Bayesian Hierarchical Modeling for Label-Free Proteomics.

Baldur

Baldur is a hierarchical Bayesian model for the analysis of proteomics data. By leveraging empirical Bayes methods, Baldur estimates hyperparameters for variance and measurement-specific uncertainty. It then computes the posterior difference in means between conditions for each peptide, protein, or PTM, and integrates the posterior to estimate error probabilities.

Features

• Hierarchical Bayesian modeling of proteomics data
• Empirical Bayes estimation of variance and uncertainty
• Posterior probability calculations for differential analysis
• Supports peptide, protein, and PTM data

Installation

Install the stable release from CRAN:

install.packages('baldur')

Or, install the development version from GitHub (after installing rstan):

Follow the instructions for installing rstan: https://github.com/stan-dev/rstan/wiki/RStan-Getting-Started

Then:

devtools::install_github('PhilipBerg/baldur', build_vignettes = TRUE)

Note:

• On Ubuntu, pandoc may be needed to build vignettes.
• On Windows, sometimes the development version of rstan is required.

Usage

For detailed examples, see the package vignettes:

vignette('baldur_yeast_tutorial')
vignette('baldur_ups_tutorial')

Main Modeling Work and Equations

Baldur implements a hierarchical Bayesian framework for label-free proteomics quantification, designed to robustly estimate differential abundance while accounting for the mean-variance relationship in mass spectrometry data. For exact details please see the original paper.

1. Observation Model

For each feature (peptide, protein, or PTM) $i$ in sample $j$, the observed intensity $y_{ij}$ is modeled as:

$$y_{ij}\sim\text{Normal}(\mu_{j},\sigma u_{ij})$$ $$\mu_{j}\sim\text{Normal}(\mu_{0j}+\eta_j\sigma,\sigma)$$

2. Mean-Variance Modeling

Gamma Regression

The measurement standard deviation $s_{j}$ is not constant, but depends on the mean intensity. This relationship is modeled with gamma regression: $$s_{j} \sim \Gamma(\alpha, \frac{\alpha}{\beta(\bar{y}_j)})$$ where:

• $\alpha$: shape parameter (estimated empirically)
• $\beta(\bar{y}_j)$: rate parameter as a function of peptide/protein mean intensity

Latent Gamma Mixture Regression

Regression Function

For each observation, the expected mean-variance relationship is modeled as:

$$\beta_i=\kappa\cdot\exp(\theta_i\cdot(I_L-S_Lx_i))+\exp(I-S\bar{y}_i)$$

where:

• $S, S_L$: slope parameters (common and latent)
• $I, I_L$: intercepts (common and latent)
• $\bar{y}_i$: mean
• $\theta_i$: feature-specific mixture parameter

Likelihood

Given the expected mean-variance, the observed standard deviation $\sigma_i$ is modeled as: $$\sigma_i\sim\Gamma(\alpha,\frac{\alpha}{\beta_i})$$ where:

• $\alpha$: gamma shape parameter
• $\beta_i$: expected mean-variance for observation $i$

Priors

• $\alpha\sim\text{Cauchy}(0,25)$
• $\eta\sim\text{Normal}(0,1)$
• $I_L\sim\text{SkewNormal}(2,15,35)$
• $\theta_i\sim\text{Uniform}(0,1)$

NRMSE (Model Fit Metric)

The normalized root-mean-square error (NRMSE) is calculated for model diagnostics.

3. Hierarchical Modeling of Condition Means

Empirical Bayes Prior

• What is it?
  A prior that is estimated from your actual data, rather than set by hand.
• How does it work?
  • Baldur looks at the spread and center of observed means across features.
  • It sets the mean hyper-prior for each group to match the average observed mean, and its uncertainty to match the variability in your data.
• Why use it?
  • Strength: Provides "shrinkage" toward realistic values, reducing noise and false positives, especially with small sample sizes.
• Mathematical form:
  $\mu_{0j}\sim\text{Normal}(\bar{y}_j,\sigma n_R)$
  • Here, $\bar{y}_j$ is the estimated mean for group $j$, and $\sigma n_R$ is the estimated standard deviation for the prior.

Weakly Informative Prior

• What is it?
  A broad, generic prior that doesn't make strong assumptions—it's like saying "I have no idea what the mean should be, but it's probably not infinite."
• How does it work?
  • The prior mean is set to zero for all groups.
  • The uncertainty (standard deviation) is set to a large value (e.g., $10$), meaning the model expects almost any value is possible.
• Why use it?
  • Strength: Maximizes flexibility and lets the data speak for itself, at the cost of potentially adding more noise or less stability if data is limited.
• Mathematical form:
  $$\mu_{0j}\sim\text{Normal}(0,10)$$
  • For group $j$, the mean is $0$ and the standard deviation is $10$.

4. Differential Abundance

For differential analysis, Baldur estimates the posterior distribution of the difference in means between conditions: $$\boldsymbol{D}\sim\mathcal{N}(\boldsymbol{\mu}^\text{T}\boldsymbol{K},\sigma\boldsymbol{\xi}),\quad \xi_{m}=\sqrt{\sum_{i=1}^{C}\frac{|k_{im}|}{n_i}}$$

where:

• $\boldsymbol{K}$: contrast matrix
• $k_{im}$: contrast coefficient for condition $i$ in contrast $m$
• $n_i$: number of samples in condition $i$
• $\boldsymbol{\xi}$: scaling factor for each contrast

The probability of error for contrast $c$ is then:

$$P(\mathrm{error}) = 2\Phi(-|\mu_{D_c} - \mu_{h_0}| \odot \tau_{D_c})$$

where:

• $\Phi$: cumulative distribution function (CDF) of the standard normal
• $\mu_{h_0}$: null hypothesis mean (often zero)
• $\boldsymbol{\tau}_{\boldsymbol{D}}$: precision (inverse standard deviation) for each contrast
• $\odot$: element-wise multiplication

Summary:
Baldur combines hierarchical modeling, mean-variance trend estimation via gamma regression, and empirical Bayes to robustly quantify differential abundance and propagate uncertainty from individual measurements to protein/PTM level, outputting interpretable error probabilities for each feature.

For full details, see the reference publication.

Reference

Berg, Philip, and George Popescu.
“Baldur: Bayesian Hierarchical Modeling for Label-Free Proteomics with Gamma Regressing Mean-Variance Trends.”
Molecular & Cellular Proteomics (2023): 2023-12.
https://doi.org/10.1016/j.mcpro.2023.100658