference

Authored explanation; editorial review pending.

Inference asks what an observation permits you to learn under a specified model. Before calculating a narrow interval, ask whether the observations distinguish the parameters at all. Brownian diffusivity constrains the product of radius and molecular number when temperature and viscosity are given; it cannot identify both members of that product separately.

An estimator is a rule applied to a finite sample. Different samples produce different estimates. Bias concerns the estimator's mean across such repetitions, not whether one particular estimate happens to equal the answer. Fitting drift from the same displacements uses information and changes the degrees of freedom available to estimate spread.

A 95% confidence procedure covers a fixed true parameter in 95% of repetitions when its assumptions hold. That statement is not a 95% posterior probability that this one fixed parameter lies inside one realized interval. Keeping the missed intervals in a repeated-trial plot is essential to understanding coverage.

A conditional interval holds the input radius, viscosity and temperature exact. A combined interval can include their uncertainty only when their measurement procedures and coverage are declared. A plausible-looking plus-or-minus bound with no coverage is not a confidence statement. The union bound adds error probabilities without requiring statistical independence between the input intervals.

A synthetic path created from a chosen molecular number is useful for checking recovery and coverage, not as independent evidence for that number in nature. With modern SI definitions, the Avogadro constant is already fixed; the analogous inversion is a consistency check. An empirical determination requires noncircular physical inputs and an admitted observation model.

## Worked example

1. Consider a purely numerical inverse exercise N = 12/D, with D estimated as 4. Its point estimate is N = 3.
2. Suppose an admitted procedure gives the D interval [3, 6]. Since division by a larger positive D produces a smaller N, the corresponding interval is [12/6, 12/3] = [2, 4]. The endpoints reverse.
3. Now suppose an extra unknown scale a enters as N = 12/(aD). The same D supports many a–N pairs. No interval for a unique N is justified until a is supplied independently.
4. Declaring a = 2 halves both the point estimate and the inverted bounds without changing any displacement. It changes an assumption, not the observations.
5. For a combined guarantee, three input procedures with at most 1% failure each and a diffusion procedure with at most 2% failure have at most 5% total failure by the union bound. Without their stated coverage, that arithmetic supplies no guarantee.

State what is identifiable, which estimator and observation model you used, what is held exact, and whether the result is synthetic recovery, an independent estimate or a consistency check.