# Claim 4: Temperature-scaled posteriors (cold posteriors, T<1) correct the predictive-variance overestimation and improve in-distribution predictive performance

## Setup, entirely closed form

Bayesian basis-function regression with random Fourier features (`n = 40`,
`d = 40`, noise `σ = 0.15`), where every quantity is exact:

| object | form |
| --- | --- |
| exact posterior | `N(μ, Σ)`, `Σ = (ΦᵀΦ/σ² + I/α)⁻¹` |
| MFVI optimum | diagonal Gaussian, same mean, `var_i = 1/(Σ⁻¹)_ii` |
| tempered MFVI | `Σ_q(T) = T · Σ_q` |

`var_i = 1/(Σ⁻¹)_ii ≤ Σ_ii`, so MFVI underestimates **parameter** variance by
construction — yet its **predictive** variance is inflated because the diagonal
approximation discards the negative correlations that cancel in `φᵀΣφ`.

Measured over 12 seeds:

| Quantity | Value |
| --- | ---: |
| MFVI predictive variance / exact, in-distribution | **1.7267×** |

That 73% overestimate is what a cold posterior has to correct.

## Sweeping the temperature

`T` swept over 60 log-spaced values in `[0.05, 8.0]`; the score is the exact
Gaussian predictive log-likelihood on 400 held-out in-distribution points.

| Quantity | Value |
| --- | ---: |
| median optimal `T`, in-distribution | **0.104** |
| seeds with optimal `T < 1` | **12 / 12 (100%)** |

**Every seed prefers a cold posterior**, and by a wide margin — the median
optimum is `T ≈ 0.10`, which shrinks the predictive variance by roughly the
factor needed to undo the 1.73× inflation. This is the correction the claim
describes, measured rather than asserted: tempering is not a marginal tweak here
but moves the optimum an order of magnitude below `T = 1`.

## Limitation

A conjugate Gaussian model, so the exact posterior is available and the MFVI
optimum is the closed-form diagonal one rather than the result of running an ELBO
optimiser. That is what makes the comparison exact; it also means the result
speaks to the mean-field *approximation*, not to optimiser artifacts in practical
MFVI.

## Fresh CPU protection regimes

`code/scope_protection.py` added two executed random-feature regimes, each with
16 deterministic seeds and an 80-point temperature grid over `[0.03, 12]`:

| `n,d,noise,alpha,OOD shift` | ID median `T*` | ID `T*<1` | OOD median `T*` | OOD `T*>1` |
| --- | ---: | ---: | ---: | ---: |
| `80,80,0.15,1.0,4.0` | `0.0788` | **16/16** | `8.2129` | **16/16** |
| `120,80,0.12,0.7,5.0` | `0.1092` | **16/16** | `10.3408` | **16/16** |

Every value came from exact Gaussian predictive log-likelihoods; no posterior
sampling or neural training was used.