Gauges for normal mean–variance mixtures

Gauges for normal mean–variance mixtures#

Status: study plan, 2026-08-19. Companion to Subordinator tracking on the S&P 500, whose VG/NIG/GH \(\kappa\) comparison motivated it. Planned code: notebooks/gauges/ (public API only; no package change is presumed).

Every normal mean–variance mixture

\[ X \stackrel{d}{=} \mu + \gamma Y + \sqrt{Y}\,Z, \qquad Z\sim N(0,\Sigma)\perp Y\ge 0, \]

carries a one-parameter family of observationally equivalent parametrizations: for \(s>0\),

\[ (\mu,\;\gamma,\;\Sigma,\;Y)\;\longmapsto\; (\mu,\;\gamma/s,\;\Sigma/s,\;sY) \]

leaves the law of \(X\) unchanged. Fitted \((\gamma,\Sigma)\) and the subordinator moments \(E[Y]\), \(\mathrm{Var}(Y)\) are therefore quoted in a gauge — a rule that picks one representative per orbit. This page asks which gauges make two comparisons meaningful:

Distribution-invariant. Fits of GH, VG, NIG, NInvG to the same data should display \(\mu\), \(\gamma\), \(\Sigma\) on the same scale.
Dimension-invariant. A fit at \(d=100\) should display \(E[Y]\), \(\mathrm{Var}(Y)\) on the same scale as a fit to a \(d=50\) column subsample — and as the ensemble of \(100\) univariate (\(d=1\)) fits, with the multivariate estimate sitting inside the ensemble spread rather than on a different scale.

It then separates two places a gauge can act: inside EM (re-gauge each iteration, as regularization= does now) versus post estimation (fit in any gauge, re-gauge once for display), states hypotheses about when the choice affects estimation at all, and pre-registers a numerical plan.

1. The orbit and its invariants#

\(\mu\) is untouched by the orbit: location is identified, and every gauge below reports the same \(\mu\). The one-dimensional freedom lives in \((\gamma, \Sigma, Y)\). A maximal invariant is

\[ \bar\gamma = E[Y]\,\gamma, \qquad \bar\Sigma = E[Y]\,\Sigma, \qquad \bar Y = Y / E[Y], \]

i.e. the parameters of the representative with \(E[\bar Y]=1\), together with the shape of the normalized subordinator. The observable moments are functions of invariants only:

\[ E[X] = \mu + \bar\gamma, \qquad \mathrm{Cov}(X) = \bar\Sigma + \mathrm{cv}^2\,\bar\gamma\bar\gamma^\top, \qquad \mathrm{cv}^2 = \frac{\mathrm{Var}(Y)}{E[Y]^2} = \mathrm{Var}(\bar Y), \]

and the tracking-page SNRs are \(\kappa_{\mathrm{lev}}=\bar\gamma^\top\bar\Sigma^{-1}\bar\gamma\) and \(\kappa=\kappa_{\mathrm{lev}}\,\mathrm{cv}^2\).

A gauge is a section: a scale functional \(T(\theta)>0\) with \(T(g_s\theta) = s\,T(\theta)\), applied as \(s = 1/T(\theta)\) so that the representative satisfies \(T=1\). Candidates, existing and proposed:

Gauge	Condition	Reads off	Exists
`'a_eq_b'`	\(a=b\) on the GIG	subordinator coordinates	GH, NIG (no-op for VG, NInvG)
`'det_sigma_one'`	\(\lvert\Sigma\rvert=1\)	\(\Sigma\)	always
`'det_sigma_x'`	\(\log\lvert\Sigma\rvert\) = initial model’s	\(\Sigma\)	always
\(E[Y]=1\) (proposed; `pin_mean_y`)	\(E[Y]=1\)	subordinator law	needs \(E[Y]<\infty\) (NInvG: \(\alpha>1\))
geometric (proposed)	\(\exp E[\log Y]=1\)	subordinator law	always
median (proposed)	\(\mathrm{med}(Y)=1\)	subordinator law	always
trace (considered)	\(\mathrm{tr}(\Sigma)=d\)	\(\Sigma\)	always

Call a section \(Y\)-side if its condition involves only the law of \(Y\) (\(a=b\), \(E[Y]=1\), geometric, median) and \(\Sigma\)-side if it involves \(\Sigma\) (both determinant modes, trace).

2. Which sections satisfy which invariance#

2.1 Distribution invariance#

Fits of two families to the same data land on nearby laws of \(X\) but on different orbits in different parameter spaces. The gauged \((\gamma,\Sigma)\) are functions of the fitted invariants \((\bar\gamma, \bar\Sigma, \mathrm{cv}^2)\); two families display comparable parameters exactly when the section is pinned by functionals of the \(X\)-law that both families are forced to match.

\(E[X]\) and \(\mathrm{Cov}(X)\) are matched by every reasonable fit, and they identify \(\bar\gamma\) (through the skewness direction and \(E[X]-\mu\)) and \(\bar\Sigma = \mathrm{Cov}(X) - \mathrm{cv}^2\bar\gamma\bar\gamma^\top\) up to a correction that is second order in \(\bar\gamma\) (negligible on equity panels where \(\kappa_{\mathrm{lev}}\sim 10^{-2}\)). Hence:

\(E[Y]=1\): distribution-invariant. The displayed \((\bar\gamma,\bar\Sigma)\) are the moment-pinned invariants themselves. There is a second, EM-specific reason it is the natural \(Y\)-side choice: at an EM fixed point each family matches the prior subordinator moments in its sufficient set to the posterior averages, and \(E[Y]\) is in that set for VG (\(E[\log Y], E[Y]\)), NIG (\(E[Y], E[1/Y]\)), and GH (all three). Only NInvG (\(E[1/Y], E[\log Y]\)) treats \(E[Y]\) as an extrapolation — and its \(E[Y]=\beta/(\alpha-1)\) may not exist (\(\alpha\le 1\); normix floors the denominator at ALPHA_MOMENT_MARGIN, so a pinned “\(E[Y]\)” would silently use the floored surrogate). The geometric gauge \(\exp E[\log Y]=1\) is the mirror image: matched for VG, NInvG, GH, extrapolated for NIG, and it always exists.
\(\lvert\Sigma\rvert=1\), \(\mathrm{tr}(\Sigma)=d\): distribution-invariant. Both are functionals of \(\bar\Sigma\) alone.
\(a=b\): not distribution-invariant. The condition lives in GIG coordinates; the implied mean \(E[Y]=K_{p+1}(a)/K_p(a)\) depends on the index \(p\), which differs across families (\(p=-1/2\) for NIG, free for GH, degenerate for VG/NInvG). Measured on the S&P panel: after 'a_eq_b', \(E[Y]=1\) for NIG but \(0.25/0.14/0.10\) for GH at \(d=5/10/50\) (Subordinator tracking on the S&P 500, family-comparison section).

One caveat applies to all of them: distribution invariance holds only to the extent the families fit comparable laws. A clamped VG (alpha_min='density') is a different estimand by construction and will disagree on \(\mathrm{cv}^2\) under any gauge.

2.2 Dimension invariance#

For \(S\subseteq\{1,\dots,d\}\) the sub-vector \(X_S\) is a mixture with the same \(Y\) and restricted \((\mu_S,\gamma_S,\Sigma_{SS})\) — closure under affine maps, Blaesild1981. This gives a clean criterion:

Proposition. A section commutes with coordinate marginalization — gauging the \(d\)-model and then restricting equals restricting and then gauging — if and only if its condition is a functional of the law of \(Y\) alone. \(Y\)-side sections pick the same \(s\) for the model and every sub-model; a \(\Sigma\)-side section picks \(s=T(\Sigma_{SS})\), which moves with \(S\).

So \(E[Y]=1\), geometric, median, and also \(a=b\) are dimension-invariant at the population level; the determinant and trace gauges are not. The failure of the determinant gauge is not small. Under equicorrelation \(\Sigma = \sigma^2[(1-\rho)I + \rho\mathbf 1\mathbf 1^\top]\), the \(E[Y]\) displayed by the \(\lvert\Sigma\rvert=1\) gauge is

\[ E[Y]\Big|_{\det} = \det(\bar\Sigma)^{1/d} = \sigma^2\,(1-\rho)^{(d-1)/d}\bigl(1+(d-1)\rho\bigr)^{1/d} \;\xrightarrow{d\to\infty}\; \sigma^2(1-\rho): \]

at \(\rho=0.35\) the same data displays an \(E[Y]\) about \(32\%\) smaller at \(d=100\) than at \(d=1\) — pure gauge artifact. The trace gauge is exact in expectation under uniformly random column subsets (\(E[\mathrm{tr}(\Sigma_{SS})/k] = \mathrm{tr}(\Sigma)/d\)) but noisy per draw.

Two caveats separate the population statement from what the proposed experiment will see.

First, once a \(Y\)-side gauge is fixed, \(E[Y]\equiv 1\) across \(d\) is trivial, and the entire content of dimension invariance moves into \(\mathrm{Var}(Y)=\mathrm{cv}^2\) — which is gauge-free. Whether \(\mathrm{cv}^2\) is stable across \(d\) is then a statistical question about the data and the model, not about the gauge. The existing NIG fits already answer it for this panel: in the \(E[Y]=1\) gauge,

\(d\)	5	10	25	50	100	200	468
\(\mathrm{cv}^2\)	1.91	1.11	0.79	0.64	0.56	0.52	0.46

a monotone decline, while the univariate NIG fits of the equal-weight index give \(\mathrm{cv}^2 = 2.19\) (\(d=50\) universe) and \(2.32\) (full panel) — above every multivariate point. The mechanism: each coordinate carries common plus idiosyncratic excess kurtosis; a univariate fit reads the total, while the \(d\)-dimensional fit identifies \(Y\) from cross-sectional co-movement only, and the posterior concentrates via \(q(x)/d\). On data with idiosyncratic volatility clocks, the shared-\(Y\) model should report \(\mathrm{cv}^2\) falling with \(d\). No gauge can remove this; a dimension-invariant gauge is what makes it visible as a model property instead of a scale artifact.

Second, VG under alpha_min='density' has \(\mathrm{cv}^2 = 1/\alpha \le 2/(d+0.2)\): the clamp forces a \(1/d\) decay of the estimand itself. Cross-\(d\) VG comparisons are structurally broken under the \(d\)-aware clamp; a \(d\)-independent absolute alpha_min restores comparability of the estimand but abandons density boundedness for \(d > 2\,\alpha_{\min}\).

2.3 Summary#

Section	Distribution-invariant	Dimension-invariant
\(E[Y]=1\)	yes (moment-pinned; matched for VG/NIG/GH)	yes (\(Y\)-side)
geometric \(\exp E[\log Y]=1\)	yes, weaker (matched for VG/NInvG/GH)	yes (\(Y\)-side)
median\((Y)=1\)	weakest (not moment-matched)	yes (\(Y\)-side)
`'a_eq_b'`	no (GH vs NIG \(E[Y]\): \(0.10\) vs \(1\))	yes (\(Y\)-side)
`'det_sigma_one'` / `'det_sigma_x'`	yes	no (\(\approx 1-\rho\) drift)
\(\mathrm{tr}(\Sigma)=d\)	yes	in expectation only
`'none'`	no (init- and path-dependent)	no

\(E[Y]=1\) is the only candidate satisfying both, with the NInvG existence caveat; the geometric gauge is the robust fallback that always exists. Both are cheap post-fit rescales along the existing orbit (NormalMixture._rescale).

3. In EM or post estimation?#

The batch EM map is gauge-equivariant. Under \(g_s\) the posterior is a relabeling, \(Y\mid x \mapsto sY\mid x\) (GIG\((p-\tfrac d2,\,a+\tilde q,\,b+q(x))\) with \(a\mapsto a/s\), \(b\mapsto sb\), \(\tilde q\mapsto\tilde q/s\), \(q(x)\mapsto s\,q(x)\)), and every family’s M-step commutes with the orbit because the subordinator families are closed under \(Y\mapsto sY\). Hence, in exact arithmetic, re-gauging after each M-step does not change the sequence of orbits EM visits — only the representative reported at the end. Gauging inside batch EM and gauging once after it are the same estimator. Equivalently: the orbit is a zero-information direction of the marginal likelihood; batch EM neither pushes along it nor restores it.

The equivalence leaks through exactly four channels, which is where the hypotheses live:

Absolute floors and clamps are not equivariant. B_POST_FLOOR (\(b_{\mathrm{post}}\ge 10^{-6}\)), SIGMA_REG, GIG_CLAMP_LO/HI are fixed constants while \(q(x)\) and the GIG parameters scale with the gauge. On daily equities the \(E[Y]=1\) gauge puts \(q(x)\approx d\) (Mahalanobis scale); the \(\lvert\Sigma\rvert=1\) gauge puts \(q(x)\sim 10^{-4}d\) — four orders closer to the \(b_{\mathrm{post}}\) floor. VG, the only family whose E-step actually leans on that floor, is the family where the in-EM gauge can plausibly change the fitted orbit.
The stopping rule is gauge-dependent. Convergence is measured on changes of \((\mu,\gamma,L_\Sigma)\) (em_convergence_params). A gauged loop quotients out the flat direction; an ungauged loop lets float-level drift along the orbit contaminate the metric. This changes when EM stops, not where the orbit goes.
Data scale interacts with (1). Fitting returns in natural units versus percent (\(\times 100\)) moves every floor’s bite point. Gauge-invariants should be unchanged; any change quantifies channel (1).
Online EM averages η across iterations. The EWMA recursion \(\eta_t = (1-w)\eta_{t-1} + w\,t(x_t)\) (Cappe2009) is an average of sufficient statistics in a fixed gauge. Re-gauging the model each step without transforming the running \(\eta\) mixes incompatible scales — that is why the tracking study ran Phase 3 without re-gauging. But with no re-gauge the stochastic updates random-walk along the zero-information direction: at \(h=21\) the gauge exploded to \(\overline{E[Y]}_t = 4772\). Neither option is right. The consistent third option transforms the running η with the same orbit map: under \(Y\mapsto sY\),

\[ \eta_1 \mapsto \eta_1 + \log s,\quad \eta_2 \mapsto \eta_2/s,\quad \eta_3 \mapsto s\,\eta_3,\quad \eta_4 \mapsto \eta_4,\quad \eta_5 \mapsto \eta_5/s,\quad \eta_6 \mapsto \eta_6/s, \]

for \(\eta = (E[\log Y], E[1/Y], E[Y], E[X], E[X/Y], E[XX^\top/Y])\). This is the one place a gauge plausibly improves estimation rather than reporting, and the one concrete package-change candidate.

4. Hypotheses#

G1 (distribution invariance). Under the \(E[Y]=1\) display gauge, NIG/GH (and unclamped VG) fits of the same universe agree on \(\gamma\) and \(\Sigma\) to within estimation error (target: \(\le 10\%\) relative \(\ell_2\)/Frobenius at \(d\le 10\)); under 'a_eq_b' the displayed \(E[Y]\) disperses by \(\ge 5\times\) (already observed). \(\lvert\Sigma\rvert=1\) also aligns families.
G2 (dimension invariance of sections). On the same data, \(E[Y]\) displayed by the \(\lvert\Sigma\rvert=1\) gauge drifts across nested \(d\) by a factor consistent with the equicorrelation prediction \(\approx(1-\rho)\); \(Y\)-side gauges show no drift by construction.
G3 (well-specified unbiasedness). On synthetic single-clock data (fitted NIG generator, matched \(T\)), \(\mathrm{cv}^2\) estimates are unbiased across \(d\in\{1,\dots,100\}\) and the \(d=100\) estimate falls inside the interquartile range of the \(100\) univariate estimates.
G4 (real-data decline is misspecification). On the S&P panel the univariate ensemble of \(\mathrm{cv}^2\) sits above the multivariate estimate at every \(d\), and the multivariate \(\mathrm{cv}^2(d)\) declines monotonically. The ensemble mid-range is not recovered by any gauge; the gap is an idiosyncratic-clock measure.
G5 (VG clamp). With alpha_min='density', VG \(\mathrm{cv}^2\) tracks the forced ceiling \(2/(d+0.2)\) across \(d\); with a fixed absolute alpha_min, the cross-\(d\) profile parallels NIG’s.
G6 (batch equivalence). For NIG/GH/NInvG, fitting with 'none', 'a_eq_b', or 'det_sigma_one' and re-gauging post-fit gives identical gauge-invariants (\(\kappa_{\mathrm{lev}}\), \(\mathrm{cv}^2\), mean log-likelihood) up to float noise (target: \(10^{-8}\) relative for NIG’s closed-form M-step, \(10^{-6}\) for GH’s solver path).
G7 (the leaks). VG violates G6 measurably through B_POST_FLOOR when the in-EM gauge compresses \(q(x)\) (\(\lvert\Sigma\rvert=1\) on raw returns); iteration counts differ across in-EM gauges through the stopping metric even where invariants agree; rescaling the data \(\times 100\) leaves invariants unchanged except where floors bind.
G8 (online EM). Per-step re-gauging with the η transform of §3(4) reproduces the no-regauge orbit path (gauge-invariant \(\kappa_t\) within Monte Carlo bands) while keeping \(\overline{E[Y]}_t\in[0.5,2]\); per-step re-gauging without the transform biases \(\kappa_t\).

5. Study plan#

Each phase states its completion criterion. All code in notebooks/gauges/ on the public API; NIG/GH/VG (+NInvG where noted); the S&P panel and universes of Subordinator tracking on the S&P 500.

Phase A — equivariance audit (G6, G7). Synthetic NIG/GH data, \(d\in\{2,10\}\), one seed. Fit from a common default_init under each in-EM gauge; record per-iteration invariants (\(\kappa_{\mathrm{lev}}\), \(\mathrm{cv}^2\), mean log-likelihood) and floor-hit counters (\(b_{\mathrm{post}}\) at floor, GIG clamps). Done when: invariant trajectories overlaid per family with agreement tolerances measured, and VG’s floor-driven divergence (or its absence) quantified.

Phase B — distribution invariance (G1). Real panel, \(d\in\{5,10,25,50\}\), seed 0: NIG, GH, VG (clamped and absolute alpha_min), NInvG. Display each fit under 'a_eq_b', \(\lvert\Sigma\rvert=1\), \(E[Y]=1\), geometric. Report cross-family relative distances of \(\gamma\) (ℓ2), \(\Sigma\) (Frobenius and log-det), and the displayed \(E[Y]\) table. Done when: the §2.3 table’s distribution-invariance column is confirmed or amended with measured numbers.

Phase C — dimension invariance (G2–G5). (i) Synthetic: draw from the fitted \(d=100\) NIG generator, \(T=2552\), \(R=10\); fit at nested \(d\in\{1,5,10,25,50,100\}\) and fit all \(100\) univariate margins on one replicate; compare \(\mathrm{cv}^2\) point estimates to truth and the univariate ensemble quantiles. (ii) Real panel: same design, NIG (+GH at \(d\le 50\); VG both clamp variants); additionally display \(E[Y]\) under \(\lvert\Sigma\rvert=1\) across \(d\) to measure the G2 drift against the equicorrelation prediction. Done when: synthetic bias/SE table across \(d\) and the real-data ensemble-vs-multivariate figure exist, with the G4 gap quantified.

Phase D — scale robustness (G7). Refit Phase B’s NIG/GH/VG at \(d\in\{10,50\}\) with returns in natural units and \(\times 100\). Compare gauge-invariants and floor-hit counters. Done when: the invariants’ sensitivity to data scale is tabulated per family × in-EM gauge, and any discrepancy is attributed to a named constant.

Phase E — online EM gauge handling (G8). Rerun the Phase 3 online loop (\(d=50\), \(h\in\{21,63\}\)) under three policies: no re-gauge (baseline), per-step re-gauge without η transform, per-step re-gauge with the §3(4) η transform. Track \(\overline{E[Y]}_t\), gauge-invariant \(\kappa_t\), direction cosine, and terminal-vs-static consistency. Done when: the three policies are compared on one figure and G8 is accepted or rejected.

6. Decision gates#

If G6 holds and G7 is confined to VG: the in-EM regularization flag is a reporting/stopping convenience for batch EM. Keep 'a_eq_b' as a cheap dimension-invariant loop gauge, add a 'mean_y_one' mode (or promote pin_mean_y) as the recommended display gauge, and standardize all cross-family and cross-\(d\) tables on it.
If G7 shows VG orbit-level sensitivity: recommend fitting VG in the gauge that keeps \(q(x) = O(d)\) (the \(E[Y]=1\) scale) and record it as a design row.
If G8 holds: the η-consistent re-gauge is the package-change candidate — an online-EM option that re-gauges model and running η together, closing the Phase 3 gauge-explosion problem without biasing the recursion.
NInvG: if the \(E[Y]=1\) gauge is adopted, its display must fall back to the geometric gauge when \(\alpha\le 1+\) margin rather than silently using the floored moment.

7. What this is not asking `normix` to do#

No change to fitting defaults before Phases A–E report. The invariants \(\kappa_{\mathrm{lev}}\), \(\kappa\), \(\mathrm{cv}^2\) need no gauge at all and remain the preferred comparison quantities; the gauge question is about displaying \((\gamma, \Sigma, E[Y], \mathrm{Var}(Y))\) themselves, about warm starts that cross families or dimensions, and about the online-EM η recursion.