IXPLORE: Bounded Ideal Point Estimation with Grid-Based Uncertainty Quantification

Fynn Bachmann

cs.LGSep 5, 2026

IXPLORE: Bounded Ideal Point Estimation with Grid-Based Uncertainty Quantification

Authors: Fynn Bachmann

Organizations: Department of Informatics, University of Zurich, Switzerland

Abstract

Ideal point estimation is widely used to analyze and visualize political data. However, selecting the corresponding spatial model involves various trade-offs: while model-based approaches such as Item Response Theory (IRT) are based on utility functions rather than optimized for predictive accuracy, most Machine Learning (ML) alternatives struggle to generalize beyond training data when embedding sparse test responses. We introduce IXPLORE, a bounded ideal point estimation algorithm that combines a predictive fit objective with a sparsity-aware likelihood function. On five benchmark datasets spanning surveys, roll calls, and deliberation, this approach surpasses model-based and ML-based algorithms on reconstruction and imputation error - especially for users with sparse responses. Furthermore, we show that non-linear feature transforms can further reduce the reconstruction error while remaining visually interpretable. To quantify uncertainty, IXPLORE applies grid-based posterior inference on a bounded 2D latent space. Available as a Python package on PyPI, IXPLORE offers a flexible framework for constructing bounded, interpretable political maps with fast inference and strong imputation performance.

Figures & tables

Dataset	Source	Train Users	Test Users	Items	Scale	Missing
Smartvote 2019	VAA (CH)	$1{,}912$ candidates	–	$75$	4-Likert	$0\%$
Smartvote 2023	VAA (CH)	$1{,}029$ candidates	$1{,}121$ voters	$75$	4-Likert	$0\%$
Voteview (S117)	Roll-call (US)	$102$ senators	–	$603$	binary	$5.2\%$
Polis (vTaiwan)	Deliberation (TW)	$1{,}921$ participants	–	$197$	ternary	$86.9\%$
EVS (2017)	Survey (EU)	$35{,}116$ respondents	–	$32$	10-Likert	$0\%$

Table 1: Datasets used in the experiments. All response values are rescaled to

[0,1]

. Where no separate test set is available, we hold out a test set from the train data.

Figure 1: IXPLORE and five reference algorithms on Smartvote 2023. Reconstruction and imputation error are reported as a function of sparsity in each of the four evaluation cells. Three points of interest are highlighted: the train reconstruction error at

u=0

corresponds to embedding candidates with full data; the test reconstruction error (e.g., at

v=0.6

) corresponds to embedding voters in the same space; the test imputation error (e.g., at

v=0.6

) corresponds to predicting voters’ remaining responses.

	Train Rec.		Train Imp.		Test Rec.		Test Imp.
Algorithm	MAE	ACC	MAE	ACC	MAE	ACC	MAE	ACC
Linear PCA	0.245	80.2%	0.274	76.4%	0.282	72.4%	0.301	69.1%
Logistic PCA	0.175	85.1%	0.206	80.9%	0.267	72.7%	0.289	69.5%
Kernel PCA	0.195	84.7%	0.223	80.4%	0.283	70.6%	0.304	67.4%
t-SNE	0.212	80.9%	0.239	76.7%	0.280	70.0%	0.300	67.0%
UMAP	0.210	81.1%	0.233	77.4%	0.284	69.6%	0.301	67.0%

Table 2: Performance comparison on Smartvote (2023). Each cell is the mean over all sparsity levels. Reference algorithms are listed in the top block; IXPLORE is run both with the continuous input and the binarized input ∗ . The best-performing algorithm is shown in bold , the runner-up in italics .

Figure 2: Reconstruction and imputation error of PCA on Smartvote 2023 as a function of the latent dimensionality

d

, averaged over all sparsity levels. The circles mark each curve’s minimum. The horizontal red lines indicate the IXPLORE performance with

\tau^{2}=0.25

, PCA initialization, and

T=10

iterations. Logistic PCA exceeds IXPLORE ’s training reconstruction error for

d\geq 3

(Linear PCA for

d\geq 21

); neither variant reaches IXPLORE ’s training or test imputation error at any

d\leq 74

Figure 3: IXPLORE embeddings of the full Smartvote 2023 candidate data at

T=20

for three different priors. The reconstruction error stays stable, while the boundary fraction (share of points within 10% of the grid boundary) increases with

\tau^{2}

Figure 4: Iteration trajectories

T\in\{0,1,2,5,10,20\}

for each prior

\tau^{2}

with PCA initialization. The marker size grows with

T

while the first and last iteration are connected with an arrow pointing from

T=0

T=20

. ( A ) The boundary fraction and train reconstruction error at

u=0

decrease over the course of iterations. ( B ) The distortion of the latent space remains stable over the course of iterations, while the test imputation error decreases (averaged over

v\in\{0.3,0.6,0.9\}

Figure 5: Effect of initialization across iterations on Smartvote 2023 at

\tau^{2}=0.25

. ( A ) The reconstruction MAE is reported at

u=0

. ( B ) The imputation MAE is averaged over

v\in\{0.3,0.6,0.9\}

Figure 6: Reconstruction and imputation error on Smartvote 2023 data for the three IXPLORE feature transforms (colored) against the reference algorithms (gray).

Figure 7: Predicted probability maps for three different feature transformations of the IXPLORE model initialized with

\tau^{2}=0.25

after

T=10

iterations on the Smartvote 2023 data. In each panel, the item with the largest accuracy spread across feature transforms is chosen (item 32263). The dots correspond to the 1,029 candidates colored by their response to this item (orange/blue

=

no/yes).

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Symbol	Default	Varied in
Prior variance	$\tau^{2}$	$0.25$	Section 5.2.1
Iterations	$T$	$10$	Section 5.2.2
Initialization	–	PCA	Section 5.2.2
Feature transform	$\varphi$	identity	Section 5.2.3
Confidence weights	$W_{nk}$	uniform	Section D.1
Weight scaling	–	False	Section D.1

Appendix

Table 3: IXPLORE hyperparameters. The defaults are used in the baseline comparison ( Section 5.1 ), while the configuration analysis ( Section 5.2 ) varies parameters as described.

Figure 8: One-dimensional slice of the two prior families at

x_{2}=0

over the grid

[-L,L]^{2}

with

L=1

. ( A ) The Gaussian prior concentrates around the origin as

\tau^{2}

shrinks. ( B ) The log-barrier prior is flat in the interior and vanishes at the grid boundary for larger

\alpha

. Both priors are symmetric in

x_{1}

and

x_{2}

$n_{\text{obs}}$	MAP	Posterior mean	Full posterior
5	0.2313	0.2255	0.2439
10	0.2111	0.2067	0.2190
15	0.2023	0.1983	0.2078
30	0.1934	0.1905	0.1961
45	0.1915	0.1894	0.1933
60	0.1894	0.1878	0.1907

Appendix

Table 4: Held-out MAE on Smartvote 2023 by point estimator and number of observed answers, at the default configuration (

\tau^{2}=0.25

T=10

). Best per row in bold .

	Boundary frac.		Recon. MAE		Recon. Acc.
$\tau^{2}$	MAP	Mean	MAP	Mean	MAP	Mean
0.05	0.000	0.000	0.1824	0.1804	0.836	0.836
0.1	0.000	0.000	0.1823	0.1804	0.836	0.836
0.25	0.006	0.000	0.1820	0.1805	0.836	0.836
0.5	0.045	0.000	0.1815	0.1806	0.836	0.836
1	0.088	0.000	0.1811	0.1807	0.836	0.836

Appendix

Table 5: Boundary fraction (share of embeddings within 5% of the grid edge) and train reconstruction MAE / accuracy on Smartvote 2023, per prior variance with a model retrained at each

\tau^{2}

(

T=10

), for the MAP and posterior-mean estimators.

Figure 9: Sampling strategies on Smartvote 2023 as a function of the number of observed answers. ( A ) The MAE is computed between each method’s per-item sample mean and the true answer on the unobserved items, for Rasch, posterior, and uniform-random sampling, with the deterministic posterior-mean prediction for reference. ( B ) The mean per-item sample standard deviation on the unobserved items is reported; higher values indicate more diverse samples.

Figure 10: Posterior trajectory and convergence on Smartvote 2023. ( A ) The test user whose final posterior mean lies furthest from the origin is shown, fitted at

\tau^{2}=0.25

. The contour is the posterior density after the user has answered

k=15

K=75

items; black dots and the connecting line trace the running posterior mean at every

k

in answer order; the dashed ellipse is the

1\sigma

contour of the moment-matched Gaussian fit to the final posterior. ( B ) The CDF of the smallest

k

at which the running posterior mean enters the final

1\sigma

ellipse is shown, evaluated over all 1,121 test users, for the tightest (

\tau^{2}=0.05

) and weakest (

\tau^{2}=10^{6}

) priors in our sweep. Dotted lines mark the 50% and 95% crossings.

	Train Rec.		Train Imp.		Test Rec.		Test Imp.
Algorithm	MAE	ACC	MAE	ACC	MAE	ACC	MAE	ACC
Linear PCA	0.239	80.0%	0.265	76.6%	0.234	81.1%	0.260	78.1%
Logistic PCA	0.176	85.0%	0.207	80.5%	0.212	80.8%	0.236	77.8%
Kernel PCA	0.199	84.0%	0.227	79.6%	0.232	79.4%	0.256	76.2%
t-SNE	0.213	80.8%	0.239	76.5%	0.221	79.7%	0.242	76.6%
UMAP	0.207	81.1%	0.228	77.6%	0.228	78.4%	0.245	76.1%

Appendix

Table 6: Performance comparison on Smartvote (2019). Each cell is the mean over all sparsity levels. Reference algorithms are listed in the top block; IXPLORE is run both with the continuous input and the binarized input ∗ . The best-performing algorithm is shown in bold , the runner-up in italics .

	Train Rec.		Train Imp.		Test Rec.		Test Imp.
Algorithm	MAE	ACC	MAE	ACC	MAE	ACC	MAE	ACC
Linear PCA	0.281	82.4%	0.307	78.5%	0.282	81.4%	0.300	79.3%
Logistic PCA	0.178	88.9%	0.240	80.9%	0.253	81.2%	0.271	79.1%
Kernel PCA	0.259	85.2%	0.297	79.2%	0.274	82.0%	0.289	79.9%
t-SNE	0.272	82.2%	0.313	75.1%	0.271	82.6%	0.290	79.2%
UMAP	0.281	82.0%	0.315	74.9%	0.324	75.5%	0.335	73.1%

Appendix

Table 7: Performance comparison on Polis (vTaiwan). Each cell is the mean over all sparsity levels. Reference algorithms are listed in the top block; IXPLORE is run both with the continuous input and the binarized input ∗ . The best-performing algorithm is shown in bold , the runner-up in italics .

	Train Rec.		Train Imp.		Test Rec.		Test Imp.
Algorithm	MAE	ACC	MAE	ACC	MAE	ACC	MAE	ACC
Linear PCA	0.231	90.6%	0.320	82.9%	0.267	90.9%	0.318	89.9%
Logistic PCA	0.053	96.8%	0.082	93.6%	0.190	84.7%	0.226	81.8%
Kernel PCA	0.121	95.2%	0.169	90.9%	0.192	85.1%	0.222	81.7%
t-SNE	0.124	95.1%	0.183	89.0%	0.243	77.4%	0.297	71.5%
UMAP	0.143	93.6%	0.208	87.0%	0.124	89.7%	0.140	88.3%

Appendix

Table 8: Performance comparison on Voteview (S117). Each cell is the mean over all sparsity levels. Reference algorithms are listed in the top block; IXPLORE is run both with the continuous input and the binarized input ∗ . The best-performing algorithm is shown in bold , the runner-up in italics .

	Train Rec.		Train Imp.		Test Rec.		Test Imp.
Algorithm	MAE	ACC	MAE	ACC	MAE	ACC	MAE	ACC
Linear PCA	0.207	75.6%	0.230	71.3%	0.207	75.9%	0.229	71.6%
Logistic PCA	0.182	79.6%	0.240	71.2%	0.199	76.2%	0.225	71.9%
Kernel PCA	0.195	76.5%	0.226	71.5%	0.202	76.1%	0.227	71.7%
t-SNE	0.204	74.8%	0.235	70.1%	0.209	74.1%	0.231	70.7%
UMAP	0.208	74.3%	0.236	69.6%	0.205	74.6%	0.229	70.9%

Appendix

Table 9: Performance comparison on European Values Study. Each cell is the mean over all sparsity levels. Reference algorithms are listed in the top block; IXPLORE is run both with the continuous input and the binarized input ∗ . The best-performing algorithm is shown in bold , the runner-up in italics .

Figure 11: Reconstruction (solid, circles) and imputation (dashed, squares) MAE on Smartvote 2019 as a function of train sparsity

u

and test sparsity

v