Anchoring Priors¶

The variable capture probability (VCP) model introduces two layers of practical non-identifiability that can degrade inference. This page describes two complementary anchoring priors that resolve them, each justified by a law-of-large-numbers concentration argument. For complete derivations, see the accompanying paper.

Layer 1: The Capture Prior¶

The degeneracy¶

The expected observed count for gene \(g\) in cell \(c\) is:

\[ \langle u_{gc} \rangle = \mu_g \cdot \nu_c, \]

where \(\mu_g\) is the biological mean expression and \(\nu_c\) is the cell-specific capture probability. The observed library size \(L_c = \sum_g u_{gc}\) constrains only the product \(\nu_c \cdot \mu_T\) (where \(\mu_T = \sum_g \mu_g\)), not each factor individually. With a flat prior on \(\nu_c\), the model is free to trade off between "high capture, low total expression" and "low capture, high total expression."

This degeneracy manifests as systematic biases when fitting multiple datasets: even when the underlying biology is identical, the optimizer can assign different capture efficiencies to each dataset, absorbing expression differences into \(\nu_c\) rather than into the biological parameters.

The fundamental relationship¶

The key insight is that the capture probability is approximately the ratio of the observed library size to the total mRNA content:

\[ \nu_c \approx \frac{L_c}{M_c}. \]

The observed library size \(L_c\) is known from the data. If we can constrain \(M_c\) (the total mRNA content), we can constrain \(\nu_c\).

Why total mRNA is nearly deterministic¶

Under the shared-\(p\) model, the total mRNA \(M_c = \sum_{g=1}^G m_g\) is itself a negative binomial with aggregate dispersion \(r_T = \sum_g r_g\). Its coefficient of variation is:

\[ \text{CV}(M_c) = \frac{1}{\sqrt{r_T \cdot p}}. \]

For a mammalian cell with \(G \approx 20{,}000\) genes and typical per-gene dispersions of order 1--10, the aggregate dispersion \(r_T\) exceeds 50,000, giving \(\text{CV}(M_c) \approx 0.5\%\). This is simply the law of large numbers over genes: summing tens of thousands of independent random variables produces a total that concentrates tightly around its expectation. Within the model, total mRNA is effectively deterministic.

The biology-informed prior¶

Given an organism-specific expected total mRNA \(M_0\) (e.g., \(\approx 200{,}000\) for mammalian cells), the prior on the working variable \(\eta_c = \log(M_c / L_c)\) is:

\[ \eta_c \sim \mathcal{N}^{+}(\log M_0 - \log L_c, \; \sigma_M^2), \]

where \(\mathcal{N}^{+}\) denotes a normal distribution truncated below at zero (enforcing the physical constraint \(\nu_c \leq 1\)). The capture probability and odds are recovered exactly:

\[ \nu_c = \exp(-\eta_c), \qquad \phi_c = \exp(\eta_c) - 1. \]

Key properties:

Cell-specific centering: cells with larger library sizes get a prior centered at higher capture efficiency, and vice versa
Biologically grounded scale: \(\sigma_M \approx 0.5\) corresponds to roughly 1.5--2x fold variation in total mRNA, consistent with known biological variation
No approximation: the transformations are exact for all capture regimes, including high-capture experiments where \(\nu_c\) approaches 0.5 or beyond

Interpretation of \(\sigma_M\)¶

\(\sigma_M\)	Behavior
\(\approx 0\)	Deterministic anchoring -- \(\nu_c = L_c / M_0\), no degrees of freedom
\(\approx 0.5\) (default)	Moderate anchoring, accommodates cell-size heterogeneity
\(> 1\)	Weak anchoring, approaching the flat prior

Organism-specific values¶

Organism	\(M_0\) (mRNA/cell)
Human	200,000
Mouse	200,000
Yeast (S. cerevisiae)	60,000
E. coli	3,000

Relationship to standard normalization¶

Standard normalization (CPM, scran, etc.) divides by library size and multiplies by a constant: \(\tilde{u}_{gc} = u_{gc} \cdot C / L_c\). This is structurally identical to \(\nu_c = L_c / C\), but with an arbitrary reference constant. The biology-informed prior improves on this by:

Anchoring to a biologically meaningful \(M_0\) instead of an arbitrary constant
Operating within the full probabilistic model, preserving uncertainty quantification
Accounting for the nonlinear coupling between capture and expression through the likelihood (the effective \(\hat{p}\) depends nonlinearly on both \(p\) and \(\nu_c\))

Layer 2: The Mean Anchoring Prior¶

The residual degeneracy¶

Even after the capture prior pins \(\nu_c\), a second non-identifiability remains. The expected count \(\langle u_{gc} \rangle \approx \mu_g \cdot L_c / M_0\) constrains \(\mu_g\) through the first moment (the mean), while the overdispersion \(\phi_g\) is constrained only through the second moment (the variance-to-mean ratio). Because the variance is inherently noisier to estimate than the mean, the likelihood surface exhibits a characteristic ridge in \((\mu_g, \phi_g)\) space: the mean is tightly constrained, but the overdispersion direction is broad.

This manifests as:

Different datasets settling on different \(\phi\) values despite identical biology
The hierarchical gene-specific \(\phi_g\) model failing in the mean-odds parameterization (where \(\phi_g\) only affects the variance, not the mean)
Poor cross-dataset parameter correspondence despite good within-model posterior predictive checks

The concentration argument¶

The sample mean of gene \(g\) across \(C\) cells,

\[ \bar{u}_g = \frac{1}{C} \sum_{c=1}^C u_{gc}, \]

concentrates around \(\mu_g \cdot \bar{\nu}\) (where \(\bar{\nu} = \frac{1}{C} \sum_c \nu_c\)) with coefficient of variation:

\[ \text{CV}(\bar{u}_g) \approx \frac{1}{\sqrt{C}} \sqrt{\frac{1}{\mu_g \bar{\nu}} + \frac{1}{r_g}}. \]

This is the law of large numbers over cells, complementing the capture prior's concentration over genes. Typical values for \(C = 1{,}000\) cells:

Gene type	\(\mu_g\)	\(r_g\)	CV
Low expression, high overdispersion	10	1	≈4.5%
Moderate expression	100	5	≈1.7%
High expression	1,000	10	≈1.0%

With \(C = 5{,}000\) cells, these improve by \(\sqrt{5}\) to sub-percent precision for all but the most extreme genes.

The data-informed prior¶

The empirical anchor for each gene is:

\[ \hat{\mu}_g = \frac{\bar{u}_g + \epsilon}{\bar{\nu}}, \]

where \(\epsilon > 0\) is a small pseudocount. The prior on \(\mu_g\) is log-normal, centered on this anchor:

\[ \log(\mu_g) \sim \mathcal{N}\!\left(\log(\hat{\mu}_g), \; \sigma_\mu^2\right). \]

Interpretation of \(\sigma_\mu\)¶

\(\sigma_\mu\)	Behavior
0.1--0.2	Tight anchoring -- constrains \(\mu_g\) within ≈10--20% of the data-implied value
0.3 (recommended default)	Moderate anchoring, accommodates finite-sample noise
\(> 1\)	Weak anchoring, approaching the uninformative prior

Why this fixes the hierarchical \(\phi_g\) failure¶

Without the anchor, the optimizer faces \(2G\) free parameters (\(\mu_g\) and \(\phi_g\) per gene) with \(G\) tight constraints (means) and \(G\) loose constraints (variances), creating \(G\) independent degeneracy manifolds.

With the anchor, \(\mu_g\) is effectively fixed (up to \(\sigma_\mu\) slack), leaving only \(\phi_g\) per gene. Any change in \(\phi_g\) now unambiguously changes the overdispersion without being compensated by a shift in \(\mu_g\). The banana-shaped ridges in \((\mu_g, \phi_g)\) space are replaced by well-defined optima along the \(\phi_g\) axis.

The Three-Layer Regularization Stack¶

The capture prior and mean anchor, combined with the hierarchical prior on overdispersion, form a complete regularization stack:

\[ \boxed{ \begin{aligned} &\text{Layer 1 (capture):} && \nu_c \approx L_c / M_0 && \text{(LLN over genes)} \\ &\text{Layer 2 (mean):} && \mu_g \approx \bar{u}_g / \bar{\nu} && \text{(LLN over cells)} \\ &\text{Layer 3 (overdispersion):} && \phi_g \sim \text{Hierarchical prior} && \text{(shrinkage across genes)} \end{aligned} } \]

Each layer constrains one factor in the expected count \(\langle u_{gc} \rangle = \mu_g \cdot \nu_c\), with the overdispersion regularized by a hierarchical prior across genes.

Using Anchoring Priors in SCRIBE¶

Capture prior (Layer 1)¶

Enable the biology-informed capture prior by passing organism-specific total mRNA information through the priors dictionary:

import scribe

# Using organism shortcut (resolves M_0 and sigma_M automatically)
results = scribe.fit(
    adata,
    variable_capture=True,
    priors={"organism": "human"},
)

# Using explicit values (for other organisms or custom M_0)
import math
results = scribe.fit(
    adata,
    variable_capture=True,
    priors={"eta_capture": (math.log(200_000), 0.5)},
)

The priors["organism"] key accepts "human", "mouse", "yeast", or "ecoli" and resolves to the appropriate \(M_0\) and \(\sigma_M\). For other organisms, pass priors["eta_capture"] directly as a tuple of (log_M_0, sigma_M).

Parameter	Description
`priors["organism"]`	Shortcut: `"human"` (\(M_0=200{,}000\)), `"mouse"`, `"yeast"` (\(60{,}000\)), `"ecoli"` (\(3{,}000\))
`priors["eta_capture"]`	Explicit `(log_M_0, sigma_M)` tuple
`capture_scaling_prior`	Data-driven shared scaling across datasets: `"none"`, `"gaussian"`, `"horseshoe"`, `"neg"`

Mean anchoring prior (Layer 2)¶

Enable the data-informed mean anchor with the expression_anchor flag:

results = scribe.fit(
    adata,
    variable_capture=True,
    priors={"organism": "human"},
    expression_anchor=True,
    expression_anchor_sigma=0.3,
)

Parameter	Default	Description
`expression_anchor`	`False`	Enable data-informed anchoring prior on \(\mu_g\)
`expression_anchor_sigma`	0.3	Log-scale standard deviation (smaller = tighter anchor)

Note

Enabling expression_anchor automatically sets unconstrained=True. For VCP models, priors["organism"] or priors["eta_capture"] must also be set so that \(\bar{\nu}\) can be estimated from library sizes and \(M_0\). For non-VCP models, \(\bar{\nu} = 1\) is used by default.

Combined usage (recommended for VCP models)¶

For the best results with variable capture models, enable both layers:

results = scribe.fit(
    adata,
    variable_capture=True,
    parameterization="mean_odds",
    priors={"organism": "human"},
    expression_anchor=True,
    expression_anchor_sigma=0.3,
)

This resolves both the \(\nu_c \cdot \mu_T\) degeneracy (Layer 1) and the \(\mu_g\text{--}\phi_g\) degeneracy (Layer 2), producing well-identified parameters that are comparable across datasets and compatible with hierarchical gene-specific \(\phi_g\) models.

For the full API details, see the API Reference.

Next steps

See the Dirichlet-Multinomial Model for the NB generative model and capture-probability derivation that these priors regularize.
See Hierarchical Priors for Layer 3 of the regularization stack — adaptive shrinkage on overdispersion using Gaussian, Horseshoe, and NEG prior families.
See Beta Negative Binomial for an orthogonal overdispersion extension whose mean structure critically depends on the mean anchoring prior.
See the Model Selection guide for NBVCP and ZINBVCP models where anchoring priors are most important.