Probabilistic Bézier Curves

• Probabilistic Bézier curves are defined as classical Bézier curves with Gaussian-distributed control points, enabling uncertainty representation in geometric modeling.
• They employ Bayesian conditioning and Gaussian process formulations to achieve closed-form updates and efficient inference for trajectory prediction.
• The framework generalizes to multi-modal and high-dimensional settings, linking geometric modeling with probabilistic inference for robust performance metrics.

A probabilistic Bézier curve generalizes the classical Bézier curve by incorporating uncertainty or stochasticity into the definition and manipulation of control points, blending classical geometric modeling with probabilistic, statistical, and Bayesian inference frameworks. This perspective enables both uncertainty-aware geometric modeling and provides a foundation for fully Bayesian sequential modeling, offering closed-form solutions and expressiveness competitive with or exceeding that of classical neural generative and probabilistic approaches (Hug et al., 2024, Hug et al., 2022, Tanaka et al., 2021, Vaitkus, 2018).

1. Probabilistic Structure of Bézier Curves

A degree-$n$ Bézier curve is classically defined for $t \in [0,1]$ using control points $P_0, \dots, P_n \in \mathbb{R}^d$ and the Bernstein basis: $$B(t) = \sum_{i=0}^n b_{i,n}(t)\,P_i, \quad b_{i,n}(t) = \binom{n}{i}(1-t)^{n-i} t^i$$ Probabilistic Bézier curves, termed $\mathcal{N}$-Curves in (Hug et al., 2022), introduce random variables for the control points: $P_i \sim \mathcal{N}(\mu_i, \Sigma_i)$ Here, $\mu_i \in \mathbb{R}^d$ denotes the mean and $\Sigma_i \in \mathbb{R}^{d\times d}$ the covariance for each control point. The point on the curve at parameter $t$ becomes a (marginal) Gaussian random variable: $$X_{t} \sim \mathcal{N} \left( \mu_{\mathcal{P}}(t),\, \Sigma_{\mathcal{P}}(t) \right )$$ with

$t \in [0,1]$0

This structure can be extended: for points $t \in [0,1]$1, the full sequence $t \in [0,1]$2 is a joint Gaussian, and in mixture models (e.g., multi-modal trajectory priors), one constructs Gaussian mixture models over full trajectory vectors (Hug et al., 2024, Hug et al., 2022).

2. Bayesian and Gaussian Process Formulation

Probabilistic Bézier curves are shown to be special cases of Gaussian processes (GPs), with their GP mean and covariance kernels determined by the choices of control point distributions and Bernstein polynomials: $t \in [0,1]$3

$t \in [0,1]$4

where $t \in [0,1]$5 is the Bézier degree (Hug et al., 2022). Thus, for any finite collection of times, the induced trajectory distributions are jointly Gaussian. In the multi-modal case, mixtures of such GPs are formed, preserving analytic tractability and facilitating Bayesian updating.

Given observations of the process at some time steps, these models admit closed-form Bayesian conditioning as in standard GP inference. In mixture ($t \in [0,1]$6-MDN) settings, each component's mean and kernel is conditioned and the posterior weights are updated via the likelihood, circumventing the need for Monte Carlo trajectory sampling (Hug et al., 2024, Hug et al., 2022).

3. Connections to Probability, Physics, and Analytic Blossoms

The probabilistic interpretation of Bézier curves is rooted in the identification of the Bernstein basis with the probability mass function (pmf) of binomial random variables, via the urn model:

• Consider $t \in [0,1]$7 i.i.d. draws with success probability $t \in [0,1]$8; $t \in [0,1]$9.
• The curve point $P_0, \dots, P_n \in \mathbb{R}^d$0 is then the expectation over possible control points selected with Bernstein probability weights.

The extension to Poisson processes arises via the limiting case $P_0, \dots, P_n \in \mathbb{R}^d$1, $P_0, \dots, P_n \in \mathbb{R}^d$2, $P_0, \dots, P_n \in \mathbb{R}^d$3, producing "Poisson curves" governed by Poisson pmfs, and their corresponding analytic "blossoms" become conditional expectations in infinite-dimensional settings.

Geometric quantization connects this theory to spin systems and harmonic oscillators in physics, encapsulating control-point geometry as images of classical moment maps and providing a unifying interpretation for the binomial, Poisson, and related bases (Vaitkus, 2018).

4. Inference, Learning, and Computation

Probabilistic Bézier curves support Bayesian inference (parameter learning and updating) through several method classes:

• Bayesian ABC (Approximate Bayesian Computation) with Wasserstein metric: The control points are learned via rejection sampling. Synthetic data generated from sampled control points is compared to observed data via the Wasserstein distance; accepted samples yield approximate posteriors, with rigorously characterized bias $P_0, \dots, P_n \in \mathbb{R}^d$4 for acceptance threshold $P_0, \dots, P_n \in \mathbb{R}^d$5 (Tanaka et al., 2021).
• Neural $P_0, \dots, P_n \in \mathbb{R}^d$6-MDNs: Deep models (LSTM encoder + dense heads) output mixture weights and control point distributions, training via negative log-likelihood on full trajectories.
• Computational trade-offs: Analytic inference (conditioning on partial observations, updating predicted futures) is efficient (matrix inversions of modest size), outperforming both classical MDNs (which lack analytic uncertainty handling) and sampling-based Bayesian neural networks in speed and tractable uncertainty management (Hug et al., 2022).

The table below summarizes computational and modeling distinctions:

Model Class	Inference Mode	Uncertainty Handling
Classical MDN	MC/Sampling	Limited (pointwise)
Bayesian RNN	MC (expensive)	Approximate (Monte Carlo)
$P_0, \dots, P_n \in \mathbb{R}^d$7-MDN + $P_0, \dots, P_n \in \mathbb{R}^d$8-GP	Analytic (closed-form)	Full/tractable

5. Applications in Trajectory Prediction and Synthetic Data

Probabilistic Bézier curves are directly applicable to multi-step trajectory prediction tasks:

• Synthetic ground truth generation: Composite probabilistic Bézier curves allow explicit generation of trajectory distributions, supporting benchmarks that admit closed-form posteriors and principled computation of expressive metrics such as the Wasserstein distance.
• Model evaluation and metrics: The true ground-truth distributions enable expressive evaluation: Negative Log Likelihood (NLL) and Wasserstein ($P_0, \dots, P_n \in \mathbb{R}^d$9) distances. Wasserstein distance better penalizes variance misestimation and is more interpretable, though computationally more intensive (about $$B(t) = \sum_{i=0}^n b_{i,n}(t)\,P_i, \quad b_{i,n}(t) = \binom{n}{i}(1-t)^{n-i} t^i$$0 slower than NLL in reported experiments) (Hug et al., 2024).
• Human trajectory prediction: In (Hug et al., 2022), trained LSTM+$$B(t) = \sum_{i=0}^n b_{i,n}(t)\,P_i, \quad b_{i,n}(t) = \binom{n}{i}(1-t)^{n-i} t^i$$1-MDN models show that conditioning predictions on more observed points (one or two) significantly reduces prediction error and improves the negative log-likelihood.

6. Generalizations and Theoretical Extensions

The probabilistic Bézier framework generalizes in several theoretically significant ways:

• Simplex and toric generalizations: Tensor-product, multivariate, or simplex-version Bernstein bases extend the probabilistic interpretation to surfaces, volumes, and higher-dimensional patches, relevant in geometric modeling and multi-objective optimization (Tanaka et al., 2021, Vaitkus, 2018).
• Stochastic basis substitution: The classical binomial/Bernstein basis can be replaced with Poisson, Beta-binomial, $$B(t) = \sum_{i=0}^n b_{i,n}(t)\,P_i, \quad b_{i,n}(t) = \binom{n}{i}(1-t)^{n-i} t^i$$2-deformed, or umbral polynomials, each introducing distinct probabilistic or combinatorial structures.
• Hierarchical, mixture, and reinforced laws: Mixture models, urn-with-reinforcement (Beta-binomial), or hierarchical urn setups flexibly instantiate new curve classes with tunable uncertainty or multimodality.

A plausible implication is that this probabilistic approach unifies trajectory modeling, uncertainty quantification, and geometric inference within a single analytic framework, extending both practical applicability and theoretical interpretability.

7. Experimental and Empirical Findings

Empirical evaluations in (Hug et al., 2024) and (Tanaka et al., 2021) consistently demonstrate that probabilistic Bézier approaches:

• Outperform deterministic methods (e.g., OLS) under observable noise, with improved robustness and generalization.
• Admit closed-form, interpretable posterior updates and evaluation metrics, such as NLL and Wasserstein distance.
• Maintain computational feasibility for moderate problem sizes ($$B(t) = \sum_{i=0}^n b_{i,n}(t)\,P_i, \quad b_{i,n}(t) = \binom{n}{i}(1-t)^{n-i} t^i$$3, $$B(t) = \sum_{i=0}^n b_{i,n}(t)\,P_i, \quad b_{i,n}(t) = \binom{n}{i}(1-t)^{n-i} t^i$$4), with analytic methods scaling better than MC-based Bayesian approaches in update scenarios.

Overall, probabilistic Bézier curves provide a flexible, theoretically principled, and computation-efficient foundation for stochastic curve modeling, bridging geometric modeling, probabilistic machine learning, and Bayesian statistical inference (Hug et al., 2024, Hug et al., 2022, Tanaka et al., 2021, Vaitkus, 2018).