Fit per-feature KDE naive-Bayes class-conditional LRT discriminator

Learns a frozen two-class discriminator from training data using a per-feature kernel density estimate (KDE) of each class-conditional marginal under a naive-Bayes (feature-independence) assumption. Each specimen is first mapped to a self-contained, per-sample robust centred-log-ratio (rCLR) representation $z$, where $z_j = \log v_j - \mathrm{mean}_{k:\,v_k>0}\log v_k$ on the nonzero parts of the specimen and $z_j = 0$ on its zero parts. Because the centring uses only that specimen's own nonzero values, $z(c\,v) = z(v)$ for any $c>0$: the representation – and therefore every downstream score – is exactly invariant to per-sample scaling (library size / input amount).

This is the marginal complement to the Gaussian-copula LRT. The copula matches out the per-feature marginals via a probability-integral transform and scores only the residual feature dependence; lrt-nbkde does the opposite – it scores the per-feature class-conditional marginal density ratio and ignores dependence (the naive-Bayes assumption). For each feature $j$ and class $c \in \{\mathrm{case}, \mathrm{control}\}$ the frozen marginal is a Gaussian-kernel KDE of the training rCLR values $z^{\mathrm{train}}_{c,j}$: $$\log f_{c,j}(t) = \mathrm{logSumExp}_{i}\!\left( -\tfrac{1}{2}\Big(\tfrac{t - z^{\mathrm{train}}_{c,j,i}}{h_j}\Big)^2\right) - \log n_{c} - \log h_j - \tfrac{1}{2}\log(2\pi),$$ evaluated directly at the specimen's own coordinate $t = z_j$ (not on a fixed grid), so it is exact and computable from a single specimen.

The per-feature bandwidth $h_j$ is frozen at fit. By default it is the Silverman rule-of-thumb stats::bw.nrd0 of the pooled rCLR values of feature $j$ across both classes (so the case and control marginals of a feature are smoothed comparably), scaled by hp$bw_adjust and floored at hp$eps: $h_j = \max(\mathrm{adjust}\cdot \mathrm{bw.nrd0}(\mathrm{pool}_j),\, \epsilon)$. If hp$bw is supplied, that positive scalar replaces bw.nrd0 for every feature.

Fitting stores the raw case/control anchor rows (so the per-feature KDEs can be re-derived consistently over any present-feature subset at scoring), the frozen training feature universe, the resolved hyperparameters, and – for inspection and tests – the full-universe class representation (per-class rCLR anchor values and the per-feature bandwidths). No test data and no test-time batch statistic are used.

Usage

fit_lrt_nbkde(X_train, y_train, meta_train = NULL, hp = list())

Arguments

X_train: Numeric matrix (samples $\times$ features) of non-negative abundance values. Columns must be uniquely named feature ids.
y_train: Integer/numeric 0/1 labels aligned to X_train; 1 is case/disease and 0 is control.
meta_train: Optional per-sample metadata. Accepted for interface uniformity and ignored by this method.
hp: List of frozen hyperparameters: bw either NULL (default; the per-feature pooled stats::bw.nrd0 bandwidth is used) or a single positive finite scalar bandwidth applied to every feature; bw_adjust positive finite multiplier on the bandwidth analogous to density(adjust=) (default 1.0); max_anchors_per_class positive-integer anchor cap per class (default 200L); min_features positive-integer feature-overlap floor at scoring (default 3L); eps positive bandwidth/density floor (default 1e-6); and seed non-negative integer used only for deterministic anchor subsampling while restoring the global RNG state (default 1L).

Value

A plain list of class lrt_nbkde_model containing case_anchors, control_anchors, feature_universe, repr (the full-universe frozen class representation: z_case, z_control, and the per-feature bw), and the resolved hp.

Details

The single-specimen score (see score_lrt_nbkde) is the naive-Bayes class-conditional marginal log-likelihood ratio $$S(z) = \sum_{j} \big[\log f_{\mathrm{case},j}(z_j) - \log f_{\mathrm{control},j}(z_j)\big],$$ a sum of per-feature class-conditional marginal log-density ratios; larger means more case-like. The bandwidth and the $-\tfrac12\log(2\pi)$ constant cancel within each feature's ratio, so only the kernel sums and the per-class log-counts $\log n_c$ survive.

Degenerate features are handled without any caller-side special-casing. A feature that is constant across a class's training anchors (e.g. zero in all of them) still yields a finite KDE: the pooled bw.nrd0 stays positive (it falls back to a unit scale for a constant input) and is in any case floored at hp$eps, and the Gaussian kernel sum is evaluated through a stable log-sum-exp so each $\log f_{c,j}$ is finite. The fit and every score therefore remain finite.

References

Silverman B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, London.

Scott D.W. (1992) Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York.

Hand D.J., Yu K. (2001) Idiot's Bayes – not so stupid after all? International Statistical Review 69(3): 385-398.

Examples

if (FALSE) { # \dontrun{
set.seed(1)
n <- 120; p <- 30
L <- matrix(stats::rnorm(n * p, 4, 0.6), nrow = n,
            dimnames = list(NULL, paste0("miR-", seq_len(p))))
y <- rep(c(0, 1), each = n / 2)
L[y == 1, 1:8] <- L[y == 1, 1:8] + 1.2   # case-only marginal location shift
X <- exp(L)
model <- fit_lrt_nbkde(X, y)
score <- score_lrt_nbkde(model, X)
} # }