## 5 Belief Model
#### NUTS (No-U-Turn Sampler) Belief Model Update  
The calibration of parameters within the Hamiltonian Monte Carlo (HMC) method, particularly the step size \(\epsilon\) and the number of steps \(L\), necessitates considerable expertise and practical experience from the user. The No-U-Turn Sampler (NUTS) [20] algorithm, a refinement of the HMC framework, automates this process, thus enhancing efficiency and user-friendliness. NUTS improves upon HMC by implementing a recursive algorithm that self-determines the appropriate number of steps, integrating a slice variable \(u\) to facilitate a more streamlined computational procedure. As an advanced Markov chain Monte Carlo (MCMC) technique, NUTS is tailored to boost sampling efficiency from complex target distributions. It diverges from the traditional HMC in that it eliminates the need for manual specification of the step parameter by introducing an adaptive mechanism for path length determination. This innovation markedly reduces the demands for manual tuning, leading to a more efficient sampling process and elevated algorithmic efficacy in intricate probabilistic models. The NUTS algorithm leverages a Hamiltonian dynamical system to generate samples, systematically selecting an initial state and traversing the Hamiltonian dynamics trajectory in both forward and reverse directions. It adaptively modifies the path length to maintain optimal efficiency and promote convergence. This process can be conceptualized as dynamically mapping a route within the state space, where the route's length is adjusted in response to the geometric properties of the target distribution. NUTS operates by commencing at an initial state and recursively building a collection of potential candidate points. It advances via Hamiltonian dynamics steps, randomly choosing between forward and backward directions, and extends the path to compile a set of prospective samples. During this process, the NUTS algorithm applies a No-U-Turn Sampler criterion--a decision heuristic based on the energy change along the trajectory--to discern when to halt path extension and avoid fruitless backward movements. This heuristic ensures that the algorithm minimizes unnecessary computation and steps while covering the target distribution effectively. By employing a recursive algorithm to construct the candidate points set, NUTS does away with the manual specification of step count parameters that characterize conventional HMC approaches. This feature significantly reduces the labor involved in parameter tuning, ensuring greater efficiency and rapid convergence. Consequently, NUTS has gained extensive recognition as a powerful and accessible MCMC algorithm, finding wide application in Bayesian statistical inference and machine learning. The No-U-Turn Sampler (NUTS) as shown in Algorithm 1 is an extension of Hamiltonian Monte Carlo (HMC) that addresses the choice of a suitable number of steps. NUTS uses a recursive algorithm to automatically determine the trajectory length so that it stops before the simulated path turns back on itself, which is the \"No U-Turn\" condition. The procedure starts by initializing the first sample, choosing a suitable step size, \(\epsilon\), and a maximum tree depth, \(D\). For each iteration in the Markov Chain, the following steps are taken:  
1. A momentum variable is sampled from a normal distribution with the same dimension as the parameter space. 2. Starting from the current position, the algorithm begins to construct a binary tree. For each level of the tree depth:The direction of exploration is decided by sampling uniformly from \(\{-1,+1\}\). * The tree is expanded in the chosen direction by simulating Hamiltonian dynamics for a series of leapfrog steps that double with each level of depth. * At any point during the tree expansion, if a U-turn is detected, which is determined by the scalar product of the momentum and the displacement being negative, the expansion stops. * The maximum depth of the tree is limited to prevent excessively long trajectories. 3.