QBIC in Statistics & Wave Physics

• The paper derives QBIC using a Laplace expansion of a marginal quasi‐likelihood for SEMs, ensuring consistent model selection under high-frequency observations.
• QBIC is a dual-concept representing both a statistical tool for penalizing model complexity and a resonant state in wave physics with high yet finite Q factors.
• Quasi-bound states in the continuum (qBIC) emerge by perturbing ideal BIC conditions, enabling ultra-high-Q resonances for enhanced sensing and modulation applications.

Qbic, more commonly written QBIC or qBIC, denotes two unrelated technical constructs in current research literature. In statistical inference, QBIC is a quasi‑Bayesian information criterion derived from a marginal quasi‑log likelihood for model selection in structural equation models (SEM) with latent diffusion or jump‑diffusion processes observed at high frequency. In wave physics, optics, and acoustics, qBIC denotes a quasi‑bound state in the continuum: a near‑BIC resonant state with very high but finite quality factor $Q$, arising when perfect decoupling from radiation is perturbed by loss, asymmetry, angle, or finite‑size effects (Kusano et al., 2024, Kusano et al., 13 Apr 2026, Wu et al., 15 Jul 2025, Damgaard-Carstensen et al., 2024, Yang et al., 29 Aug 2025).

1. Terminological scope and disambiguation

The same acronym is used in two different research domains. The statistical usage is capitalized as QBIC and is modeled on BIC in the Schwarz sense. The wave‑physics usage is commonly written qBIC and is modeled on BIC as bound states in the continuum.

Notation	Expansion	Research setting
QBIC	quasi‑Bayesian information criterion	SEM model selection from diffusion or jump‑diffusion observations
qBIC / QBIC	quasi‑bound state in the continuum	optical, acoustic, and metasurface resonances with high but finite $Q$

A recurrent source of confusion is that the two meanings are not variants of a single theory. In the SEM literature, QBIC is an information criterion built from a quasi‑likelihood and justified by a Laplace expansion of a marginal quasi‑likelihood. In the metasurface and resonator literature, qBIC is a resonant state that approximates an ideal BIC, which would have zero leakage and $Q \to \infty$ (Kusano et al., 2024, Wu et al., 15 Jul 2025).

2. QBIC as a quasi‑Bayesian information criterion in SEM

In the statistical literature, QBIC is formulated for SEMs in which observable processes are generated by latent continuous‑time dynamics. Both the diffusion and jump‑diffusion papers use two observable blocks, a factor structure, and a latent structural equation of the form

$$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$

with $\Psi_0 = I_{k_2} - B_0$ assumed nonsingular. The observable vector is stacked as $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$, and candidate models are indexed by $m=1,\dots,M$, each with parameter $\theta_m$ and a positive definite covariance structure $\Sigma_m(\theta_m)$ determined by loadings and volatility matrices (Kusano et al., 2024, Kusano et al., 13 Apr 2026).

The high‑frequency asymptotic scheme is fixed‑horizon: observations are taken at

$t_i = i h_n,\qquad T = n h_n \text{ is fixed},$

with $Q$0 and $Q$1. In the diffusion case, the quasi‑likelihood is Gaussian and covariance‑based, using increments $Q$2 and the SEM‑implied covariance matrix. The resulting quasi‑log likelihood is

$Q$3

Equivalently, with empirical quadratic variation $Q$4,

$Q$5

The quasi‑MLE is $Q$6 (Kusano et al., 2024).

For jump‑diffusion SEMs, the same covariance‑based strategy is combined with increment truncation. A jump is declared on $Q$7 when $Q$8, and only increments with $Q$9 are retained. The paper assumes $Q \to \infty$0 for the theoretical results. The quasi‑log likelihood becomes

$Q \to \infty$1

and $Q \to \infty$2 maximizes $Q \to \infty$3 over $Q \to \infty$4 (Kusano et al., 13 Apr 2026).

3. Laplace expansion, penalty structure, and model‑selection consistency

The defining feature of QBIC in the SEM papers is that it is not introduced ad hoc. It is derived from an asymptotic expansion of a marginal quasi‑likelihood

$Q \to \infty$5

or, in the jump‑diffusion paper,

$Q \to \infty$6

The expansion is controlled by an identifiability condition based on a population contrast. In the diffusion paper this contrast is

$Q \to \infty$7

with quadratic identifiability

$Q \to \infty$8

In the jump‑diffusion paper, the Laplace‑type expansion is

$Q \to \infty$9

which yields the BIC‑type form (Kusano et al., 2024, Kusano et al., 13 Apr 2026).

Two diffusion‑SEM criteria are proposed. The first is

$$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$0

where $$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$1 is the normalized observed quasi‑information. The second is the simplified

$$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$2

For jump‑diffusion SEMs, the proposed criterion is

$$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$3

In both settings, the selected model is the one with the smallest criterion value (Kusano et al., 2024, Kusano et al., 13 Apr 2026).

The theoretical model‑selection target is the smallest correctly specified model. A model is correctly specified if its covariance matrix can reproduce the true covariance, and misspecified otherwise. The diffusion paper proves that both $$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$4 and $$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$5 are model‑selection consistent: asymptotically they select the uniquely defined smallest correctly specified model $$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$6. The jump‑diffusion paper proves the analogous result for $$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$7: misspecified models are asymptotically rejected, and correctly specified but over‑fitted nested models are asymptotically excluded (Kusano et al., 2024, Kusano et al., 13 Apr 2026).

The same papers contrast QBIC with QAIC,

$$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$8

or

$$\eta_{0,t} = B_0\,\eta_{0,t} + \Gamma_0\,\xi_{0,t} + \zeta_{0,t},$$9

Because the QAIC penalty does not scale with $\Psi_0 = I_{k_2} - B_0$0, it is not model‑selection consistent for nested models. In the jump‑diffusion paper, the failure is formalized by

$\Psi_0 = I_{k_2} - B_0$1

In the diffusion paper’s simulation with 10,000 replications, $\Psi_0 = I_{k_2} - B_0$2, and candidate models of dimensions $\Psi_0 = I_{k_2} - B_0$3, $\Psi_0 = I_{k_2} - B_0$4, and $\Psi_0 = I_{k_2} - B_0$5, $\Psi_0 = I_{k_2} - B_0$6 selected Model 1 9991 times, $\Psi_0 = I_{k_2} - B_0$7 selected Model 1 9995 times, and QAIC selected the over‑fitted Model 2 1600 times. In the jump‑diffusion simulation, 10,000 data sets were generated for each $\Psi_0 = I_{k_2} - B_0$8 with $\Psi_0 = I_{k_2} - B_0$9, $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$0, and $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$1; the reported pattern is that QBIC increasingly selects the optimal Model 1, selects the over‑fitted Model 2 less often as $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$2 grows, and essentially never selects the misspecified Model 3 for large $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$3 (Kusano et al., 2024, Kusano et al., 13 Apr 2026).

4. qBIC as a quasi‑bound state in the continuum

In wave physics, a bound state in the continuum is a localized mode whose eigenfrequency lies inside the radiation continuum but does not radiate. In the ideal case, the leakage rate is zero, the linewidth is zero, and $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$4. A quasi‑BIC is the experimentally relevant counterpart: the mode remains very weakly radiative, but material loss, fabrication imperfections, finite‑size effects, or deliberate symmetry breaking make $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$5 large rather than infinite (Wu et al., 15 Jul 2025, Damgaard-Carstensen et al., 2024, Yang et al., 29 Aug 2025).

Several mechanisms appear in the cited literature. A Friedrich–Wintgen BIC arises from destructive interference between two modes that couple to the same radiation channel. In the acoustic‑solid paper this is expressed through the non‑Hermitian effective Hamiltonian

$X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$6

with a BIC condition

$X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$7

The vanishing of the continuum coupling sets the attenuation factor $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$8 to zero. In practice, the condition is only approximately met, giving a Friedrich–Wintgen QBIC with very small but nonzero $X_{0,t} := (X_{1,0,t}^T,X_{2,0,t}^T)^T$9 (Wu et al., 15 Jul 2025).

A second route is symmetry protection and controlled perturbation. In the LiNbO$m=1,\dots,M$0 electro‑optic metasurface, a BIC is a guided mode of the planar LiNbO$m=1,\dots,M$1 waveguide at a frequency within the free‑space continuum that remains non‑radiative under particular symmetry or angle conditions. A slight shift from normal incidence produces a near‑BIC with high $m=1,\dots,M$2, while grating coupling and loss convert the inaccessible BIC into a measurable qBIC resonance (Damgaard-Carstensen et al., 2024).

A third route is environmental asymmetry. The permittivity‑asymmetric metasurface defines the asymmetry factor as

$m=1,\dots,M$3

and numerically verifies the canonical scaling law

$m=1,\dots,M$4

When $m=1,\dots,M$5, the environment becomes symmetric and the structure returns to a BIC. When $m=1,\dots,M$6, the mode becomes a qBIC with finite radiative coupling (Yang et al., 29 Aug 2025).

These studies also delimit a common misconception. A qBIC is not simply a high‑$m=1,\dots,M$7 resonance. Its defining feature is genealogical: it is a radiative state that originates from a BIC condition and inherits the BIC’s decoupling mechanism in perturbed form (Wu et al., 15 Jul 2025, Damgaard-Carstensen et al., 2024, Yang et al., 29 Aug 2025).

5. Implementations in acoustic cavities and electro‑optic metasurfaces

One concrete realization is a semi‑closed acoustic cavity in which thin circular aluminum plates seal side openings and mediate acoustic–solid coupling. The main cavity has $m=1,\dots,M$8, tunable $m=1,\dots,M$9, and depth $\theta_m$0. The cavity mode and a Fabry–Pérot‑type plate mode interfere through the same acoustic continuum. At a height/length ratio $\theta_m$1, the attenuation factor $\theta_m$2 of mode M14 becomes zero, indicating the BIC point, and around that point the simulated $\theta_m$3 reaches $\theta_m$4. The same study reports experimental $\theta_m$5-factors of about $\theta_m$6 at optimal thickness $\theta_m$7, and explicitly compares this with a “typical value (around 400)” for open‑system QBICs (Wu et al., 15 Jul 2025).

The same paper identifies the design levers governing the quasi‑closed acoustic QBIC. Port diameter near $\theta_m$8 and aluminum thickness near $\theta_m$9 maximize the simulated $\Sigma_m(\theta_m)$0; as plate thickness tends to zero, the system approaches a conventional open cavity, where the Fano resonance persists but the resonance is broader and the $\Sigma_m(\theta_m)$1-factor is much lower. This suggests that the quasi‑closed configuration suppresses leakage not only through interference but also through channel selectivity imposed by the elastic plates (Wu et al., 15 Jul 2025).

A second realization is an electro‑optic LiNbO$\Sigma_m(\theta_m)$2 metasurface operating in reflection. The device consists of a $\Sigma_m(\theta_m)$3 gold back reflector, a $\Sigma_m(\theta_m)$4 Cr adhesion layer, an $\Sigma_m(\theta_m)$5 $\Sigma_m(\theta_m)$6-cut LiNbO$\Sigma_m(\theta_m)$7 layer, and a gold nanoridge grating. For the fabricated symmetric grating, the period is $\Sigma_m(\theta_m)$8, ridge width $\Sigma_m(\theta_m)$9, ridge thickness $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$0, and the number of periods is $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$1. The qBIC appears as a deep reflection minimum near $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$2–$t_i = i h_n,\qquad T = n h_n \text{ is fixed},$3, with loaded $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$4 and intrinsic $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$5 at critical coupling (Damgaard-Carstensen et al., 2024).

Electro‑optic tuning uses the Pockels effect,

$t_i = i h_n,\qquad T = n h_n \text{ is fixed},$6

with applied bias $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$7 across the $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$8 film. The paper reports a modulation depth reaching $t_i = i h_n,\qquad T = n h_n \text{ is fixed},$9, modulation of $Q$00 of the total incident power, and a measured $Q$01 electrical bandwidth of $Q$02; with a smaller electrode, the measured cutoff increases to $Q$03. Using the propagation length $Q$04 as a minimal pixel size, the estimated potential bandwidth is $Q$05 (Damgaard-Carstensen et al., 2024).

6. Sensing architectures, restored states, and cross‑domain significance

The high‑$Q$06 character of qBICs is directly exploited for sensing. In the acoustic–solid system, changing the gas composition shifts the resonance frequency because the sound speed and acoustic impedance change. The study reports that for the M31 QBIC mode the resonance moves from about $Q$07 in air to about $Q$08 in pure CO$Q$09. The sensitivity is defined as

$Q$10

and the numerical result is about $Q$11 per $Q$12 change in CO$Q$13 concentration, while the experimental result near $Q$14 CO$Q$15 is about $Q$16 per $Q$17. The same work states that simulations indicate the potential to detect a $Q$18 concentration change in principle, given enough frequency resolution (Wu et al., 15 Jul 2025).

In the optical sensing literature, the permittivity‑asymmetric qBIC metasurface uses the environment itself as the asymmetry control variable. Its single‑wavelength sensing signal is the transmittance change

$Q$19

with intensity sensitivity

$Q$20

The paper reports a competitive transmittance sensitivity of $Q$21 under single‑wavelength conditions and a linear window area that is 104 times larger than that of a geometry‑asymmetric qBIC. It also reports that environmental permittivity asymmetry can restore a geometry‑asymmetric qBIC to an ultra‑high‑$Q$22 state, with $Q$23 over $Q$24, approaching a BIC condition (Yang et al., 29 Aug 2025).

A distinctive concept in that work is the restored symmetry‑protected BIC. Starting from a geometry‑broken qBIC, local dielectric compensation can re‑establish cancellation of the net dipole moment and thereby suppress radiation again. The article’s framing is precise: unlike traditional BICs, which are typically inaccessible once perturbed, the permittivity‑restored BIC becomes accessible through environmental perturbations. This suggests a sensing architecture in which the analyte does not merely shift a resonance horizontally but also changes the radiative coupling that defines the qBIC linewidth and depth (Yang et al., 29 Aug 2025).

Across the two domains, the commonality is only structural at a very abstract level. Statistical QBIC uses a quasi‑likelihood to penalize model dimension and recover the smallest correctly specified SEM. Physical qBIC uses incomplete decoupling from radiation to create an ultra‑narrow resonance for modulation or sensing. The shared acronym therefore marks a convergence of terminology rather than a convergence of method (Kusano et al., 2024, Wu et al., 15 Jul 2025).