We consider the problem of estimation and structure learning of high dimensional signals via a normal sequence model, where the underlying parameter vector is piecewise constant, or has a block structure. We develop a Bayesian fusion estimation method by using the Horseshoe prior to induce a strong shrinkage effect on successive differences in the mean parameters, simultaneously imposing sufficient prior concentration for non-zero values of the same. The proposed method thus facilitates consistent estimation and structure recovery of the signal pieces. We provide theoretical justifications of our approach by deriving posterior convergence rates and establishing selection consistency under suitable assumptions. We also extend our proposed method to signal de-noising over arbitrary graphs and develop efficient computational methods along with providing theoretical guarantees. We demonstrate the superior performance of the Horseshoe based Bayesian fusion estimation method through extensive simulations and two real-life examples on signal de-noising in biological and geophysical applications. We also demonstrate the estimation performance of our method on a real-world large network for the graph signal de-noising problem.

翻译：我们考虑了通过正常序列模型对高维信号进行估计和结构学习的问题,在正常序列模型中,基本参数矢量是小数不变的,或具有块状结构。我们开发了一种贝叶斯混合估计方法,在使用马蹄之前,对平均参数的相继差异产生强烈的缩小效应,同时对非零值规定足够的先前集中,从而有利于对信号片进行一致估计和结构恢复。我们通过得出后游趋同率和在适当假设下建立选择一致性,为我们的方法提供了理论依据。我们还扩展了我们所建议的方法,在提供理论保证的同时,对任意图形进行分辨,并开发高效的计算方法。我们通过广泛的模拟和生物和地球物理应用中信号分辨的两个真实生命实例,展示了基于马丘的波斯聚评估方法的优异性。我们还展示了我们方法在图形信号脱氮问题真实世界大型网络上的估计性能。