1. Introduction
Understanding driver behaviour is a cornerstone of improving road safety, optimising traffic management, and developing autonomous vehicle systems. Human drivers exhibit complex decision-making processes influenced by numerous factors with a high degree of uncertainty, including environmental conditions, traffic scenarios, and psychological states among others. To reflect human behaviour, a probabilistic model is required that can both account for its complexity and capture the uncertainty associated with it. Bayesian networks (BNs) provide a robust framework to address this challenge.
A BN1 is a probabilistic model within a graphical framework to represent stochastic relationships between random variables. These variables are shown as nodes, and the edges between them encode probabilistic conditional dependencies, allowing the BN to assess probabilities associated with different outcomes based on data. From a modelling perspective, BNs provide a flexible framework for inference and prediction, because as Pearl and Russell2 state “they are direct representations of the world, not of reasoning processes”. By incorporating probabilistic reasoning, BNs are powerful tools for modelling complex systems. This approach is particularly suitable for analysing driver behaviour, as it can effectively accommodate the unpredictability inherent in human decision-making processes.3
A driver behavioural model (DBM)4 is a broad framework that integrates data from various sources, such as sensors, traffic systems, or driver profiles, to examine how different factors (such as speed, road conditions, and distractions) influence driver actions. This research is a part of the BERTHA EU project (BEhavioural ReplicaTion of Human drivers),5 which aims to develop a comprehensive modular DBM capable of explaining and predicting human driving behaviour in a wide range of contexts relevant for connected, cooperative, and automated mobility (CCAM). The DBM proposed in BERTHA integrates multiple dimensions of human driving—including cognitive, affective, perceptual, and contextual factors—within a unified probabilistic framework. By combining empirical data from controlled experiments and real-world driving, BERTHA seeks to capture the inherent variability of human driving.
In this paper, we focus on a part of the affective module of the DBM of the BERTHA project, designed to capture and model the emotional and cognitive states that influence the behaviour of a driver. Specifically, we will discuss the intensity of some mental states, such as active fatigue and mental load, that significantly impact a driver’s response in various scenarios. Furthermore, the model in this paper could easily be extended to explore how mental states influence driving actions, offering a more comprehensive understanding of driver behaviour under dynamic and uncertain conditions.
The remainder of this paper is organised as follows. Section 2 outlines the foundational principles of BNs in a Bayesian inferential context, details the database for the BN model and presents its structure. Section 3 presents the results in general terms, illustrating them with predictive assessments of active fatigue and mental load. Finally, Section 4 briefly discusses the extension of the model and areas for further research.
2. Methods
2.1 Bayesian networks and Bayesian inference
BNs are probabilistic models whose stochastic behaviour can be depicted graphically by using directed acyclic graphs (DAGs). Graphs are mathematical structures that can be used to connect pairs of nodes by edges. These elements are random variables and probability distributions in probabilistic DAGs, respectively. In addition, DAGs must adhere to two general properties: all edges must have a probabilistic direction, and the graph must be acyclic. In a DAG, the parents of a node are all those nodes in the network structure that have directed edges pointing to it.
Figure 1 illustrates the DAG corresponding to the set of random variables
, where the random variables
and
are the parents of
(pa
and
is the only parent of
(pa
). The four variables are connected through three pointed edges that represent the probability distribution of
given its parents and that of
given
. Nodes
and
have no parents.
Figure 1. Basic direct acyclic graph example.
Probabilistic models that depend on unknown parameters
need the statistic framework to estimate them. In this paper, this will be the Bayesian inferential methodology. This approach requires the full specification of a probability distribution for all random variables in the model, together with a prior distribution for all the unknown parameters. Formally, a BN2 is a joint probability distribution
over a set of random variables
, representing the node set of the network, and a vector
of their subsequent parameters as follows:
where
is the conditional distribution of
given
and
is a prior distribution for
. The BN satisfies the factorization property. This means that the conditional joint probability distribution
decomposes in terms of the univariate conditional distributions
being
the conditional distribution of node
given its parents and
. Data
are the food of the statistical model that updates the information about
through Bayes’ theorem to obtain its posterior distribution
. The computation of this posterior distribution is approximated via Markov chain Monte Carlo (MCMC) simulation.
6 This results in a subsequent approximate random sample that allows to approximate the joint posterior predictive distribution
from which any posterior predictive marginal or conditional distribution of interest can be approximated.
2.2 Data structure and variables
Data for the construction of a BN for the affective module of the DBM came from Instituto de Biomecánica de València (IBV), the leading institution of the European BERTHA project. IBV performed an experimental study employing a driving simulator to characterise key mental states in car drivers. Mental workload and active fatigue were evaluated through a structured protocol combining subjective assessment with multimodal physiological monitoring. Participants completed validated mental workload and active fatigue scales at baseline and after each cognitive block, while physiological activity was recorded using ECG and respiratory signals via the PLUX Biosignals platform.
The resulting dataset encompasses a collection of demographic and physiological variables, together with responses to experimental questionnaires specifically designed to quantify mental load and active fatigue. The dataset comprises observations collected from 17 participants, with the following basic covariate information:
•
: Sex; 8 women (
) and 9 men (
).
•
: Age, in years; ranging from 24 to 61 (mean: 44.59 years).
•
: Body mass index; ranging from 17.65 to 35.16 (mean: 24.35).
Physiological information was assessed through variables related to heart and respiration function:
•
: Standard Deviation of the Time elapsed between two successive normal heartbeats (RR intervals) in milliseconds (ms); ranging from 19.45 to 191.76 (mean: 63.07 ms).
•
: Standard Deviation of the successive Differences between adjacent RR intervals in milliseconds (ms); ranging from 14.38 to 104.16 (mean: 53.17 ms).
•
: Mean Heart Rate in beats per minute (bpm); ranging from 46.08 to 152.09 (mean: 73.94 bpm).
•
: Ratio of the variance of the heart rate variability (HRV) in the Low-frequency to the High-frequency band; ranging from 0.17 to 29.91 (mean: 3.51).
•
: Mean of the Number of Breaths per minute; ranging from 9.42 to 23.38 (mean: 15.26 breaths/min).
Following the experimental tasks, participants also completed a series of questionnaires regarding mental load, (
), and active fatigue, (
). Original responses were transformed into a binary format with two outcomes: 1, indicating that the participant entered the specified mental state, and 0 otherwise.
2.3 A Bayesian network for assessing active fatigue and mental load in drivers
The model was formulated as a BN representing the probabilistic relationships among mental states, physiological responses, covariates and parameters.
Figure 2 presents the DAG of the analysed BN. It illustrates the relationships among mental load (
), active fatigue (
), and the set of physiological variables:
,
,
,
, and
. For simplicity, covariates and parameters have been omitted from the graph. The nodes corresponding to the two variables of interest,
and
, have a direct influence on
,
, and
. The node
affects
, and both
and
contribute to
. These three variables
,
, and
are parents of
, which in turn is a parent of
. These interdependencies illustrate the complex relationships within the network. In terms of interpreting these variables, it can be stated that, according to the model, mental states influence heart function, which in turn affects respiration. The subsequent conditional joint distribution of all nodes in the network given the set
of parameters factorizes as the product
(1)
Figure 2. Bayesian Network for examining active fatigue and mental load.
For clearer visualization, the nodes have been identified exclusively by the acronyms of their subindices.
This decomposition allows each univariate conditional distribution in (1) to be modelled independently. Mental load (
) and active fatigue (
) are Bernoulli variables whose probability of success is modelled using Bayesian logistic regression (BLR). We analysed the conditional distribution of each of the continuous physiological measures,
,
,
,
and
in (1) using a Bayesian regression model (BRM) with the subsequent conditional mean expressed as a linear function of the three base covariates and the parents variables of the response variable considered. For example, in the case of the modelling of
we assume a normal distribution with mean
expressed as a linear function of the three base covariates,
,
, and
, as well as nodes
,
,
as the parents of
.
(2)
where
is an indicator variable with value 1 if the observation comes from a man and zero otherwise and
’s are the regression coefficients, parameters, associated with the covariates.
The prior distributions for the model parameters
assume prior independence in a non-informative environment that gives all inferential prominence to the data. All regression coefficients
’s are assigned normal distributions with mean 0 and variance 25 in all BLR and BRM models. The standard deviation parameters of the BRM models are elicited as uniform distributions in the interval
.
3. Results
The posterior distribution
for all parameters of the BN was approximated by MCMC through JAGS software. The subsequent posterior MCMC sample was used to approximate the joint posterior predictive distribution. This results in a sample from which we can extract the behaviour of any predictive element of interest.
By way of illustration,
Figure 3 shows the average posterior probability that a driver exhibits each of the four conditions defined by active fatigue and mental workload in relation to their gender, age, and BMI, and the range of possible values for the
and
nodes. In a real application, conditional posterior probabilities could be derived; for example, for a female driver aged 20 with a BMI of 22 and a mean breathing rate of 20 breaths per minute, the probability of experiencing mental workload is estimated to be approximately 0.72. Predictions seem to indicate that age has a protective effect against active fatigue and mental workload, but not gender, which does not appear to be relevant.
Figure 3. Average posterior probability of each joint state of active fatigue (AF) and mental workload (ML) as a function of driver gender and age, with BMI held constant at 22, shown across
and
values.
4. Discussion
The ability of BNs to estimate probabilities associated with specific mental states, such as mental load or active fatigue, under different physiological conditions of the driver is an important aspect of this study. This information would enable real-time assessments of the driver’s mental state, with important implications for road safety.7 For example, a vehicle could suggest a break to a driver showing signs of mental fatigue. This model could help optimize the design of systems to prevent accidents caused by cognitive fatigue or mental overload.
The results of this study show promising results, but the complexity of the subject requires further research and more data. They were obtained from a fairly small sample, and it would be desirable to work with larger samples that could provide greater power and predictive efficacy. It could be interesting to explore additional markers, such as facial expressions, which could provide additional information about the driver’s mental state.
Ethics and consent
Ethical approval for this study was obtained from the Research Ethics Committee of the Universitat Politècnica de València (Approval no. P01_28-05-2024 and P06_23-10-2024). All procedures involving human participants complied with the ethical standards of the institutional research committee and with the Declaration of Helsinki (https://www.wma.net/policies-post/wma-declaration-of-helsinki/). All participants were adults and provided written informed consent prior to their participation in the study.
