>[!warning] >This content has not been peer reviewed. # The limit on precision from finite data **Cramér–Rao bound:** The variance of any unbiased estimator of a parameter $\theta$ is bounded below by the inverse of the **Fisher information**: $\mathrm{Var}(\hat{\theta}) \geq 1/I(\theta)$. So with finite data (finite sample size, finite signal-to-noise), there is a **hard limit** on how precisely you can know $\theta$ (Cramér 1946; Rao 1945; Fisher 1925). The Fisher information measures how much the data distribution changes with $\theta$; more information means smaller minimum variance. --- ## What it is - **Fisher information:** $I(\theta) = \mathbb{E}[(\partial \ln p(x|\theta)/\partial \theta)^2]$. It is additive for independent samples and sets the best possible precision (Fisher 1925). - **Cramér–Rao:** No unbiased estimator can beat $\mathrm{Var}(\hat{\theta}) \geq 1/(n I(\theta))$ for $n$ samples (Cramér 1946; Rao 1945). So “resolution” of a parameter is **bounded by information**. - **Implication:** Any substrate that “estimates” or “encodes” a quantity from finite resources (bits, samples, energy) has a **precision limit** given by the equivalent Fisher information. There is no infinite resolution. --- ## How RRT/RST uses it - **A2 (Resolution), [[Fidelity]]:** The substrate’s “resolution” — how well it can distinguish states — is limited by the signal and noise it has access to. The **[[Resource Triangle]]** ($W^n = \Omega^n + N^n$) and the fidelity curve $\mu(\eta,n)$ describe how much “signal” is available for a given noise floor. So the **precision** of any quantity derived from the substrate (e.g. acceleration, flux) is bounded by the equivalent of Fisher information: the format’s capacity to discriminate. RST does not re-derive Cramér–Rao; it **uses** it as the reason resolution cannot be infinite and why $\mu < 1$ in general. - **Format:** The **[[Format]]** (A1) has finite capacity; that finiteness, in estimation-theoretic terms, is the Fisher limit. So “the limit on precision from finite data” is the **hard-knowledge** statement that justifies bounded resolution in RST. --- ## Links | Concept | Note | |:---|:---| | Rendering quality $\mu(\eta,n)$ | **[[Fidelity]]** | | Budget, signal vs noise | **[[Resource Triangle]]** | | Finite substrate | **[[Format]]** | --- ## References - Fisher, R. A. (1925). *Theory of statistical estimation.* Proc. Camb. Philos. Soc. **22**, 700–725. [DOI](https://doi.org/10.1017/S0305004100009580) - Cramér, H. (1946). *Mathematical Methods of Statistics.* Princeton University Press, Princeton. (Cramér–Rao inequality.) - Rao, C. R. (1945). *Information and the accuracy attainable in the estimation of statistical parameters.* Bull. Calcutta Math. Soc. **37**, 81–91. (Reprinted in *Breakthroughs in Statistics*, Springer 1992.)