Alzheimer's disease (AD) can be diagnosed by utilizing traditional logistic regression models to fit magnetic resonance imaging (MRI) data of brain, which is regarded as a vector of covariates. But its parameter estimation is inefficient and computationally extensive due to ultrahigh dimensionality and complicated structure of MRI data. To overcome this deficiency, this paper proposes a tensor logistic regression model (TLRM) for AD's MRI data by regarding MRI tensor as covariates. Under this framework, a tensor candecomp/parafac (CP) decomposition tool is employed to reduce ultrahigh dimensional tensor to a high dimensional level, a novel Bayesian adaptive Lasso method is developed to simultaneously select important components of tensor and estimate model parameters by incorporating the Plya-Gamma method leading a closed-form likelihood and avoiding the usage of the Metropolis-Hastings algorithm, and Gibbs sampler technique in Markov chain Monte Carlo (MCMC). A tensor's product technique is utilized to optimize the calculation program and speed up the calculation of MCMC. Bayes factor together with the path sampling approach is presented to select tensor rank in CP decomposition. Effectiveness of the proposed method is illustrated on simulation studies and an MRI data analysis.
Keywords: Bayesian adaptive Lasso; MRI data; Póly-Gamma distribution; tensor CP decomposition; tensor logistic regression.