Possibilistic Radial Transport for Approximate IM Inference

Jungeum Kim; Percy Zhai

stat.MLOct 7, 2026

Possibilistic Radial Transport for Approximate IM Inference

Authors: Jungeum Kim, Percy Zhai

Organizations: North Carolina State University · University of Chicago Booth School of Business

Abstract

Probing the hypothesis space after seeing the data remains valid under possibilistic inferential models (IMs), provided the significance level stays fixed. The price is computation, as each plausibility is a supremum of the possibility contour over the hypothesis, and the contour itself is approximated at each queried parameter value. We propose a possibilistic radial transport, which hides the contour value of a parameter in the radius of its source point. When a transport that maximizes within-shell entropy is picked, sampling parameters covering a confidence cut becomes a matter of truncating the radius. We provide a deep learning algorithm that enforces the contour depth condition while maximizing the entropy within each shell. Our amortization makes coverage and power assessments of the learned approximation practical as well as predictive check of new datasets. We also use the sampler to construct a Bel-Pl spectrum for comparing and selecting interpretable hypotheses that satisfy a prescribed Bel-Pl decision criterion. In simulations the learned contours match or improve on ellipsoidal approximations to the cuts, while the coverage and power track the exact reference. Finally, we probe hypotheses about ovarian aging using synthetic AMH records, asking for each woman how many more years her median AMH level will remain above a specified reference value.

Figures & tables

Figure 1 : The nested structure of our trained radial transport $T^{\mathrm{MIC}}$ . Transported samples from source balls of radius $1-\alpha$ approximately fill the corresponding $\alpha$ -cuts. Black dotted lines indicate the estimated boundaries of the target cuts $C_{\alpha}$ .

Figure 2 : (Example 1) Target contour $\pi$ (left), IMMC estimate (middle) and MIC estimate (right) for Gamma, Weibull and S-shear. White curves are the level sets $\alpha\in\{0.05,0.1,0.2,0.5\}$ of each panel.

Model	Method	$D_{+}$	$D_{-}$	MSE	max err.	Spearman	$C_{.05}$ cov.
Gamma	IMMC	0.070	0.026	0.0002	0.116	0.988	0.983
	MIC (ours)	0.075	0.063	0.0001	0.069	0.989	1.000
Weibull	IMMC	0.153	0.025	0.0021	0.159	0.987	0.982
	MIC (ours)	0.136	0.056	0.0002	0.152	0.986	1.000
S-shear	IMMC	0.034	0.136	0.0016	0.110	1.000	0.543
	MIC (ours)	0.057	0.049	0.0001	0.040	1.000	0.982

Table 1 : (Example 1) depth calibration, contour accuracy, and spatial coverage.

Panel of Figure 3 — Figure 3 : Contours of $\pi_{z}$ for three data realizations $z$ (cases 1–3). (a–c) Samples from a single amortized MIC sampler. (d–f) IMMC samples, with a separate IMMC approximation fitted for each dataset. (g) Contour approximation errors and depth-calibration statistics.

Phase	Method	90% coverage	95% coverage	99% coverage	95% area
$0$	Reference	0.920 [0.874, 0.950]	0.955 [0.917, 0.976]	0.995 [0.972, 0.999]	11.764 [11.764, 11.764]
	MIC	0.945 [0.904, 0.969]	0.980 [0.950, 0.992]	0.995 [0.972, 0.999]	13.103 [12.410, 13.552]
	IMMC	0.990 [0.964, 0.997]	0.995 [0.972, 0.999]	1.000 [0.981, 1.000]	79.417 [23.612, 136.648]
$\pi/2$	Reference	0.900 [0.851, 0.934]	0.950 [0.910, 0.973]	0.985 [0.957, 0.995]	11.764 [11.764, 11.764]
	MIC	0.920 [0.874, 0.950]	0.965 [0.930, 0.983]	0.990 [0.964, 0.997]	13.029 [12.290, 13.501]
	IMMC	0.980 [0.950, 0.992]	0.995 [0.972, 0.999]	1.000 [0.981, 1.000]	86.669 [23.751, 137.184]

Table 2 : (Amortized S-shear) Repeated inference across datasets, Coverage entries are estimates [pointwise 95% Wilson CI]; area entries are medians [10th, 90th percentiles] across datasets. Reference is computed by using the exact, closed form.

Figure 4 : Empirical power for testing $H_{0}:\theta=(0,0)$ in the S-shear model at $\alpha=0.05$ . We generate data from $\theta_{\eta}=(\eta,-2\sin(2.5\eta))$ in the left panel and $\theta_{\eta}=(0,\eta)$ in the right panel, 100 times for each $\eta$ . The empirical type-I error rate corresponds to $\eta=0$ .

Figure 5 : Predictive screening for model-generated data ( $\sigma=2.5$ ; left), inflated deviations ( $\times 1.5$ ; middle), and a Gaussian mixture with approximately matched marginal variances (right). All share the same sample mean and contour. Red: $z_{0}$ to children (predictive) distances. Gray: children to grand children distances. p is a diagnostic score.

Figure 6 : The Bel-Pl spectrum for a hypothesis $H$ .

Figure 7 : The primal candidate hypotheses in Example 3. Their 15 nonempty unions are considered as candidates.

Figure 8 : The Bel-Pl spectrum comparison for the selected hypothesis. (a) MIC and the reference selected $H_{4}$ , while IMMC estimates showed $H_{4}$ as not meeting the constraint. (b) IMMC selected $H_{1,2,3,4}=\cup_{i=1}^{4}H_{i}$ . Stars located at the estimated $1-Bel(H_{\cdot})$ .

Figure 9 : Left: The true remaining years for which median AMH stays above the threshold. Center: Inference results from MIC. Right: Inference results from IMMC. A larger green area (closer to the truth) means more informative inference.

Figure 10 : MIC yields tighter $\alpha$ -cuts. Left: Empirical coverage of $C_{\alpha}$ for the true parameter $\theta$ across 160 synthetic women. Middle: The widths of the projected remaining-time intervals. Right: Fraction of women for whom judgment is withheld at each future horizon, with $\alpha=0.05$ .

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

\widehat{\rho}_{t}(\widetilde{\Theta}_{ij})=\frac{1}{K}\sum_{k=1}^{K}\sigma\!\left(\frac{R(\widetilde{\Theta}_{ij})-R(\widetilde{\Theta}_{k})}{\tau}\right).

Appendix

Algorithm 1 MIC Sampler Learning.

\widehat{\rho}_{z_{l}}(\widetilde{\Theta}_{ijl})=\frac{1}{K}\sum_{k=1}^{K}\sigma\!\left(\frac{R(\widetilde{\Theta}_{ijl},z_{l})-R(\widetilde{\Theta}_{kl},z_{l})}{\tau}\right).

Appendix

Algorithm 2 Amortized MIC Sampler Learning

Panel of Figure 11 — Figure 11 : The comparison between the original and the synthetic trajectory. Top: The original, copied from de Kat et al. (2016) . Bottom: From our synthesized data.

Figure 12 : (Example 1) Samples from MIC (blue) and IMMC (red) generated with the various source level $\alpha$ . The black dotted lines are estimated target $C_{\alpha}$ .

Figure 13 : (Example 2) The source radius to contour depth alignment (top) and the credal calibration (bottom). The contour depth is computed based on the known true contour and the stitched estimation.

Method	$H_{S}$	Bel	Pl	$Q(H)$	Gap	$\bar{\pi}$
Reference	$H_{4}$	0.93172	1.00000	0.98540	0.05368	$\mathbf{0.50702}^{\star}$
MIC	$H_{4}$	0.93140	1.00000	0.95814	0.02674	$\mathbf{0.51230}^{\star}$
IMMC	$H_{4}$	0.79780	1.00000	0.83044	0.03264	0.58206 †
Reference	$H_{1,2,3,4}$	0.99873	1.00000	0.99998	0.00125	0.50001
MIC	$H_{1,2,3,4}$	0.99980	1.00000	1.00000	0.00020	0.49190
IMMC	$H_{1,2,3,4}$	0.90160	1.00000	0.96544	0.06384	$\mathbf{0.51622}^{\star}$

Appendix

Table 3: (Example 3) Selected unions among all 15 candidates, with

\widehat{\mathrm{Bel}}(H)\geq 0.90

and

\widehat{Q}(H)-\widehat{\mathrm{Bel}}(H)\leq 0.30

. Gap denotes

\widehat{Q}(H)-\widehat{\mathrm{Bel}}(H)

{\dagger}

indicates infeasibility, and

\star

marks the selected hypothesis. MIC and Reference selected

H_{4}

and IMMC selected

H_{1,2,3,4}

Figure 14 : (Ovarian Application) Comparison to the MC reference decisions. MIC mostly aligns to the MC reference.

Horizon (years)	MIC	IMMC	MC ref.	MIC	IMMC	MC ref.
	Withhold rate (%)			Women with an error
1	68.8	91.2	68.8	0	0	0
3	86.9	100.0	86.9	0	0	0
5	88.1	96.2	89.4	0	0	0
10	83.1	88.1	76.2	0	0	0
All horizons	46.0	54.4	43.9	0	0	0

Appendix

Table 4: (Ovarian Application) Hypothesis decisions at

\alpha=0.05

for the 160 women of the primary cohort. An error is an incorrect acceptance or rejection of

H_{k}

. In the “All horizons” row, the withhold rate is averaged over the full grid

k=1,\ldots,30

, and the error count records women with at least one error anywhere on that grid. MC ref. is the direct Monte Carlo contour on the parameter grid.

Figure 15 : (Ovarian Application) Predictive check result for three chosen (simulated) women. It gives an illustration of the predictive check when the data is well explained by the likelihood model and when the amortized sampler is well-trained. Their diagnostics value is far from 0 and the red line (self-children relationship) resembles the other gray lines (children-grand-children relationship).

Figure 16 : (Ovarian Application) A few chosen individual examples. Left: Multiple hypotheses and estimated contours. Right: Bel and Pl values showing acceptance/rejection with $\alpha=0.05$ .

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