feat(analysis): Multinomial and Dirichlet-Multinomial models — Songbird-like multinomial regression, DM for overdispersed compositions  #17

Scientific Value

Current v1 has NB_GLM (counts per taxon independent, not compositional) and CLR/ILR+LM (compositional but Gaussian on log-ratios, not count-based). Multinomial models treat the vector of counts per sample as compositional directly:

• Multinomial (MN): Sample counts ~ Multinomial(total, p) where log(p_j / p_ref) = X beta. Ranks taxa by association with covariates (like Songbird). Value: interpretable as log-fold change in relative abundance, handles compositionality without pseudocount (zeros handled via count likelihood, not log(0)).
• Dirichlet-Multinomial (DM): Adds overdispersion to MN via Dirichlet prior on p, accounts for extra-multinomial variation (common in microbiome, technical + biological variance). Value: more accurate standard errors than MN, reduces false positives from overdispersion. Used in HMP, La Rosa et al. 2012.
• Use case: Diet intervention where total load unchanged but composition shifts — NB_GLM may call many taxa differential due to library size confounding, MN/DM correctly identifies compositional shift.

Impact: Bridges gap between count-based and compositional, provides effect sizes that are compositionally coherent (sum to zero in log-ratio space), avoids pseudocount tuning.

Scope (Deferred)

• New methods in AnalysisConfig v2: multinomial, dirichlet_multinomial, songbird (alias for multinomial with TensorFlow)
• Normalization: MN/DM use total as offset, not size_factors; normalization.method = none or multinomial (total as denominator)
• Zero handling: MN handles zeros naturally (likelihood includes zero count), but needs epsilon for log(p) when p=0 in optimization — use same epsilon as AdvancedConfig
• Implementation options:
  • Pure Julia: Turing.jl or Optim.jl for MN/DM MLE, DirichletMultinomial from DirichletMultinomial.jl or custom
  • R: MGLM::MGLMreg for MN/DM, HMP::DM.MoM for DM moments
  • Python: songbird via PythonCall.jl or subprocess (multinomial regression with TensorFlow)
• AdvancedConfig: add mn_reference_taxon, dm_overdispersion_method (mom, mle), mn_penalty (L1 for Songbird-like)
• Nickel: new enum values multinomial, dirichlet_multinomial, contracts for reference taxon existence
• DEED: (method :name "multinomial" :reference-taxon "Bacteroides")
• Frontend: context_help explains MN vs DM vs NB_GLM vs CLR, when to use

Difficulty

Hard — requires:

• Optimization: MN is convex (multinomial logistic regression) but high-dimensional (p = n_taxa x n_covariates), need L1 penalty or filtering (max_features). For 10k taxa x 100 samples x 5 covariates = 50k parameters, need efficient solver (e.g., MLJ or GLMNet).
• DM: non-convex, needs EM or Newton-Raphson, may have local optima. Must test convergence.
• Zero handling: MN likelihood with p_j=0 and count>0 is -Inf, so need to ensure p_j>0 via softmax, but optimization may still push p_j→0. Need epsilon and bounds.
• Performance: MN with 10k taxa, 100 samples, 5 covariates, L-BFGS ~ minutes, not seconds. Must be behind Advanced Analysis, with benchmark and estimated runtime warning.
• Memory: DM covariance matrix n_taxa x n_taxa if full, but diagonal approximation feasible. For 10k taxa, full covariance 10k^2 ~ 800MB, too large — must use diagonal or low-rank.
• Validation: compare coefficients vs R MGLM and Python songbird for 3 datasets.

Risks

• Performance regression: MN/DM 10-100x slower than NB_GLM, may timeout in CI. Must fail loudly if runtime >10x, not silently. Need separate benchmark lane, not part of main CI gate for existing methods (but still fail if existing methods regress >10%).
• Scientific controversy: Compositional methods debated — MN/DM assume compositionality but not absolute abundance. Need balanced context help, not claiming MN solves all compositional issues. Risk of users over-interpreting MN as absolute.
• Numerical instability: Softmax with large logits overflows, need log-sum-exp trick. DM with small overdispersion → MN, with large → unstable. Need heavy validation of overdispersion parameter in (0, Inf), warning if >100.
• Dependency: If using Python Songbird, adds TensorFlow dependency (non-deterministic, GPU vs CPU, seed). Must record seed, version, and make deterministic, otherwise DOI bundle not reproducible. Risk of breaking reproducibility.
• Reference taxon: MN requires reference taxon (e.g., last taxon or user-specified). Choice affects interpretation (log-ratio vs reference). If reference is rare or zero in many samples, coefficients unstable. Need validation: reference must have min_prevalence >=0.5 and min_abundance >0, otherwise refuse.
• Provenance: Must store reference taxon, penalty, overdispersion method, seed, otherwise not reproducible.

Acceptance Criteria

• New methods multinomial, dirichlet_multinomial in AnalysisConfig v2, with mn_reference_taxon, dm_overdispersion_method, mn_penalty
• Validation: reference taxon exists, prevalent, not zero-inflated; penalty >=0; overdispersion in (0, Inf)
• BH mandatory, DANGER banner preserved
• Tests: coefficients match R MGLM and Python songbird within 1e-3 for 3 datasets (mock, gut, soil) with 100 taxa subset
• Benchmark: runtime and memory for 100, 1000, 10000 taxa, with warning if >5 min, fail CI if existing methods regress >10%
• Nickel, DEED, JSON schemas updated
• Context help explains MN vs DM vs NB_GLM vs CLR, with citations (Morton et al. 2019 Songbird, La Rosa et al. 2012 DM, Gloor et al. 2017 compositional)
• Frontend: Advanced Analysis expander, reference taxon selector with prevalence filter, estimated runtime
• Docs: explains compositional coherence, reference choice, overdispersion, when to use vs NB_GLM

Related

• Blocked by: AnalysisConfig v1, TSS/CSS/RSS (needs exact normalization comparison)
• Blocks: Advanced compositional (ANCOM-BC, ALDEx2 comparison)
• References: Morton et al. 2019 mSystems Songbird, La Rosa et al. 2012 Biostatistics DM, Gloor et al. 2017 Front Microbiol compositional