Graph–Smoothed Bayesian Black-Box Shift Estimator and Its Information Geometry

We would like to thank you for your very constructive comments.

When is the label-shift model appropriate?

We first summarize the major distribution shift settings as follows.

The last column (our focus) includes

• medical testing (pathogen prevalence rises or falls, but radiograph physics stays the same),
• recommender or ad-click systems (item popularity drifts while the encoding of text or images stays fixed),
• traffic scenes across seasons or geographies.

These settings share the property that collecting a handful of new labels is expensive, whereas unlabeled target images or clicks are plentiful.

A principled diagnostic test for label shift

Define

• $\hat{r} = 1 / n’$ (histogram of target predictions),
• $\hat{C}$ (empirical confusion on the validation split),
• $\hat{p}$ (empirical source prior).

Under the null $H_0 : q = p$ (no label shift),

Hence the likelihood-ratio ($\chi^2$ statistic):

Because $\hat{p}$ enters only through the plug‑in diagonal and the null does not constrain it, the usual Bartlett correction is unnecessary; asymptotically we retain an ordinary $\chi_{K-1}^2$.

If the $p$-value is large, there is no statistical evidence for label shift, and no correction suffices.

If the null is rejected, we proceed to label shift correction approach, and the test is free because all three inputs $\hat{r}$, $\hat{C}$, $\hat{p}$ are already computed.

Power of the diagnostic

Let $\delta = q - p$ be the true prior change.

Under the alternative,

i.e., a $\chi^2$ distribution with non-centrality parameter $\lambda$.

The test therefore has asymptotic power

whenever $q \neq p$, so even moderate sample sizes suffice to detect practically relevant prior changes.

Does label-shift require the causal arrow $Y \to X$?

The assumption $P_{te}(X \mid Y) = P_{tr}(X \mid Y)$ is a statement about invariance of a conditional distribution, not about causal direction. It can arise in two distinct ways.

• Causal: $Y \to X$ and data-generation mechanism $f\colon Y, \epsilon \mapsto X$ unchanged, only $\Pr(Y)$ perturbed (an intervention on the root node $Y$ in Pearl’s SCM sense).
• Anticausal, selection: $X \to Y$ and labeling policy $\Pr(Y \mid X)$ unchanged, but we sub-sample on $Y$.

Hence, label shift is valid whenever i) the mapping $Y \mapsto X$ (causal case) or the annotation policy $X \mapsto Y$ (anticausal case) is stable across domains, and ii) only the base rate $\Pr(Y)$ drifts.

Either arrow therefore fits, the key is mechanism invariance, not direction.

Let an SCM be

with exogenous $U, V$. A shift that changes $\Pr(U)$ (but leaves $f$ fixed) yields exactly label shift.

Conversely, in an anticausal model

we can obtain label shift by re-weighting samples according to $Y$ after data collection, again $P(X \mid Y)$ is preserved.

How this perspective guides practice

• If you can argue mechanism stability, e.g., “the radiography device and organ physiology are unchanged while disease prevalence drifts” or “the feature extractor is frozen”, then label shift is defensible regardless of causal direction.
• If mechanism stability is doubtful, run the $\chi^2$ diagnostic and inspect mis-fit. Large residuals against both label shift and covariate shift tests indicate a more complex general shift.

The robustness bound in our manuscript is therefore agnostic to causal direction: only the invariance of $P(X\mid Y)$ enters through $\bm F_0$.

Computational cost

We measure work in floating-point operations (flops) and write $K$ = number of classes and $|E|$ = number of graph edges.

For the $k$-nearest neighbor graphs we use, $|E| = kK$ with a small, $K$-independent constant $k$ (4 for CIFAR‑10, 8 for CIFAR‑100).

• Vanilla BBSE: The point estimator solves a single linear system $\hat{q} = \tilde{C}^{-1}\tilde{r}$, where $\tilde{C}\in\mathbb R^{K\times K}$ is dense once any amount of Laplace or Tikhonov regularisation is applied. Gaussian elimination therefore costs $\mathcal{O}(K^3)$ flops and $\mathcal O(K^{2})$ memory.
• GS‑B$^3$SE (one Newton–CG sweep): Write $\Phi = (\phi_1,\dots,\phi_K)^\top \in \mathbb{R}^{K\times (K-1)}$ and $\theta \in \mathbb{R}^{K-1}$, both constrained to the centered subspace $\\{v^\top 1=0 \\}$. At each sweep we perform three algebraic operations:
  • update each column $\Phi \leftarrow \Phi - \mathcal{H}_{\Phi}^{-1}\nabla\mathcal{L}$,
  • update $\theta$ analogously, and
  • Gamma shape/scale updates for $\tau_q$, $\tau_C$.

Adding them and using $|E|=kK$, one sweep = $\mathcal{O}(|E|K + K^2) - \mathcal{O}(kK^2)$, and memory usage is $\mathcal{O}(|E| + K^2)$.

Overall complexity.

Thm. 2 establishes that $F(q)$ is $\alpha$-strongly geodesically convex with $\alpha = n’\lambda_{\min}(F_0) + \tau_q \lambda_2(L) > 0$. Consequently a back-tracking Newton method requires $T = \mathcal{O}(\ln 1/\epsilon)$ sweeps to reach tolerance $\epsilon$.

Hence the total complexity is

Because $k$ is at most 8 and $\ln 1/\epsilon \le 10$ in all our runs, the crossover happens already at $K\ge 25$.

Therefore the dominant term grows only quadratically in the class count $K$ (the sparse Laplacian never densifies), whereas BBSE’s one‑off inverse is cubic.

Next, we report the wall-clock runtimes (single NVIDIA T4, PyMC 5.10, float32).

As you suggested, we believe these results help users to consider the accuracy-complexity trade-off.

All times are <1 min on a single T4; in practice the wall‑clock is dominated by the CNN forward‑pass when $n'$≈10 k.

We would like to thank you for your very constructive comments.

Ablations on graph construction and the role of CLIP

We provide the additional ablations on graph construction.

In this experiment, we consider the following strategies.

• CLIP-text (already in the current manuscript): cosine $k$-NN on text prompts.
• Penultimate-image: Euclidean $k$‑NN on frozen ResNet‑18 penultimate features (identical backbone as the classifier).
• Random-embeddings: same sparsity pattern as CLIP‑text but edge weights i.i.d. $U(0, 1)$ (destroys semantics, preserves Laplacian spectrum scale).
• Identity graph: $L = 0$, switches the hierarchy off (reduces to independent logistic‑Normal prior and hence Bayesian BBSE with no smoothing).

The results are as follows.

Remarks

• Any connected graph ($\lambda_2(L) > 0$) already helps over the frequentist point estimate; cf. the “random” line.
• The more task-aligned the similarity (CLIP > penult-img > random), the lower the bias term.
• The identity graph reproduces Bayesian BBSE with no smoothing and is almost identical to MLLS. Its gap to CLIP shows the specific benefit of our graph-coupled hierarchy.

Why do the graph ablations behave as the above?

Recall the posterior variance bound we already proved:

which holds for any connected graph.

If we write the mean-squared error as

then, it shows the variance term shrinks by a factor $1 / \lambda_2(L)$.

Even a $k$-NN graph whose edge weights are i.i.d. $\mathcal{U}(0, 1)$ has, with high probability,

for $k \geq c’ \ln K$. Hence a purely random graph already cuts the class-wise standard error by $\sqrt{K / (ckN)}$ compared with the identity graph $\lambda_2 = 0$. This explain why the “random” row in the ablation table still beats the deterministic frequentist BBSE (it keeps all the variance relief while paying only a bias penalty).

Inconsistent graph construction (text vs. image embeddings)

Reviewer’s concern is “Why text on CIFAR but image on MNIST?”

• On MNIST the class names are single digits (0, 1, 2, …), so CLIP-text gives almost zero variance (every digit is equally dissimilar). Image-space features do capture visual adjacency (e.g., 3 ↔ 8 ↔ 9)
• On CIFAR the opposite holds: CLIP’s joint vision-language pre-training encodes high-level semantics missing from raw pixel features (e.g., “truck” closer to “bus” than to “cat”).

Multinomial assumption

Recap the notation:

• $n_i^S$: # source-validation examples whose true class is $i$.
• $N_i^S$: # predicted labels produced by the frozen classifier $\hat{h}$ among those $n_i^S$ examples.
• $\tilde{N}$: # predicted labels on the $n’$ unlabeled target points}.

These are integer-valued random vectors by construction. The standard sampling model is therefore

where each realization of a multinomial distribution is, of course, a vector in $\mathbb{N}^k$.

The assumption is therefore fully consistent with the data’s discrete nature.

Later, we normalize these integers, and those are vectors in the simplex (non-negative reals summing to 1), not counts. The reviewer’s remark likely stem from seeing these normalized quantities and mistaking them fro the random objects modeled in the manuscript. We will add the explicit sentence to avoid ambiguity.

For completeness, if one wishes to work directly with the normalized vectors, two rigorously equivalent formulations are available:

• Dirichlet-multinomial posterior: conditioning a Dirichlet prior on the multinomial counts yields a Dirichlet posterior for the probability vector.
• Gaussian approximation (large $n$ delta method): $\sqrt{n}(\hat{r} - r) \to \mathcal{N}(0, \text{diag}(r) - rr^\top)$.

Our Laplacian‑Gaussian hierarchy can be re‑expressed in either language.

Missing notation

Throughout the manuscript, we use

i.e., the centered log-odds of a prior vector $q$.

Therefore, $\theta_0 = \theta(q_0)$ is simply the centered log-odds of the true target prior $q_0$.

Edge weights don’t seem to matter; can we optimize $L$?

Why the weights do matter

The upper bound we proved is

Here,

• Algebraic connectivity $\lambda_2(L)$ (the smallest strictly-positive eigenvalue) appears in the denominator because it is the worst-case curvature of the quadratic penalty along any tangent direction.
• The numerator contains the full matrix-difference $(L - L_0)\theta_0$. This encodes the signed, edge-wise mismatch between our chosen weights and the unknown ground-truth graph $L_0$. If we perturb a single edge weight, it changes in direct proportion to that entry. Hence the pattern of weights does affect the bias term, and $\lambda_2(L)$ merely the concise way of writing the spectral part of the denominator.

Can we optimize $L$?

Yes, nothing in the hierarchy forbids putting a prior on the weights.

Two concrete extensions are feasible:

• Empirical Bayes (plug-in refinement): treat $L$ as hyper-parameter and maximize the marginal likelihood $\mathcal{L}(L) = \log p(\text{data} \mid L)$. Because the posterior over $(q, C)$ is log-concave given $L$, the Laplace approximation $mathcal{L}(L) \approx \log p(\text{data} \mid \hat{q}(L), \hat{C}(L), L) - \frac{1}{2}\log \det H^{-1}$ is cheap, and the Hessian $H$ is block-diagonal with Laplacian plus Fisher terms.
• Fully Bayesian weight optimization: place independent Gamma or log-normal priors on each edge weight $w_{ij}$, and the quadratic form becomes $\frac{\tau}{2}\sum_{(i,j)\in E}w_{ij}(\theta_i - \theta_j)^2$. The conditional for $w_{ij}$ is conditionally log-Gaussian given $\theta$, and a single Gibbs slice or HMC dimension is enough (dimensionality equals number of edges $\ll$ number of parameters in $C$). The resulting chain mixes well because $w_{ij}$ appears only linearly in the quadratic exponent.

With random-weight priors identifiability still holds because the confusion-matrix block stays full-rank as before, $q$ remains on the simplex, and the new edge-weight parameters enter only through the positive-definite quadratic term, so the joint Fisher information expands to

which is full-rank provided at least one $(\theta_i - \theta_j) \neq 0$.

When does optimizing $L$ help?

Using our upper bound, we can give a crisp sufficient condition:

Hence fine-tuning is valuable whenever a small decrease in mismatch can be achieved without collapsing algebraic connectivity.

We would like to express our utmost appreciation for your very detailed comments on our manuscript.

Graph quality & sensitivity

Recall Prop. 2, the following table summarizes the bias and variance of vanilla BBSE versus Bayesian shrinkage.

Here, $\kappa_i$ is the $i$-th sensitivity of the matrix inverse, which can be tiny for rare classes, hence the variance blows up. Thus

• Vanilla BBSE is unbiased but pays no attention to sampling noise, and its variance grows like the squared condition number of $\tilde{C}$.
• Bayesian shrinkage version trades a bit of bias $b_i$ for a potentially huge reduction in variance. Specifically, amplitude factor is zero when the graph is correct, and otherwise it measures how much the misspecified graph distorts $\theta_0$ along non-null Laplacian directions. In addition, damping $N^{-1}$ and $\lambda_2(L)$ ensure the bias decays with i) more data and ii) greater algebraic connectivity. That is, even for a badly misspecified graph the leading term cannot exceed $\tau_1 / (N \lambda_{\min}(F_0))||\theta_0||_2$, i.e., the same order as the frequentist standard error.

Below, we report the additional experimental results on the edge-corruption study.

Protocol. We start from the CLIP-derived $k$-NN graph $L_0$ (k = 4 for K = 10, k = 8 for K = 100). A corruption level $\rho$ means we randomly re‑wire $\rho\cdot |E|$ undirected edges (preserving degrees); the resulting Laplacian is L. All other hyper-parameters are kept fixed.

What does a “completely random” graph do?

In the above table, $\rho = 1.00$ means the completely random graph, but the resulting performance is still competitive against MLLS or other methods. To understand this observation, we can give the following discussion.

Take the Laplacian $L$ obtained by connecting each label to $k$ random neighbors (uniform among the other $K - 1$ labels). Then,

1. For an Erdős–Rényi‑like graph the second eigenvalue satisfies $\lambda_2(L) \approx k$ with high probability, so the variance bound $\text{Var}(\bar{q}_i) \leq C / \lambda_2(L)N$ is smaller than for the identity-Laplacian ($\lambda_2 = 0$). In other words, random edges provide extra shrinkage.
2. Because the edge directions are symmetric, each row of $L - L_0$ has positive and negative entries that tend to cancel when dotted with the centered log-odds $\theta_0$: $\mathbb{E}[(L - L_0)\theta_0] = 0$, and the random fluction has magnitude $O_P(1)$. Therefore, $||b_i|| = O_P(N^{-1})$, i.e., the bias is the same order as the stochastic error we cannot avoid anyway.
3. Putting the pieces together, we have the following which is precisely the rare-class / ill-conditioned regime where BBSE collapses.

Therefore, even though the random graph carries no semantic signal, its connectivity slashes variance enough to compensate for the negligible $N^{-1}$ bias, so the total error remains well below BBSE.

In addition, we provide the additional ablations on graph construction.

• CLIP-text (already in the current manuscript): cosine $k$-NN on text prompts.
• Penultimate-image: Euclidean $k$‑NN on frozen ResNet‑18 penultimate features (identical backbone as the classifier).
• Random-embeddings: same sparsity pattern as CLIP‑text but edge weights i.i.d. $U(0, 1)$ (destroys semantics, preserves Laplacian spectrum scale).
• Identity graph: $L = 0$, switches the hierarchy off (reduces to independent logistic‑Normal prior and hence Bayesian BBSE with no smoothing).

Remarks

• Any connected graph ($\lambda_2(L) > 0$) already helps over the frequentist point estimate; cf. the “random” line.
• The more task-aligned the similarity (CLIP > penult-img > random), the lower the bias term.
• The identity graph reproduces Bayesian BBSE with no smoothing and is almost identical to MLLS. Its gap to CLIP shows the specific benefit of our graph-coupled hierarchy.

Why do the graph ablations behave as the above?

Recall the posterior variance bound we already proved:

which holds for any connected graph.

If we write the mean-squared error as

then, it shows the variance term shrinks by a factor $1 / \lambda_2(L)$.

Even a $k$-NN graph whose edge weights are i.i.d. $\mathcal{U}(0, 1)$ has, with high probability,

for $k \geq c’ \ln K$. Hence a purely random graph already cuts the class-wise standard error by $\sqrt{K / (ckN)}$ compared with the identity graph $\lambda_2 = 0$. This explain why the “random” row in the ablation table still beats the deterministic frequentist BBSE (it keeps all the variance relief while paying only a bias penalty).

Summarizing the above discussion, we obtain the following remark.

Shrinkage beats instability. BBSE pays no price for bias but suffers large sampling variance, especially when classes are imbalanced and $\tilde{C}$ is noisy. Graph-based Bayesian pooling, even with arbitrary edges, acts like a data-dependent ridge penalty that stabilises all estimates.

Computational cost

To address this point, we first summarize the per-iteration algebraic cost as follows.

Therefore the dominant term grows only quadratically in the class count $K$ (the sparse Laplacian never densifies), whereas BBSE’s one‑off inverse is cubic.

Next, we report the wall-clock runtimes (single NVIDIA T4, PyMC 5.10, float32).

As you suggested, we believe these results help users to consider the accuracy-complexity trade-off.

Design choice

The identifiability relation used by all BBSE‑type estimators is $\tilde{q} = Cq$, where $\tilde{q} \in \Delta^{K-1}$ is the prediction histogram on the target domain, the unknown parameter $q \in \Delta^{K-1}$ is the target prior, and the $i$-th column $c_i = C_{\cdot,i}$ is $c_i = [p(\hat{h}(X) = j \mid Y = i)]^K_{j=1}$. Here, only the columns $c_i$ multiply the unknown vector $q$, and the rows $r_j = C_{j:}^\top$ never appear in the above, so they matter only if one later inverts $C$. Hence, any prior that tries to stabilize the estimation of $q$ should regularize the $c_i$’s, not the $r_j$’s.

Next, write a first-order delta expansion of the posterior mean

where $F_0 = \text{diag}(r_0) - r_0 r_0^\top$ is the Fisher matrix of the multinomial likelihood for $\tilde{q}$, $\delta c_i = c_i = c_{0,i}$ denotes the estimation noise on the $i$-th column, and $e_i$ is the $i$-th standard basis vector. Taking variances and using independence of source and target samples,

Thus,

• if we Laplacian-shrink the columns $c_i$, it shows a direct variance reduction by a factor $(\tau_C\lambda_2(L))^{-1}$, exactly the bound proved in Cor. 1 of the manuscript, and
• if we instead Laplacian-shrink the rows $r_j$, the random variable $\langle \delta c_i, e_i \rangle = \delta C_{i,i}$ does not get penalized, so the leading variance term is unaffected. In other words, row smoothing hardly helps for the quantity we ultimately care about, $q$.

Semantic interpretation

• Predictive columns $c_i$ capture how much class $i$ is confused with every other class in the frozen classifier. Two semantically close source classes (e.g., truck and bus) tend to produce similar confusion fingerprints, so a Laplacian on $i$-index is natural.
• Rows $r_j$ describe where each predicted label actually comes from in the source distribution, a statistic that never appears in the above identifiability and is only weakly correlated with the target-prior calculation.

Overall, the rebuttal has strengthened the paper by clarifying theoretical underpinnings and providin new data. While the practical trade-offs regarding the dependency on high-quality graphs for peak performance and the increased computational cost remain relevant points for discussion, the core contributions of the paper are sound and well-defended. The rebuttal has successfully addressed the most critical theoretical questions, even if some practical limitations persist.

First of all, we would like to thank you for your very constructive comments. The reviewer listed three questions with numbers, and we will respond to each of them.

1. “How is the similarity graph built, and what if it is noisy?”

We build a k‑NN graph once from off-the-shelf embeddings (CLIP for CIFAR, averaged penultimate‑layer features for MNIST). Noisy edges are inevitable, and that is exactly why the prior places a continuous Laplacian-precision $\tau_q L$ instead of a hard constraint.

Specifically, Prop 2 (Robustness to Laplacian misspecification) quantifies the bias when $L \neq L_0$:

so the bias shrinks as $N^{-1}$ and is damped when the graph is well‑connected ($\lambda_2(L)\uparrow$). Thus moderate graph noise merely slows convergence, and it does not break consistency.

Based on your feedback, we realized that we could better clarify the motivation for our formulation by adding the following additional direct corollary.

Corollary. From Prop. 2 set $N\to \infty$ to bound the asymptotic bias by $\tau_q\lambda_2(L)^{-1}\|(L-L_0)\theta_0\|$.

In addition, we provide the additional ablations on graph construction. In this experiment, we consider the following strategies.

• CLIP-text (already in the current manuscript): cosine $k$-NN on text prompts.
• Penultimate-image: Euclidean $k$‑NN on frozen ResNet‑18 penultimate features (identical backbone as the classifier).
• Random-embeddings: same sparsity pattern as CLIP‑text but edge weights i.i.d. $U(0, 1)$ (destroys semantics, preserves Laplacian spectrum scale).
• Identity graph: $L = 0$, switches the hierarchy off (reduces to independent logistic‑Normal prior and hence Bayesian BBSE with no smoothing).

The results are as follows.

Remarks

• Any connected graph ($\lambda_2(L) > 0$) already helps over the frequentist point estimate; cf. the “random” line.
• The more task-aligned the similarity (CLIP > penult-img > random), the lower the bias term.
• The identity graph reproduces Bayesian BBSE with no smoothing and is almost identical to MLLS. Its gap to CLIP shows the specific benefit of our graph-coupled hierarchy.

Why do the graph ablations behave as the above?

Recall the posterior variance bound we already proved:

which holds for any connected graph.

If we write the mean-squared error as

then, it shows the variance term shrinks by a factor $1 / \lambda_2(L)$.

Even a $k$-NN graph whose edge weights are i.i.d. $\mathcal{U}(0, 1)$ has, with high probability,

for $k \geq c’ \ln K$. Hence a purely random graph already cuts the class-wise standard error by $\sqrt{K / (ckN)}$ compared with the identity graph $\lambda_2 = 0$. This explain why the “random” row in the ablation table still beats the deterministic frequentist BBSE (it keeps all the variance relief while paying only a bias penalty).

2. “Computational complexity vs. vanilla BBSE?”

We measure work in floating-point operations (flops) and write $K$ = number of classes and $|E|$ = number of graph edges.

For the $k$-nearest neighbor graphs we use, $|E| = kK$ with a small, $K$-independent constant $k$ (4 for CIFAR‑10, 8 for CIFAR‑100).

• Vanilla BBSE: The point estimator solves a single linear system $\hat{q} = \tilde{C}^{-1}\tilde{r}$, where $\tilde{C}\in\mathbb R^{K\times K}$ is dense once any amount of Laplace or Tikhonov regularisation is applied. Gaussian elimination therefore costs $\mathcal{O}(K^3)$ flops and $\mathcal O(K^{2})$ memory.
• GS‑B$^3$SE (one Newton–CG sweep): Write $\Phi = (\phi_1,\dots,\phi_K)^\top \in \mathbb{R}^{K\times (K-1)}$ and $\theta \in \mathbb{R}^{K-1}$, both constrained to the centered subspace $\\{v^\top 1=0 \\}$. At each sweep we perform three algebraic operations:
  • update each column $\Phi \leftarrow \Phi - \mathcal{H}_{\Phi}^{-1}\nabla\mathcal{L}$,
  • update $\theta$ analogously, and
  • Gamma shape/scale updates for $\tau_q$, $\tau_C$.

Adding them and using $|E|=kK$, one sweep = $\mathcal{O}(|E|K + K^2) - \mathcal{O}(kK^2)$, and memory usage is $\mathcal{O}(|E| + K^2)$.

Overall complexity.

Thm. 2 establishes that $F(q)$ is $\alpha$-strongly geodesically convex with $\alpha = n’\lambda_{\min}(F_0) + \tau_q \lambda_2(L) > 0$. Consequently a back-tracking Newton method requires $T = \mathcal{O}(\ln 1/\epsilon)$ sweeps to reach tolerance $\epsilon$.

Hence the total complexity is

Because $k$ is at most 8 and $\ln 1/\epsilon \le 10$ in all our runs, the crossover happens already at $K\ge 25$.

Here, we report the wall-clock runtimes (single NVIDIA T4, PyMC 5.10, float32).

3. “Behavior with very rare classes; isn’t everything prior-driven?”

For BBSE every class $i$ is treated in isolation: the plug-in variance satisfies

so as soon as the validation set contains $n_i^{\mathrm S}=O(1)$ points the error explodes.

In our hierarchy the Gaussian-Markov random field ties all columns of the confusion matrix and the prior vector together through the graph Laplacian; information can therefore flow from frequent to infrequent neighbors. Algebraically this shows up in the curvature of the log-posterior:

with the data Fisher block $N F_0$ pooling global target counts, and the prior blocks injecting $\lambda_2(L)$-weighted curvature even when $n_i^{\mathrm S}=0$.

Corollary 1 states, for any class $i$,

Thus the worst-case posterior variance still decays like $N^{-1}$, and the only penalty for a vanishing $n_i^{\mathrm S}$ is the $1/\lambda_2(L)$ factor, i.e. how well the rare label is connected to the rest of the ontology.

Specifically, we can analyze the algorithms under a Zipf-like class-frequency law.

Suppose that source class priors follow a Zipf law

The labelled-source sample sizes are therefore

Write $N = N_S+n’$ with $N_S/N\to\rho\in(0,1)$. Target priors $q$ are arbitrary but $\min_i q_i>0$, and the frozen classifier's confusion matrix $C$ is assumed invertible with condition number $\kappa:=||C^{-1}||_2 || C ||_2<\infty$.

Denote $\hat{C}$ and $\hat{r}$ the empirical confusion matrix and target prediction histogram. Linearizing the BBSE estimator around the truth gives the usual delta-method expansion:

with $||R_N||_2 = o(N^{-1/2})$.

Here, $\hat{r} \sim \text{Mult}(n' r)$ with $r= C q$, and hence

For label $i$ the column $\hat{C_{\cdot, i}}$ is a multinomial vector with size $n^S_i$ and mean $C_{\cdot,i}$. Let

Because only column $i$ depends on $n_i^{\mathrm S}$, its contribution to the target‑prior error is

For a rare label $i$ we typically have $n_i^S \ll n’$, and we obtain

Thus the MSE deteriorates polynomially in the rank $i$.

For our graph-smoothed Bayesian estimator, on the other hand, from Cor.1 and Prop.2 we have

Thus, the risk no longer depends on the tail exponent $b$ or the rank $i$.

Hello again,

While I understand that you are not an expert in this topic, please read the Authors' feedback and prepare a suitable short reply to them. It will be really helpful.

Regards,

AC