Transportation systems — from urban traffic networks to autonomous vehicles — generate massive streams of sensor data that must be interpreted under uncertainty. Cameras misidentify objects, GPS signals bounce off buildings, radar returns are ambiguous, and the behavior of other road users is unpredictable. Bayesian methods provide the mathematical framework for combining these noisy information sources into coherent probabilistic estimates of the state of the world, enabling decisions that account for what is known and what is not.
Bayesian Sensor Fusion
Autonomous vehicles rely on multiple sensor modalities — LiDAR, cameras, radar, ultrasonic sensors, GPS/IMU — each with different strengths, failure modes, and noise characteristics. Bayesian sensor fusion combines these modalities into a unified posterior estimate of the vehicle's environment: where are the other vehicles, pedestrians, cyclists, and obstacles, and how are they moving?
Where state = {positions, velocities, classes of all objects}
zᵢ = observation from sensor i (LiDAR point cloud, camera detection, radar return)
P(zᵢ | state) = sensor-specific observation model
The extended Kalman filter, unscented Kalman filter, and particle filter provide the computational machinery for sequential Bayesian fusion, tracking objects over time as new sensor data arrive at rates of 10-100 Hz. Each sensor observation updates the posterior, and the posterior uncertainty drives downstream decisions about path planning and collision avoidance.
Simultaneous Localization and Mapping (SLAM)
SLAM — the problem of building a map of the environment while simultaneously localizing within it — is fundamentally Bayesian. The vehicle maintains a joint posterior distribution over its own pose (position and orientation) and the locations of landmarks and obstacles. As the vehicle moves and observes, this posterior is updated sequentially. Graph-based SLAM, FastSLAM, and other modern algorithms are all approximate inference methods for this Bayesian problem, trading exactness for computational tractability.
Occupancy grid maps divide the environment into cells and maintain a posterior probability of occupancy for each cell. As LiDAR sweeps through the environment, cells hit by beams have their occupancy probability increased, and cells that beams pass through have their occupancy probability decreased — a direct application of Bayes' theorem. The resulting probabilistic map represents not just where obstacles are, but how confident the system is about each location, enabling conservative navigation through uncertain areas.
Traffic State Estimation
Urban traffic management uses Bayesian methods to estimate traffic density, speed, and flow from heterogeneous data sources — loop detectors, floating car data, Bluetooth sensors, video cameras. Bayesian data assimilation combines these observations with traffic flow models (cell transmission model, kinematic wave model) to estimate the current traffic state and predict its short-term evolution. The posterior distribution of traffic conditions drives adaptive signal control, route guidance, and congestion pricing.
"An autonomous vehicle does not see the world — it maintains a probabilistic belief about the world, updated with every sensor reading. Safe autonomy requires not just accurate beliefs but honest uncertainty — knowing what you do not know." — Sebastian Thrun, pioneer of autonomous vehicles
Behavior Prediction and Planning
Predicting the future behavior of other road users — will the pedestrian cross? will the car change lanes? — is inherently uncertain. Bayesian models maintain a posterior distribution over the intentions and goals of other agents, updated by observing their actions. Inverse reinforcement learning recovers the reward functions that explain observed driving behavior, and Bayesian planning algorithms select actions that perform well across the range of plausible futures represented by the posterior.
Safety Assurance and Validation
Demonstrating the safety of autonomous vehicles requires showing that the system's failure rate is acceptably low — but this is a rare-event estimation problem. Bayesian methods combine simulation data, test track data, and on-road experience to estimate the posterior distribution of crash rates, accelerating validation by incorporating prior knowledge from component-level testing and related systems. The posterior probability that the system meets a safety threshold (e.g., fewer than 1 fatal crash per 100 million miles) directly informs regulatory approval decisions.