📊 statistics

Extending Evidence Accumulation Models to Bounded Continuous Self-report Data

This paper introduces and compares two new evidence accumulation models, the Half-Circular Diffusion Model and the Beta Drift Diffusion Model, for analyzing bounded continuous self-report data using amortized Bayesian inference to enable robust parameter estimation and model selection without requiring analytical likelihood functions.

Original authors: Yufei Wu, Tamás Szűcs, Agnes Moors, Francis Tuerlinckx

Published 2026-04-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yufei Wu, Tamás Szűcs, Agnes Moors, Francis Tuerlinckx

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine your brain is a busy factory trying to make a decision. For decades, scientists have used a tool called an Evidence Accumulation Model (EAM) to understand how this factory works. Traditionally, this tool only worked for simple "Yes/No" or "Left/Right" choices. It imagined a worker gathering noisy clues until they had enough to shout "Go!" and hit a stop button.

But what happens when the choice isn't just "Left" or "Right," but something on a sliding scale? Think of rating your mood on a slider from "Terrible" to "Amazing," or saying how much you like a song from 1 to 100. These are bounded continuous self-reports. The old tools couldn't handle this; they were like trying to measure the temperature of a soup with a ruler.

This paper introduces two new, specialized tools designed specifically for these sliding-scale decisions. The authors call them the Half-Circular Diffusion Model (HCDM) and the Beta Drift Diffusion Model (BDDM).

Here is how they work, using simple analogies:

1. The Two New Tools

The Half-Circular Diffusion Model (HCDM): The "Bouncing Ball"
Imagine a ball rolling on a flat floor that is shaped like a half-moon (a semi-circle).

The Goal: The ball starts somewhere in the middle and rolls toward the curved edge.
The Rule: If the ball tries to roll off the flat, straight edge of the half-moon, it doesn't fall off; it bounces back (reflects) and stays inside the valid area.
The Decision: The moment the ball hits the curved edge, the decision is made. Where it hits tells you what the answer is (e.g., closer to "Negative" or "Positive"), and how long it took tells you how fast the decision was.
Why it works: It's a clever way to force a circular movement (which is mathematically easy) to fit a straight, bounded line (like a slider).

The Beta Drift Diffusion Model (BDDM): The "Crowded Room"
Imagine a long hallway with 101 people standing in a line, each representing a different point on your slider scale.

The Goal: Instead of one ball rolling, everyone in the line is gathering evidence at the same time.
The Shape: The "drift" (the speed at which they gather evidence) isn't the same for everyone. It forms a hill shape (like a bell curve or a mountain). The peak of the hill is where the person is most confident.
The Decision: The moment anyone in the line gathers enough evidence to cross a finish line, the decision is made.
Why it works: This model is more flexible. It can handle situations where people are very unsure (the hill is flat and wide) or very sure (the hill is tall and narrow). It's like having a whole team of workers instead of just one.

2. The Problem: The "Black Box" of Math

Usually, to use these models, scientists need a specific math formula (a "likelihood function") to tell them how well the model fits the data. But for these new, complex models, that formula is too messy to write down. It's like trying to solve a puzzle where the pieces keep changing shape.

The Solution: The "Training Robot" (Amortized Bayesian Inference)
Since they couldn't write the formula, the authors used a clever workaround. They built a neural network (a type of AI) and trained it like a student:

Simulation: They ran millions of fake experiments on a computer, creating thousands of "fake" decision scenarios with known answers.
Learning: They showed these fake scenarios to the AI, teaching it: "When you see this pattern of data, the answer is that."
Application: Once the AI was trained, they fed it real human data. The AI didn't need the messy math formula; it just recognized the patterns it had learned and gave them the answer.

3. What They Found

The authors tested these tools on real data where people played a game, won or lost money, and then rated their feelings on a continuous slider.

Both tools worked: They could accurately figure out how fast people were thinking and how confident they were.
The "Bouncing Ball" (HCDM) was great for most people. It was fast to compute and worked well for standard responses.
The "Crowded Room" (BDDM) was the "super-tool." It was better at handling the extremes. If a person was extremely consistent (always picking the exact same spot) or extremely scattered (jumping all over the scale), the BDDM fit their data better. This is because the BDDM has a special "noise control" knob that lets it handle very smooth or very jagged decision paths.
The Trade-off: The BDDM is more powerful but much slower to train. It took the computer about 13 minutes to train the "Bouncing Ball" model, but nearly 7 hours to train the "Crowded Room" model.

4. The Limitations

The authors noted that both tools struggled when people acted strangely. If a participant kept clicking the exact same spot over and over (like always clicking "Neutral" or always clicking the very end), the models got confused. The models assume people are moving smoothly along the scale, so they couldn't perfectly predict these "stuck" behaviors.

Summary

In short, this paper gives psychologists two new, specialized rulers for measuring decisions on a sliding scale. They used a smart AI training method to make these complex rulers usable. One is a quick, standard ruler (HCDM), and the other is a high-precision, flexible ruler (BDDM) that handles extreme behaviors better but takes longer to calibrate. This allows researchers to finally study the "speed and confidence" of feelings and opinions, not just simple yes/no choices.

1. Problem Statement

Evidence Accumulation Models (EAMs), such as the Drift Diffusion Model (DDM), are the standard framework for analyzing binary choice and reaction time (RT) data. However, a significant gap exists in modeling bounded continuous self-report data (e.g., visual analogue scales for emotion intensity, confidence, or risk preference).

Limitations of Existing Models:
- Circular Diffusion Model (CDM): Designed for rotationally symmetric tasks (e.g., angle estimation). It is unsuitable for bounded linear scales because it assumes a full circular decision space.
- Spatially Continuous Diffusion Model (SCDM): Designed for perceptual estimation where the "correct" stimulus location is externally known (e.g., identifying a color). It cannot handle subjective self-reports where the target location is not externally specified and must be inferred from the response itself.
The Challenge: Standard EAMs require analytical likelihood functions for parameter estimation. The proposed adaptations for bounded continuous data involve complex stochastic processes that lack tractable likelihoods, rendering traditional Maximum Likelihood Estimation (MLE) or standard Bayesian inference infeasible.

2. Methodology

The authors propose two novel model adaptations and a likelihood-free inference workflow to overcome the intractable likelihood problem.

A. Proposed Models

Half-Circular Diffusion Model (HCDM):
- Concept: An adaptation of the CDM where the decision space is restricted to a semi-disc ( $D'$ ) rather than a full circle.
- Mechanism: Implements a specular reflection mechanism at the diameter boundary ( $x_2 = 0$ ). If the diffusion trajectory attempts to cross this flat edge, the remaining displacement is reflected back into the valid region.
- Parameterization: Uses a polar coordinate system for the drift rate vector $(l, \phi)$ , where $l$ is magnitude and $\phi$ is the angle relative to the decision arc. This allows the starting point to vary without constraining the valid drift directions.
- Output: The response is the polar angle $\phi$ at the first passage time to the curved boundary.
Beta Drift Diffusion Model (BDDM):
- Concept: An adaptation of the SCDM where the drift rate is defined over a continuous spatial domain but is not tied to an external stimulus.
- Mechanism: The drift rate function is modeled as a scaled Beta probability density function (PDF) parameterized by mean ( $\mu$ ), concentration ( $\kappa$ ), and height ( $h$ ).
- Noise: Uses a spatial Gaussian process with a squared exponential kernel to model correlated noise across the decision space, ensuring biologically plausible accumulation.
- Output: The response is the spatial location $u$ where the accumulated evidence first crosses the boundary.

B. Inference Framework: Amortized Bayesian Inference (ABI) & Comparison (ABMC)

Since analytical likelihoods are unavailable, the authors employ likelihood-free inference:

Simulation-Based Training: Large datasets are simulated from the models using known "ground truth" parameters.
Neural Networks:
- Summary Network: Compresses variable-sized data into fixed-size sufficient statistics (using SetTransformers).
- Inference Network: Learns the mapping from data to the posterior distribution of parameters (for ABI) or model probabilities (for ABMC).
Model Comparison (ABMC): Uses Deep Ensembles (multiple independently trained networks) to estimate Posterior Model Probabilities (PMPs) and mitigate sensitivity to model misspecification. Contaminated data (5% noise) is added during training to ensure robustness.

3. Key Contributions

Theoretical Extension: Successfully extends the EAM framework to bounded continuous self-report data, bridging the gap between binary decision models and continuous subjective reporting.
Methodological Workflow: Demonstrates a complete pipeline for fitting complex, likelihood-free EAMs using ABI and ABMC, making these models accessible for practical psychological research.
Open Science: Provides fully documented code, example datasets, and trained neural network weights to facilitate adoption in the experimental psychology community.
Parameter Interpretability: Establishes clear psychological interpretations for the new parameters (e.g., drift rate magnitude as processing speed, boundary as caution, and Beta parameters as evidence clarity).

4. Results

The models were applied to an empirical dataset involving a probabilistic reward task where participants reported affective states on a continuous slider.

Parameter Recovery: Both HCDM and BDDM demonstrated high accuracy in recovering ground-truth parameters from simulated data (correlations $r > 0.95$ for most parameters).
Empirical Fit:
- HCDM: Parameters correlated strongly with empirical summary statistics. Drift angle ( $\phi$ ) predicted mean affect ratings ( $r=0.992$ ), and boundary ( $a$ ) predicted mean RT ( $r=0.615$ ).
- BDDM: Showed similar correlations but offered superior flexibility. The mode of the drift rate ($modedrift$) predicted mean ratings ( $r=0.992$ ), and the inter-quartile range ( $IQR_{drift}$ ) predicted rating variability.
Model Comparison:
- ABMC Results: Approximately 70 participants strongly favored the BDDM, while ~40 favored the HCDM.
- Driver of Selection: The BDDM outperformed the HCDM primarily for datasets with extreme response distributions (either highly concentrated or highly dispersed). The BDDM's kernel parameter ( $\rho$ ) allows it to model jagged or flat noise trajectories, capturing extreme variability that the HCDM (which assumes moderate spread) cannot.
- Goodness-of-Fit (GOF): The $\Delta$ GOF (difference in fit) aligned perfectly with the PMPs; participants favoring BDDM showed lower prediction errors for extreme rating distributions.
Limitations: Both models struggled with multimodal, clustered distributions (e.g., participants anchoring strictly on "neutral" or extreme endpoints), violating the models' continuity assumptions.

5. Significance and Implications

New Toolkit for Psychologists: This work provides a user-friendly toolkit for modeling the cognitive dynamics of continuous self-reports, moving beyond simple categorical or binary analyses.
Computational Trade-off: The study highlights a trade-off between flexibility and efficiency. The BDDM offers superior fit for complex data but requires significantly more computational resources (400 mins vs. 13 mins for training) due to its discretized 101-accumulator architecture.
Future Directions: The authors suggest future work should address discrete/anchored response strategies (e.g., via zero-and-one inflated models) and link these parameters to neural measures.
Generalizability: The use of Amortized Bayesian methods demonstrates that complex, non-standard EAMs can be effectively deployed in real-world research without requiring closed-form likelihood solutions.