human: Given a node-centered hypergraph: <hypergraph>, where nodes represent papers and hyperedges represent groups of papers sharing a common citing paper.

Details of the hypergraph:
Center node: "Estimating L 1 Error of Kernel Estimator: Monitoring Convergence of Markov Samplers"
  Abstract: In many Markov chain Monte Carlo problems, the target density function is known up to a normalization constant. In this paper, we take advantage of this knowledge to facilitate the convergence diagnostic of a Markov sampler by estimating the L 1 error of a kernel estimator. Firstly, we propose an estimator of the normalization constant which is shown to be asymptotically normal under mixing and moment conditions. Secondly, the L 1 error of the kernel estimator is estimated using the normalization constant estimator, and the ratio of the estimated L 1 error to the true L 1 error is shown to converge to 1 in probability under similar conditions. Thirdly, we propose a sequential plot of the estimated L 1 error as a tool to monitor the convergence of the Markov sampler. Finally, a 2-dimensional bimodal example is given to illustrate the proposal, fl Bin Yu is Assistant Professor, Department of Statistics, University of California, Berkeley, CA 94720-3860. Research supported in part by the Junior Faculty Research Grant from University of California at Berkeley, grants DAAL03-91-G-007 and DAAH04-94-G-0232 from the Army Research Office, and grant DMS-9322817 from the National Science Foundation. The author is very grateful to Professors Peter Bickel and Andrew Gelman for many helpful discussions and their comments on the draft. Special thanks are due to Mr. Sam Buttrey for his help on simulation, to Professor Per Mykland and Mr. Karl Broman for commenting on the draft, and to two anonymous and two Markov samplers are compared in the example using the proposed diagnostic plot.

Connected through hyperedge HE-1 (induced by source paper "A simulation approach to convergence rates for Markov chain Monte Carlo algorithms"):
  with members:
    - "Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review"
    - "Bounding Convergence Time of the Gibbs Sampler in Bayesian Image Restoration"
    - "Theoretical rates of convergence for Markov chain Monte Carlo"
    - "Rate of Convergence of the Gibbs Sampler by Gaussian Approximation  SUMMARY"
    - "Analysis of the Gibbs sampler for a model related to James-Stein estimators"
    - "Convergence properties of perturbed Markov chains"
    - "Rates of convergence of the Hastings and Metropolis algorithms"
    - "A note on convergence rates of Gibbs sampling for nonparametric mixtures"

Connected through hyperedge HE-2 (induced by source paper "Rates of convergence of the Hastings and Metropolis algorithms"):
  with members:
    - "Bounding Convergence Time of the Gibbs Sampler in Bayesian Image Restoration"
    - "Theoretical rates of convergence for Markov chain Monte Carlo"
    - "A simulation approach to convergence rates for Markov chain Monte Carlo algorithms"
    - "Rate of Convergence of the Gibbs Sampler by Gaussian Approximation  SUMMARY"
    - "Analysis of the Gibbs sampler for a model related to James-Stein estimators"
    - "EXACT TRANSITION PROBABILITIES FOR THE INDEPENDENCE METROPOLIS SAMPLER"
    - "A note on acceptance rate criteria for CLTs for Hastings-Metropolis algorithms"
    - "Exponential Convergence of Langevin Diffusions and Their Discrete Approximations"

Question: Please tell me which class the center node belongs to. The 7 classes are: Neural_Networks, Reinforcement_Learning, Probabilistic_Methods, Theory, Genetic_Algorithms, Case_Based, Rule_Learning. Directly output the class name.

gpt: Probabilistic_Methods