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In many scientific experiments involving the multi-arm bandits, the data is collectedand are genuinely dependent. As a result, many commonly used the statistical inference methods could be problematic. For example, the ordinary least square estimator, which is widely used in the regression literature, will produce a biased estimator in the contextual disjoint linear models. As a result, any further statistical inference methods such as the hypothesis testing and confidence interval based on this biased estimator could be either invalid or conservative. We develop approaches in two stages: pre-data bias mitigation (pre-BM) and post-data bias mitigation (post- BM) to correct this. In Chapter 2, we propose an alternative approach named the randomized Multi-Arm Bandits (rMAB) that combines a randomization step with a chosen MAB algorithm. The proposed rMAB can achieve the optimal regret asymptotically if choosing randomization probability appropriately. It is shown numerically that the magnitude of the bias of the sample mean based on the rMAB is substantially smaller than that of competing methods. In Chapter 3, we first explicitly derive the exact bias formula for a family of estimators. It is shown that the bias term depends on the average number of the times that a particular arm is pulled and the covariance between the estimator and this number. To get a data-driven version method, we introduce the RBA, a Resampling-based Bias Adjustment method, to calculate this bias term. It is numerically shown that the RBA performs better than its competitors. / Statistics

In modern statistical applications when the dimension is relatively large, it is a common practice to reduce the dimension using methods such as principal component analysis (PCA), sliced inverse regression and others before applying any statistical models. In this article, we synthetically combine these two steps by considering three Bayesian multi-index models: Bayesian multi-index additive model (BMIAM) for continuous response variable, Bayesian single-index model for binary response variable, and Bayesian multi-index model for categorical response variable. The indexes are parametrized by the hyper-spherical coordinates. The ridge functions are modeled using the Bayesian B-splines, which could be easily extended to other non-parametric methods. We have shown that the posterior consistency holds under certain conditions for the BMIAM. Further, we have developed the Markov chain Monte Carlo (MCMC) algorithm to sample the posterior of the proposed methods. It has been demonstrated through both simulation and real data analysis that the proposed methods provide a reliable estimation of indexes, dimension reduction space and good predictions for the responses. / Statistics

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