A Worked Interaction-Prior Analysis Across Molecular Layers

From Curated Interactions to Cross-Layer Estimates

A curated molecular interaction is useful to an integration model only after its role is made explicit. A directed transcriptional regulatory record can inform the prior plausibility of a regulator-to-gene link; a protein-protein interaction record can inform the plausibility of a protein-level association. Neither record, by itself, establishes that a link operates in the tissue being studied or that the linked molecules are associated in the current samples.

The analysis below uses matched samples, so a molecular measurement on one layer can be paired with a measurement on another layer for the same sample. It separates two uncertainties: whether a candidate edge is included in the model, and, conditional on its inclusion, the size and direction of its coefficient. The numerical examples are hypothetical.

Definition: For a candidate molecular link $e$, let $Z_e=1$ mean that the link is included in the statistical model and $Z_e=0$ mean that it is absent. The prior inclusion probability $P(Z_e=1)$ represents uncertainty before using the current molecular measurements. A Bayes factor for the link, $BF_{10}$, is the ratio of the marginal likelihoods under the included-link and absent-link models (data densities for continuous measurements). The posterior inclusion probability is $$ P(Z_e=1\mid D)=\frac{P(Z_e=1)BF_{10}}{P(Z_e=0)+P(Z_e=1)BF_{10}}. $$

A curated evidence score is not automatically a prior probability or a Bayes factor. To use it quantitatively, one needs a stated calibration or a transparent modeling choice. For example, a likelihood ratio for curated evidence can update baseline prior odds, while a confidence score may instead be mapped to a prior probability using an externally justified calibration. The examples state their calibration assumptions rather than treating database scores as probabilities.

Calibrating a Regulatory-Edge Prior

Suppose the candidate edge is from the protein encoded by a transcription-factor gene to the RNA expression of a target gene. The direction of the proposed edge comes from the regulatory evidence; the regression coefficient estimated from observational data remains an association and should not be described as a causal effect. A regulatory record can support a nonzero prior probability for the edge, while uncertainty about whether the evidence applies in the measured context remains.

Worked Example: Turning Regulatory Evidence Into Prior Odds

Assume a hypothetical analysis begins with prior inclusion probability $0.10$ for a particular regulator-to-target edge. A separately calibrated evidence source assigns the curated regulatory record a likelihood ratio of $4$ for a true edge versus no edge. These values are assumptions for this example, not universal properties of regulatory databases.

The baseline prior odds are $$ \frac{0.10}{1-0.10}=\frac{0.10}{0.90}=0.1111. $$

Multiplying these odds by the evidence likelihood ratio gives posterior odds after considering the curated record: $$ 0.1111(4)=0.4444. $$

Converting those odds back to a probability gives $$ P(Z_e=1\mid \text{curated evidence})= \frac{0.4444}{1+0.4444}=0.3077. $$ Equivalently, the odds are $4/9$, so the probability is $4/(4+9)=4/13=0.3077$. The curated evidence raises the prior inclusion probability from $0.10$ to about $0.308$, but does not make the edge certain.

This calculation is justified only if the likelihood ratio has a meaningful calibration for the evidence and the target setting. If only an uncalibrated confidence score is available, reporting a probability such as $0.308$ would imply more precision than the evidence supports.

Estimating a Cross-Layer Coefficient

For an included candidate edge, consider a simple paired-sample model in which $x_i$ is the standardized abundance of a regulator protein in sample $i$, and $y_i$ is the standardized abundance of a target protein. A curated regulatory record for the corresponding regulator and target can motivate this cross-layer candidate, but it does not guarantee that the protein measurements will covary. To keep the worked calculation transparent, suppose the data are centered, the residual variance is known, and there are no additional covariates.

Model and assumptions: For $i=1,\ldots,n$, assume $y_i=\beta x_i+\epsilon_i$, where the errors are independent and $\epsilon_i\sim N(0,\sigma^2)$. Under the included-edge model, assign $\beta\sim N(0,\tau^2)$; under the absent-edge model, set $\beta=0$. The predictor and response are centered and standardized on a common scale, the samples are correctly paired, and $\sigma^2$ is treated as known for the calculation. The zero-centered prior permits either positive or negative associations; it shrinks estimates toward zero rather than fixing a sign.

With $S_{xx}=\sum_i x_i^2$ and $S_{xy}=\sum_i x_i y_i$, the likelihood contributes precision $S_{xx}/\sigma^2$ for $\beta$, and the normal prior contributes precision $1/\tau^2$. Thus, conditional on inclusion, the posterior is normal:

\beta\mid D,Z_e=1\sim N\left( \frac{S_{xy}/\sigma^2}{S_{xx}/\sigma^2+1/\tau^2}, \frac{1}{S_{xx}/\sigma^2+1/\tau^2} \right).

Worked Example: Updating a Cross-Layer Relationship

Consider eight hypothetical matched samples. The table gives regulator-protein measurements $x_i$ and target-protein measurements $y_i$, standardized using an external reference scale; both are centered within these samples.

Sample	$x_i$	$y_i$
1	1	1.5
2	1	-0.5
3	1	1.5
4	1	-0.5
5	-1	-0.5
6	-1	-0.5
7	-1	-0.5
8	-1	-0.5

Both variables are centered. The assumed residual variance is $\sigma^2=1$, and the prior variance is $\tau^2=0.25$. The data summaries are $$ S_{xx}=8,\qquad S_{xy}=1(1.5)+1(-0.5)+1(1.5)+1(-0.5) +(-1)(-0.5)+(-1)(-0.5)+(-1)(-0.5)+(-1)(-0.5)=4. $$

The posterior variance conditional on inclusion is $$ \frac{1}{S_{xx}/\sigma^2+1/\tau^2} =\frac{1}{8/1+1/0.25} =\frac{1}{12}=0.08333, $$ so the posterior standard deviation is $\sqrt{0.08333}=0.2887$. The posterior mean is $$ \frac{S_{xy}/\sigma^2}{S_{xx}/\sigma^2+1/\tau^2} =\frac{4/1}{8/1+4} =\frac{4}{12}=0.3333. $$

Conditional on the edge being included, the estimated association is therefore $0.3333$ standardized target-protein units per one standardized unit of regulator-protein abundance, with posterior standard deviation $0.2887$. Under the normal posterior, the probability that this coefficient is positive is $\Phi(0.3333/0.2887)=\Phi(1.155)\approx0.8759$. This is a probability about the coefficient under the stated model and conditional on inclusion, not a probability that the regulatory mechanism is causal.

Updating the Probability That a Link Is Present

The coefficient calculation alone does not tell us how plausible it is that the edge belongs in the model. To update that question, compare the marginal data probability under the included-edge model, which averages over the prior distribution of $\beta$, with the data probability under $\beta=0$. For the normal model above, this ratio has a closed form. Integrating the normal likelihood against the normal prior, or equivalently completing the square in $\beta$, gives

BF_{10} =\sqrt{\frac{\sigma^2}{\sigma^2+\tau^2 S_{xx}}} \exp\left\{ \frac{\tau^2 S_{xy}^2} {2\sigma^2(\sigma^2+\tau^2 S_{xx})} \right\}.

This expression assumes the same centered, one-predictor model, known $\sigma^2$, and the specified normal prior under the included model. A Bayes factor above one favors the included model relative to the absent model; its magnitude depends on the prior variance as well as on the data.

Worked Example: Combining the Prior and the Cross-Layer Data

Use the prior probability $4/13=0.3077$ from the regulatory-evidence example, and the data summaries $S_{xx}=8$, $S_{xy}=4$, $\sigma^2=1$, and $\tau^2=0.25$ from the matched-sample example. The Bayes factor is $$ BF_{10} =\sqrt{\frac{1}{1+0.25(8)}}\, \exp\left\{\frac{0.25(4^2)}{2(1)(1+0.25(8))}\right\} =\sqrt{\frac{1}{3}}\exp\left(\frac{4}{6}\right) \approx 0.5774(1.9477)=1.125. $$

The prior odds are $(4/13)/(9/13)=4/9=0.4444$. The posterior odds are $$ \frac{4}{9}(1.1247)=0.4999, $$ and the posterior inclusion probability is $$ \frac{0.4999}{1+0.4999}=0.3333. $$

The data provide only modest evidence in favor of inclusion, since the Bayes factor is near one. The prior probability of about $0.308$ consequently changes only slightly, to a posterior probability of about $0.333$. Combining this with the conditional probability of a positive coefficient gives $P(Z_e=1,\beta>0\mid D)\approx0.3333(0.8759)=0.2919$. This last quantity concerns a positive, included edge under this model; it is not a causal probability.

Protein-Interaction Evidence and Uncertain Links

Protein-interaction evidence needs different interpretation from a directed regulatory record. A protein-protein interaction is generally undirected: it does not identify which protein regulates the other, and it does not imply a positive expression coefficient. It can inform a prior for a protein-level link or help prioritize a candidate cross-layer association, but the model must still specify which measurements and edge are being assessed.

Worked Example: A Protein-Interaction Link With Uncertain Support

Suppose a hypothetical protein-level candidate has prior inclusion probability $0.50$, based on the specified evidence assessment. A new, relevant molecular data summary has probability $0.08$ under the included-link model and probability $0.04$ under the absent-link model. These are model probabilities for the same observed summary, so $$ BF_{10}=\frac{0.08}{0.04}=2. $$

The prior odds are $0.50/0.50=1$; the posterior odds are therefore $1(2)=2$. Converting back to probability, $$ P(Z_e=1\mid D)=\frac{0.50(2)}{0.50+0.50(2)} =\frac{1}{1.5}=\frac{2}{3}\approx0.6667. $$

The posterior probability of inclusion is about $0.6667$, not certainty. The result does not imply a two-thirds probability that the null hypothesis is true or false; it is a model-based probability for link inclusion under the stated prior and likelihoods. Nor does it provide a direction for the interaction.

Diagnostics, Alternatives, and Limitations

A useful interaction-prior analysis checks whether the assumptions that make the calculations interpretable are plausible. In particular, verify that measurements are matched by sample, variables have the stated transformations and scale, the candidate direction matches the regulatory evidence, and the residual model is not contradicted by strong dependence or outliers. In the worked regression, $\sigma^2=1$ is assumed known; that simplification should not be silently carried into an application where the residual variance is uncertain.

Compare the prior-informed result with an analysis that uses a weak or neutral prior, and examine how posterior inclusion and coefficient estimates change across defensible prior strengths. If small changes to the prior lead to large changes in the conclusion, the link is prior-sensitive and should be reported as such. A Bayes factor near one, as in the main example, means that the observed data do little to distinguish the two specified models; it is not evidence that the prior is correct.

The main alternatives encode knowledge differently. A model without an interaction prior lets the current data determine the comparison without this source of prior information, though it may be less stable with limited samples. A hard constraint forces an edge to be present or absent; it does not represent uncertainty in the curated evidence. A soft prior retains uncertainty and allows the data to update it. These approaches answer different questions, so comparisons should use the same target link and measurement scale.

Common failure modes include treating a database confidence score as a calibrated probability, using an undirected protein interaction to claim a regulatory direction, and interpreting an included-model coefficient as causal. Another error is to report only a conditional coefficient estimate while omitting the probability that the edge is included. In a mixture model with a point mass at zero, an estimate conditional on inclusion is not the same as an unconditional summary. Report the inclusion probability and the conditional coefficient uncertainty separately.

Key takeaway: Curated regulatory and protein-interaction evidence can inform explicit, uncertain priors. The current matched measurements update those priors through a stated likelihood; neither the prior nor the posterior alone establishes a causal mechanism.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A candidate edge has baseline prior probability $0.20$, and calibrated curated evidence has likelihood ratio $3$. What is the updated probability after the curated evidence?
For the main regression example, what is the posterior variance of $\beta$ conditional on inclusion if $\tau^2=0.25$, $S_{xx}=8$, and $\sigma^2=1$?
A candidate edge has prior inclusion probability $0.25$ and Bayes factor $3$. Calculate its posterior inclusion probability.
Why does a protein-protein interaction record not, by itself, establish a positive directed effect of one protein on another?

Solutions. For question 1, the prior odds are $0.20/0.80=0.25$. Updating gives odds $0.25(3)=0.75$, hence probability $0.75/(1+0.75)=0.4286$. For question 2, the posterior variance is $1/(S_{xx}/\sigma^2+1/\tau^2)=1/(8+4)=1/12=0.08333$. For question 3, the prior odds are $0.25/0.75=1/3$, so posterior odds are $(1/3)(3)=1$, giving probability $1/(1+1)=0.50$. For question 4, a protein-protein interaction is generally undirected and does not establish a regulatory direction or coefficient sign; those claims require an appropriately specified model and supporting evidence.