Use Bayesian Inference to measure uncertainty and Lipschitz to cap consequences of that uncertainty
The issue with what I see as Bayesian Inference to mimic general intelligence is that it doesn't have a means to focus on an issue in a dynamic or active way. I discovered Lipschitz robustness which I believe solves this issue.
Real-world environments are messy. Sensors are noisy, objects move unpredictably, and one bad control command can mean a collision or instability. To build robots that can adapt to any domain, we need both:
- Bayesian inference — to predict what is likely to happen.
- Lipschitz robustness — to guarantee what is allowed to happen.
This post shows how combining these two ideas gives us robots that are both adaptive and provably safe.
Lipschitz Robustness: Motion with Certified Safety
Lipschitz continuity ensures that small changes in the input cannot cause huge jumps in the output:
‖f(x') − f(x)‖ ≤ L · ‖x' − x‖In control terms, it lets us say:
- If my state estimate or sensor reading is off by a little,
- then the controller’s output will only change by a bounded amount.
Now think about obstacles. We can wrap each obstacle in an expanded safety zone that accounts for all possible disturbances and estimation errors.
If the robot’s next step would enter this expanded region, the Lipschitz-safe controller simply rescales or redirects the motion until the step is certified safe. No step is executed unless it’s safe in the worst case.
Bayesian Inference: Predicting What’s Likely
Bayesian inference gives robots a principled way to maintain beliefs over hidden states (like a target’s true position and velocity). Instead of a single guess, the robot carries a distribution over possibilities.
With a Bayesian filter (such as a Kalman filter), the robot can:
- Fuse noisy sensor measurements.
- Update beliefs with each new observation.
- Predict where a moving target is likely to be at the next time step.
This predictive ability is crucial for tracking people, other robots, or dynamic obstacles.
Two-Brain Architecture: Prediction + Safety
We end up with a neat two-layer architecture:
- Bayesian Predictor (e.g., Kalman filter, particle filter)
- Learns motion patterns.
- Predicts future positions with uncertainty.
- Lipschitz Safety Layer
- Takes the desired motion from the predictor.
- Enforces distance and smoothness constraints.
- Guarantees that the robot never violates safety margins.
This makes the robot both smart (Bayesian) and trustworthy (Lipschitz-safe).
Probable vs Safe Regions
In the figure above:
- The inner filled region is the high-probability Bayesian belief about where something (e.g., a target) will be.
- The outer dashed ring is the Lipschitz-certified safe zone for robot motion.
The robot’s simple policy is:
Stay inside the safe ring, while steering toward the probable region.
This pattern generalizes beautifully to warehouses, hospitals, homes, drones, and search-and-rescue tasks.
Demo: Bayesian Predictor + Lipschitz-Safe Robot
You can implement this idea in a few dozen lines of Python. The example below uses:
- A 2D Kalman filter (= Bayesian inference) to track and predict a moving target.
- A Lipschitz-style controller to ensure the robot stays a safe distance from circular obstacles.
(This is standard Python + NumPy; no fancy dependencies beyond that.)
import math
import numpy as np
from dataclasses import dataclass
# ==============================
# Kalman Filter (Bayesian predictor)
# ==============================
class ConstantVelocityKF2D:
"""
State: [px, py, vx, vy]^T
px, py: position; vx, vy: velocity
"""
def __init__(self, dt=1.0, process_var=0.05, meas_var=0.5):
self.dt = dt
# State transition
self.F = np.array([
[1, 0, dt, 0],
[0, 1, 0, dt],
[0, 0, 1, 0],
[0, 0, 0, 1],
], dtype=float)
# Observation: measure position only
self.H = np.array([
[1, 0, 0, 0],
[0, 1, 0, 0],
], dtype=float)
# Process / measurement noise covariances
q = process_var
r = meas_var
self.Q = q * np.array([
[dt**4/4, 0, dt**3/2, 0 ],
[0, dt**4/4, 0, dt**3/2 ],
[dt**3/2, 0, dt**2, 0 ],
[0, dt**3/2, 0, dt**2 ],
], dtype=float)
self.R = r * np.eye(2)
# Initialize state and covariance
self.x = np.zeros((4, 1))
self.P = np.eye(4) * 10.0 # uninformative prior
def initialize(self, p0, v0=(0.0, 0.0)):
self.x = np.array([[p0[0]], [p0[1]], [v0[0]], [v0[1]]], dtype=float)
def predict(self):
self.x = self.F @ self.x
self.P = self.F @ self.P @ self.F.T + self.Q
return self.x.copy(), self.P.copy()
def update(self, z):
z = np.array(z, dtype=float).reshape(2, 1)
y = z - (self.H @ self.x)
S = self.H @ self.P @ self.H.T + self.R
K = self.P @ self.H.T @ np.linalg.inv(S)
self.x = self.x + K @ y
I = np.eye(self.P.shape[0])
self.P = (I - K @ self.H) @ self.P
def predict_next_position(self, steps_ahead=1):
# Roll forward 'steps_ahead' times without updating
Fpow = np.linalg.matrix_power(self.F, steps_ahead)
x_pred = Fpow @ self.x
return x_pred[0, 0], x_pred[1, 0]
# ==============================
# Environment & Robot (Lipschitz-safe)
# ==============================
@dataclass
class Obstacle:
center: np.ndarray # shape (2,)
radius: float
class LipschitzSafeRobot2D:
"""
Point robot with dynamics: x_{t+1} = x_t + u_t + w_t, ||w_t|| <= d_max
We enforce: dist(x_t+u_t, obs) >= obs.radius + r_safe + d_max for all obstacles.
"""
def __init__(self, x0, max_speed=0.8, d_max=0.2, r_safe=0.3, obstacles=None):
self.x = np.array(x0, dtype=float)
self.max_speed = float(max_speed)
self.d_max = float(d_max)
self.r_safe = float(r_safe)
self.obstacles = obstacles or []
def _dist_to_obstacle(self, p, obs: Obstacle):
return np.linalg.norm(p - obs.center) - obs.radius
def _is_safe(self, x_next):
# worst-case safety margin
needed = self.r_safe + self.d_max
for obs in self.obstacles:
if self._dist_to_obstacle(x_next, obs) < needed:
return False
return True
def _project_to_safe(self, x_des):
"""
Try to scale the step down and gently redirect away from nearest obstacle
until the worst-case Lipschitz safety condition holds.
"""
step = x_des - self.x
norm = np.linalg.norm(step)
if norm < 1e-9:
return self.x.copy()
# Gentle obstacle avoidance direction (potential-field-like)
def avoidance_dir(xp):
v = np.zeros(2)
for obs in self.obstacles:
vec = xp - obs.center
dist = np.linalg.norm(vec) - obs.radius
if dist <= 0:
# inside: push strongly outwards
v += vec / (np.linalg.norm(vec) + 1e-9) * 10.0
else:
# repulsive potential decays with distance
v += (vec / (np.linalg.norm(vec) + 1e-9)) / (dist + 1e-6)
n = np.linalg.norm(v)
return v / n if n > 1e-9 else v
# Start with desired step; backtrack if unsafe
alpha = 1.0
for _ in range(50):
x_try = self.x + alpha * step
if self._is_safe(x_try):
return x_try
# Redirect away from obstacles a bit
redir = avoidance_dir(x_try)
x_try2 = x_try + 0.2 * redir # small nudge
if self._is_safe(x_try2):
return x_try2
alpha *= 0.7 # backtrack
# give up: stay put if cannot find a safe step
return self.x.copy()
def step_to(self, goal_point):
# Propose a motion toward the goal
vec = np.array(goal_point) - self.x
dist = np.linalg.norm(vec)
if dist < 1e-9:
return self.x.copy()
u = vec / dist * min(self.max_speed, dist) # clip by max speed
x_des = self.x + u
# Enforce Lipschitz-style worst-case safety margin
x_next = self._project_to_safe(x_des)
self.x = x_next
return self.x.copy()
# ==============================
# Simulation tying both together
# ==============================
def simulate(
T=40,
robot_start=(0.0, 0.0),
target_start=(6.0, 5.0),
target_vel=(0.1, -0.05),
obstacles=None,
d_max=0.2,
r_safe=0.3,
max_speed=0.8,
process_var=0.02,
meas_var=0.25,
seed=7
):
rng = np.random.default_rng(seed)
# World / robot
robot = LipschitzSafeRobot2D(robot_start, max_speed=max_speed,
d_max=d_max, r_safe=r_safe, obstacles=obstacles)
# Target ground-truth (moves with small process noise)
p_t = np.array(target_start, dtype=float)
v_t = np.array(target_vel, dtype=float)
# Bayesian predictor (KF)
kf = ConstantVelocityKF2D(dt=1.0, process_var=process_var, meas_var=meas_var)
kf.initialize(p0=p_t, v0=target_vel)
traj_robot = [robot.x.copy()]
traj_target_true = [p_t.copy()]
traj_target_pred = [p_t.copy()]
for _ in range(T):
# --- target evolves (hidden state) ---
v_t = v_t + rng.normal(0, math.sqrt(process_var), size=2) * 0.0 # keep velocity mostly constant
p_t = p_t + v_t + rng.normal(0, math.sqrt(process_var), size=2)
# --- sensor measurement ---
z = p_t + rng.normal(0, math.sqrt(meas_var), size=2)
# --- Bayesian update/predict (KF) ---
kf.predict()
kf.update(z)
p_pred_next = kf.predict_next_position(steps_ahead=1)
# --- Lipschitz-safe motion towards predicted target ---
robot.step_to(p_pred_next)
traj_robot.append(robot.x.copy())
traj_target_true.append(p_t.copy())
traj_target_pred.append(np.array(p_pred_next))
return np.array(traj_robot), np.array(traj_target_true), np.array(traj_target_pred)
if __name__ == "__main__":
obstacles = [
Obstacle(center=np.array([2.5, 1.2]), radius=0.8),
Obstacle(center=np.array([4.0, 3.0]), radius=0.9),
]
r_path, t_true, t_pred = simulate(
T=50,
robot_start=(0.0, 0.0),
target_start=(6.0, 5.0),
target_vel=(0.10, -0.05),
obstacles=obstacles,
d_max=0.2,
r_safe=0.35,
max_speed=0.8,
process_var=0.02,
meas_var=0.25,
seed=42
)
print("Final robot position:", r_path[-1])
print("Final target (true) :", t_true[-1])
print("Final target (pred) :", t_pred[-1])
Animation: Bayesian Prediction + Lipschitz-Safe Motion
You can also visualize the behavior. Here is a GIF animation showing:
- The robot (circle) moving toward the predicted target.
- The true target (x) wandering through the environment.
- An obstacle the robot must keep a safe distance from.
This is the same concept scaled up: a Bayesian brain that predicts, plus a Lipschitz safety shield that ensures nothing catastrophic happens—even when the world is uncertain.
That combination is the key to robots that can genuinely adapt to any domain while remaining trustworthy.
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