Key assumptions
The MetaHunt pipeline rests on four assumptions. We state them in
applied-reader voice; the manuscript contains the formal statements and
proofs.
A1. Low-rank cross-study heterogeneity
We assume the true study-level functions all live inside the convex
hull of a small number \(K\) of shared
latent basis functions. Each study is a convex combination of
the bases, with non-negative weights that sum to one. Geometrically,
every study is a point inside a \((K-1)\)-simplex whose vertices are the
basis functions; the bases are identifiable as the vertices of that
hull.
Formally, there exist basis functions \(g_1, \ldots, g_K\) with \(K < m\) such that, for every study \(i \in \{0, 1, \ldots, m\}\), \[f^{(i)}(\boldsymbol{x}) \;=\; \sum_{k=1}^{K}
\pi_{ik}\, g_k(\boldsymbol{x}), \qquad \boldsymbol{x} \in
\mathcal{X},\] with weight vector \(\boldsymbol{\pi}_i = (\pi_{i1}, \ldots,
\pi_{iK})^\top \in \Delta_{K-1}\), the \((K-1)\)-simplex. The bases are
non-degenerate, i.e. \(g_2 - g_1, \ldots, g_K
- g_1\) are linearly independent in \(L^2(\mu)\).
What this buys you: the entire \(m\)-by-grid table of study functions
collapses to \(K\) shared shapes plus
an \(m\)-by-\(K\) table of weights, so heterogeneity
becomes low-dimensional and shareable.
A2. Weight model
The mixing weight \(\boldsymbol{\pi}_i\) for study \(i\) is drawn from some conditional
distribution given that study’s covariates \(\boldsymbol{W}_i\). The distribution can be
anything you can estimate — a Dirichlet regression, an RKHS-Dirichlet, a
multinomial logit, a nearest-neighbour smoother. MetaHunt does not
commit you to a specific functional form.
Formally, for \(i = 0, 1, \ldots,
m\), the weight vectors are independent draws \[\boldsymbol{\pi}_i \mid \boldsymbol{W}_i
\;\stackrel{\text{ind.}}{\sim}\; \mathcal{P}_{\boldsymbol{\pi} \mid
\boldsymbol{W}}(\,\cdot \mid \boldsymbol{W}_i),\] where \(\mathcal{P}_{\boldsymbol{\pi}\mid\boldsymbol{W}}\)
is an arbitrary distributional map from the covariate space \(\mathcal{W}\) to the simplex \(\Delta_{K-1}\).
What this buys you: once you can map metadata to weights,
predicting the function for a brand-new target population reduces to
predicting its weights and re-mixing the recovered bases.
A3. Exchangeability
The studies (including the target) are exchangeable units drawn from
a common data-generating process: their joint distribution is invariant
under reordering. This is much weaker than i.i.d. and is the standard
condition under which conformal prediction is valid.
Formally, the site-level covariates \(\boldsymbol{W}_0, \boldsymbol{W}_1, \ldots,
\boldsymbol{W}_m\) are exchangeable. Combined with A1–A2, this
implies that the triples \[\bigl(\boldsymbol{W}_i,\, \boldsymbol{\pi}_i,\,
f^{(i)}\bigr), \qquad i = 0, 1, \ldots, m,\] are jointly
exchangeable across \(i\).
What this buys you: the calibration step in conformal
prediction inherits a finite-sample, distribution-free coverage
guarantee with no need for parametric error models.
A4. Estimation error control
Each study reports a noisy version of its true function, \(\hat f^{(i)} = f^{(i)} + \epsilon^{(i)}\).
We assume the within-study estimation error vanishes uniformly in the
within-study sample size, and that the number of studies \(m\) does not grow too fast relative to
those sample sizes. In words: every study should be reasonably
well-estimated, and you should not have a large number of studies whose
individual sample sizes \(n_i\) are
small.
Formally, write \(\hat
f^{(i)}(\boldsymbol{x}) = f^{(i)}(\boldsymbol{x}) +
\epsilon^{(i)}(\boldsymbol{x})\). We assume \[\sup_{\boldsymbol{x} \in \mathcal{X}}
\mathbb{E}\!\left[\epsilon^{(i)}(\boldsymbol{x})^2\right] =
O\!\left(n_i^{-r}\right) \quad \text{for some } r > 0,\] and
that \(m = o\bigl(\inf_i
n_i^{\,a}\bigr)\) for some \(0 < a
< r\), where \(n_i\) is study
\(i\)’s sample size.
What this buys you: the noise in \(\hat F\) does not accumulate through the
pipeline, basis recovery is consistent, and conformal intervals retain
their asymptotic coverage despite a multi-stage estimator.
The three-step pipeline
metahunt() is a thin wrapper around three exported
steps. You can also call them individually if you want fine-grained
control.
Step 1. Basis hunting via d-fSPA
dfspa() extends the Successive Projection Algorithm to
functions. It iteratively picks the study whose current residual norm is
largest, projects the rest onto the orthogonal complement, and repeats
\(K\) times. A denoising step averages
each study with its near neighbours before the search; this trades a
small bias for substantially smaller variance when the per-study
estimates are noisy.
Step 2. Fitting weight model
For each study, project_to_simplex() finds the convex
weights that best reconstruct its observed function from the recovered
bases. fit_weight_model() then regresses these weight
vectors on the study-level covariates W using a method of
your choice (default: Dirichlet regression).
Step 3. Target prediction
predict.metahunt() takes a new metadata row \(W_0\), predicts its weight vector through
the fitted weight model, and returns the convex combination of the
recovered bases. With wrapper = mean (or any other
reduction) it returns a scalar summary — for example, an ATE under a
uniform grid weighting.
A worked end-to-end example
For the rest of the vignette we work with a simulated
(F_hat, W) so the truth is known. In your own data, replace
this block with the data-prep onramp described in
vignette("data-prep").
G <- 40; m <- 120; K_true <- 3
x <- seq(0, 1, length.out = G)
basis <- rbind(sin(pi * x), cos(pi * x), x) # 3 true bases on the grid
W <- data.frame(w1 = rnorm(m), w2 = rnorm(m)) # study-level covariates
beta <- cbind(c(1, -0.8), c(-0.5, 1.2), c(0, 0))
pi_true <- exp(as.matrix(W) %*% beta)
pi_true <- pi_true / rowSums(pi_true)
F_hat <- pi_true %*% basis + matrix(rnorm(m * G, sd = 0.05), m, G)
dim(F_hat)
#> [1] 120 40
In real data, F_hat[i, ] would be
predict(model_i, newdata = grid) and W[i, ]
would be that centre’s metadata. The data-prep vignette describes a
one-line lm-based onramp.
Fit the full pipeline and inspect the recovered bases:
fit <- metahunt(F_hat, W, K = 3)
fit
#> MetaHunt fit
#> m (studies): 120
#> G (grid size): 40
#> K (bases): 3
#> weight method: dirichlet
#> predictors: w1, w2
plot(fit, x_axis = x,
col = c("#0072B2", "#D55E00", "#009E73"))
Predict the target functions for three new metadata profiles and plot
them:
W_new <- data.frame(w1 = c(0, 1, -1), w2 = c(0, -0.5, 1),
row.names = c("baseline", "high w1, low w2", "low w1, high w2"))
f_pred <- predict(fit, newdata = W_new)
dim(f_pred)
#> [1] 3 40
oldpar <- par(mar = c(4, 4.5, 3, 1))
matplot(x, t(f_pred), type = "l", lty = 1,
col = c("#0072B2", "#D55E00", "#009E73"),
xlab = "x", ylab = expression(tilde(f)(x)),
main = "Predicted target functions")
legend("topright", legend = rownames(W_new),
col = 1:3, lty = 1, bty = "n")
Pass a wrapper for a scalar summary per target (an ATE
under uniform grid weights):
predict(fit, newdata = W_new, wrapper = mean)
#> [1] 0.3318159 0.5559631 0.1020037
References
Shi, W., Imai, K., and Zhang, Y. (2024). Privacy-preserving
meta-analysis through low-rank basis hunting.
Zhang, Y., Huang, M., and Imai, K. (2024). Minimax regret
estimation for generalizing heterogeneous treatment effects with
multisite data. arXiv:2412.11136.