# Bayesian Matrix Learning by Principle Eigenvector for Completing Missing Medical Data
## 1 Introduction  
In daily life, the doctor judges the patient's disease according to the symptoms monitoring and then formulates a medication plan for the patient based on the curative effect of different medication regimens. However, this process is mainly based on the statistical data of past cases [1; 2; 3]. The accuracy of the medication regimen directly affects the development of the patient's condition and is the most critical link in the treatment process, while due to the diversity of diabetes complications and the diversity of diabetes-related drugs, the statistics from clinical tests often include missing values [4; 5; 6]. Completing these missing values based on the rules of the datasets can greatly enhance the accuracy of doctors' judgments. The research on missing value completion in the existing literature mainly focuses on the low-rank matrix completion methods: [PERSON] et al. applied low-rank SVD to advance the current matrix completion algorithms and developed a new algorithm that can deal with the Netflix competition data [7]. [PERSON] et al. used matrix completion to provide full data available for Computer Aided Diagnosis using machine learning [8]. [PERSON] et al. proposed a Matrix Completion for Planning Diabetes Treatment by using nonlinear convex optimization to generate information [9]. [PERSON] proposed two new estimators, based on singular value threshold and nuclear norm minimization, to recover the sparse matrix under this assumption [10]. However, in medical cases, the causal relationship between different symptoms and multiple medication regimens forms a full-rank Bayesian matrix, so the previous low-rank matrix completion methods are no longer applicable. Therefore, this paper develops a different matrix completion algorithm assuming the matrix has a full rank to provide better support for the choice of diabetes medication plan. This article establishes a method to complete a probabilities matrix by one eigenvector. Through actual case statistics, we find that it is easier to obtain the eigenvector with the DTBM eigenvalue of 1, that is, the probability of different symptoms in diabetic patients and the probability of each medication regimen being used. The data isobtained from medical tests after decades of sample statistics, which are the stable probability distributions of the symptoms and medication regimens in people with diabetes, and it mathematically conforms to the stable state theory of the Markov matrix. Based on the above information, we give the matrix completion algorithm for the DTBM, which accurately completes the 56 missing values by one eigenvector in Section 6. At the same time, the 56 completed values can be fixed into the two blocks of the DTBM, one block is the probabilities of medications given multiple symptoms, and another is the probabilities of symptoms given medications. The new algorithm can complete an empty matrix from one eigenvector which can help doctors make a medication plan with limited information. This original work completes a DTBM with 56 missing data representing relations between symptoms and medications to support doctors making medical plans. The remainder of this article is organized as follows. In the second Section, this paper will introduce and explain two types of sample data about diabetic symptoms and regimens. Also, the article mentions the collecting and processing of these two kinds of sample data that are separated into causes and effects in one matrix. Section 3 will analyze the properties of the principle eigenvector of the DTBM for computing missing data in the sample data. Section 4 will demonstrate the method for completing missing data of one column in the DTBM using the principal eigenvector of the DTBM.