No-U-Turn Sampler (NUTS) is an adaptive extension of Hamiltonian Monte Carlo that automatically tunes the trajectory length by building a balanced binary tree of leapfrog steps and stopping when the trajectory begins to turn back on itself (a "U-turn"), eliminating HMC's most critical and difficult-to-tune hyperparameter. NUTS also adapts the step size during warm-up to achieve a target acceptance rate, making it a nearly tuning-free MCMC algorithm.

Why NUTS Matters in AI/ML: NUTS removes the primary barrier to practical HMC usage—trajectory length tuning—making efficient gradient-based MCMC accessible to practitioners without expertise in sampler configuration, and enabling it as the default algorithm in probabilistic programming frameworks like Stan, PyMC, and NumPyro.

• U-turn criterion — NUTS detects when a trajectory starts returning toward its origin by checking whether the dot product of the momentum with the displacement (p · (θ - θ₀)) becomes negative, indicating the trajectory has begun to curve back and further simulation would waste computation • Doubling procedure — NUTS builds the trajectory by repeatedly doubling its length (1, 2, 4, 8, ... leapfrog steps), alternating between extending forward and backward in time; this exponential growth efficiently finds the right trajectory length without trying every possible value • Balanced binary tree — The doubling procedure creates a balanced binary tree of states; the next sample is drawn uniformly from the set of valid states in the tree (those satisfying detailed balance), ensuring proper MCMC semantics • Dual averaging step size adaptation — During warm-up, NUTS adjusts the step size ε using dual averaging (Nesterov's primal-dual method) to achieve a target acceptance probability (typically 0.8 for NUTS), automatically finding the largest stable step size • Mass matrix estimation — NUTS estimates the posterior covariance during warm-up to construct a diagonal or dense mass matrix that preconditions the Hamiltonian dynamics, matching the sampler's geometry to the posterior shape

NUTS is the breakthrough algorithm that made gradient-based MCMC practical for everyday Bayesian analysis, automatically adapting trajectory length and step size to achieve near-optimal sampling efficiency without manual tuning, establishing itself as the default MCMC algorithm in modern probabilistic programming and enabling routine Bayesian inference for complex hierarchical models.