Bayesian Funnel Model

• Bayesian funnel model is a probabilistic graphical model that jointly models sequential decision processes and selective censoring to yield unbiased risk estimation.
• It incorporates stage-specific thresholds and latent risk scores, addressing non-random missing outcomes in contexts such as clinical triage and digital marketing.
• Inference via HMC and related methods demonstrates superior parameter recovery and predictive performance compared to standard regression techniques.

A Bayesian funnel model is a probabilistic graphical model designed to accurately estimate outcomes in sequential, multi-stage decision processes that induce selective or multi-stage censoring of ground-truth labels. Such funnel structures are common in domains where individuals progress through a decreasing sequence of stages—each with associated decision-points and increasing information or cost—culminating in an outcome observed only among those passing a specified threshold. The canonical setting is clinical triage, but analogous scenarios arise in online conversion funnels, hiring, and lending. Statistically, these structures induce complex, non-ignorable censoring that can bias estimation and calibration unless modeled jointly with the decision rules (Sadhuka et al., 12 Nov 2025, Iyengar et al., 2024).

1. Funnel Structure and Selective Multi-stage Censoring

A funnel comprises $K$ sequential stages (e.g., emergency department [ED] triage $\to$ hospital admission $\to$ ICU), each characterized by an evolving covariate set $X_{i,k}$ for patient $i$ at stage $k$. At each stage, a discrete decision $$D_{i,k} \in \{ \text{“discharge”}, k+1, \ldots, K \}$$ is taken. Advancement depends on binning a latent risk score $p_{i,k}$; the ground-truth outcome $Y_i$ (e.g., in-hospital mortality) is only revealed if the individual passes a specified stage $S$ (typically, hospital admission). Patients exiting earlier have their outcomes censored, introducing a non-random missingness pattern that standard regression methods cannot address without bias.

This funnel architecture is mirrored in other domains: in digital marketing, a consumer may traverse viewing, clicking, cart, and purchase stages, each contingent on prior behavior and firm intervention; only some consumers’ ultimate outcomes (conversion or quit) are realized (Iyengar et al., 2024).

2. Probabilistic Specification and Graphical Model

The full generative process is as follows:

• For each patient/stage, covariates $X_{i,k}$ inform a discriminant score:

$\phi_{i,k} = \sigma(\alpha + X_{i,k}^T\beta)$

with $\sigma$ denoting the sigmoid, $\beta \in \mathbb{R}^d$, $\alpha \in \mathbb{R}$.

• A risk variable $p_{i,k}$ is sampled from a distribution $R(\phi_{i,k},\delta_k)$ on $[0,1]$, which captures risk calibration and discriminability.
• Stage-specific thresholds $0 = t_0 < t_1 < \dots < t_{K-1} < t_K =1$ define bins for decision outcomes: admit/discharge. The action $D_{i,k}$ is categorical with probability mass

$$A_{i,k \rightarrow m} = \mathbb{P}(t_{m-1} \leq p_{i,k} < t_m) = \int_{t_{m-1}}^{t_m} R(p|\phi_{i,k}, \delta_k)dp$$

• The outcome $Y_i \sim \operatorname{Bernoulli}(p_{i,s_i})$ is observed only if the patient advances to stage $s_i \ge S$; otherwise, $Y_i$ is censored.

The joint posterior over all parameters $$\Theta = \{\alpha, \beta, \delta_1:\!K-1, t_1:\!K-1\}$$ is given by:

$$p(\Theta | D, Y_\text{obs}) \propto p(\alpha)p(\beta)p(\delta)p(t) \prod_{i,k} p(D_{i,k}|\Theta)\prod_{i:s_i\geq S} p(Y_i|\Theta)$$

This plates-structured graphical model explicitly encodes the staged conditional dependence and selective label observability (Sadhuka et al., 12 Nov 2025).

3. Inference Methods

Bayesian inference is performed via Hamiltonian Monte Carlo (HMC), using Stan with GPU acceleration and the cmdstanpy interface. Standard settings are 4 chains with 500 warmup and 500 sampling iterations, convergence monitored at $\hat{R} \leq 1.05$. Posterior samples are drawn for all generative parameters, providing full uncertainty quantification for risk scores, thresholds, and calibration parameters.

The pipeline readily generalizes to any number of funnel stages and is robust to high-dimensional covariates, provided sufficient sample size at each progressive stage to support parameter estimation.

4. Performance in Synthetic and Real Data

In controlled synthetic data experiments, the funnel model demonstrates superior recovery of true generative parameters and improved predictive performance on censored outcomes relative to three baselines: (i) logistic regression predicting a final-stage decision (e.g., ICU admit), (ii) logistic regression on observed outcomes only, and (iii) logistic regression imputing a default outcome for censored cases.

Mean absolute error (MAE) in recovering intercept $\alpha$ and coefficients $\beta_{1:6}$ in 1,000 simulations:

Method	$\alpha$ MAE	$\beta$ MAE
LogReg (target ICU)	2.45	1.33
LogReg (target $Y$)	0.69	0.21
LogReg (imputed $Y$)	1.74	0.51
Funnel Model	0.16	0.06

Predictive metrics for censored cases (AUROC $\uparrow$, ECE $\downarrow$):

Method	AUROC	ECE
Random Forest	0.625	0.347
LogReg (ICU)	0.743	0.155
LogReg ($Y$)	0.722	0.302
LogReg (impute $Y$)	0.739	0.166
Funnel Model	0.786	0.040

Application to $\sim$425,000 real-world ED visits (MIMIC) supports these gains, especially in calibration. Separate funnel models fitted to female and male subpopulations revealed statistically significant differences in admission thresholds and triage patterns:

• Admission risk thresholds: $t_{\text{ICU,F}} = 0.051$ (95% CI [0.048, 0.053]) vs $t_{\text{ICU,M}} = 0.045$ (95% CI [0.043, 0.047]); men are admitted to ICU at lower estimated mortality risk.
• In triage regression, at identical estimated risk, women are assigned slightly lower acuities: $\beta_F = 2.967$ vs $\beta_M = 2.940$.

The model’s interpretability permits direct interrogation of such disparities (Sadhuka et al., 12 Nov 2025).

5. Connections to Conversion-Funnel MDPs and Attribution-based Learning

Bayesian funnel models have close analogs in large-scale conversion-funnel Markov Decision Processes, such as in digital marketing (Iyengar et al., 2024). In these settings:

• States correspond to dynamic consumer stages; actions to interventions (emails, ads).
• Absorbing states represent conversion or quit.
• The “funnel” nature arises from increasing probabilities of absorption and decreasing pool size—closely paralleling patient attrition in clinical flows.
• Outcome observability is selective and multi-stage censored.

Iyengar & Singal (Iyengar et al., 2024) demonstrate that intractability of the full Bayesian update over high-dimensional transition dynamics can be circumvented by model-free approximate Bayesian learning (MFABL): it maintains independent Beta posteriors on state-action conversion values, updating via interpretable “attribution” rules. This yields optimality and efficiency not possible with standard bandit or full model-based methods (e.g., PSRL, QL-UCB), supporting convergent value estimation even in massive state spaces.

This suggests that the funnel modeling principle—jointly representing sequential decisions, selective observability, and outcome inference—has broad applicability beyond healthcare, especially in any domain characterized by stage-wise attrition and outcome censoring.

6. Pathologies Induced by Funnel Geometry and Addressing Them

A distinct but related funnel phenomenon arises in Bayesian hierarchical models, where the so-called Neal’s funnel induces sharply varying posterior geometry, challenging standard MCMC. The pathology is due to a global scale parameter $y$ controlling the variance of $n$ local variables $x_i$, generating a “throat” region of high curvature and poor mixing.

Multi-stage sampling (MSS) addresses these sampling bottlenecks by:

• Introducing a generalized higher-dimensional hyper-model that “flattens” the funnel,
• Using normalizing flows to fit the induced hyper-marginal distribution,
• Projecting back onto the original constrained space of interest via exact 1D sampling.

Empirically, MSS achieves comparable effective sample size (ESS) and posterior accuracy as prior-reparameterized methods, but with potentially reduced implementation complexity when analytic reparameterizations are costly or intractable (Gundersen et al., 14 Oct 2025).

A plausible implication is that when generalized higher-dimensional models are available, MSS may be a practical approach to circumventing highly nonstandard posterior geometries that routinely arise in funnel-shaped statistical problems.

7. Advantages, Limitations, and Prospective Extensions

The Bayesian funnel model’s primary advantages are:

• Joint modeling of human decisions and ground-truth outcomes, yielding unbiased risk estimation under multi-stage censoring.
• Recovery of generative parameters with low bias, and substantial predictive and calibration improvements in censored subpopulations.
• Provision of interpretable, population-specific decision thresholds, facilitating quantitative analysis of disparities (e.g., by gender or race).

Limitations include:

• The model does not by itself identify the causal drivers of decision disparities (e.g., resource constraints, insurance effects).
• Static threshold assumptions $t_k$, while pragmatic, may ignore temporal drift or operational changes.
• Unmeasured confounding beyond explicit risk estimation may persist.
• In massive-scale or high-dimensional MDP-like funnels, model-free approximate Bayesian updates may be required for computational tractability.

Natural extensions under active investigation include time-varying thresholds, more granular multi-stage structures (lab $\to$ imaging $\to$ biopsy $\to$ diagnosis), and application domains beyond healthcare, such as hiring, lending, and sequential consumer engagement (Sadhuka et al., 12 Nov 2025, Iyengar et al., 2024).

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References:

• “A Bayesian Model for Multi-stage Censoring,” (Sadhuka et al., 12 Nov 2025)
• “Model-Free Approximate Bayesian Learning for Large-Scale Conversion Funnel Optimization,” (Iyengar et al., 2024)
• “Escaping Neal's Funnel: a multi-stage sampling method for hierarchical models,” (Gundersen et al., 14 Oct 2025)