The theory of priors has been proved as one of the theories on the regression coe cient of the normal linear models for testing hypotheses and construct the model selection when proper priors are considered for the simpler null hypotheses in the computations of Bayes factors. For the class of normal linear models, it has long been known that for comparison pairwise of nested models in the frequency senses a decision based on the Bayes factors produces a consistent model selection Casella and Moreno (2009). In this sense, the Multivariate Cauchy prior used in Zellner and Siow (1980), the intrinsic priors and Smooth Cauchy priors introduced by Berger and Pericchi (1996) , the power expected-posterior (PEP) priors have been introduced and developed by Fouskakis, Ntzoufras and Draper (2015) in objective Bayesian approach to study the consistency. The consistency is one of the important reasons that Berger and Pericchi (2001) discuss the advantage and motivation for using a Bayesian approach to the model comparison problem over the classical approach. In this consideration, Casella et al (2009) proved that the Bayes factor for intrinsic priors is consistent when the number of parameters does not grow with the sample size n: Moreno et al (2010) proved the Bayes factor for intrinsic prior is consistent for the case where the number of parameters of both models is of order O(nb) for b < 1 and for b = 1 is consistent except for a small set of alternative model. By El as Moreno, F. Javier Gir on and George Casella (2010) the properties of the Bayes factor is not totally understood, they show that Schwartz approximation and Bayes factor are not asymptotically equivalent. They nd the Schwartz approximation and the Bayes factor for intrinsic priors are consistent when the rate growth and the dimension of the bigger model is O(nb) for b < 1; for the case where b = 1 Schwartz approximation is always inconsistent under the alternative model while the Bayes factor for intrinsic priors is consistent except for a small set of alternative model. In order to produce a minimally informative priors that reduce the e ect of the training sample on the PEP approach, by combining the ideas from the power prior and unitinformation- prior methodologies Dimitris Fouskakis and Ioannis Ntzoufras (2016) show hat the Bayes factor under the J-PEP is consistent under very mild conditions on the design matrix. Also, they prove that the asymptotic of J-PEP Bayes factor is equivalent to those Bayesian Information criteria (BIC) ensuring consistency of the PEP approach to model selection. As the BIC was developed as an asymptotic approximation to the Bayes factor between models, this approximation is valid only under certain conditions and then Stone (1979) proved that BIC can fail to be asymptotically consistent and that by a counterexample, he shows a situations in which the BIC is not an adequate approximation. Here for the case where the linear models are nested, we examine the Bayes factors for the Intrinsic Priors, Je reys Power Expected Posterior Priors, Multivariate Cauchy Priors and Smooth Cauchy Priors. We show that they are consistent with the case where the dimension of both models selected are xed, they are consistent except for a small set of alternative model when the dimension of the reduced model is xed and the dimension of the full model increases as the sample size grows to in nity. The reason for choosing the theme Bayes Factors Consistency for Nested Linear Models with Increasing Dimension is to explore the Bayesian procedures for variable selection by computing the Bayes factor under the new priors concept when the number of parameters increases as the sample size grows to in nity, using the methods of encompassing to study the validity of its asymptotic consistency by comparing nested models of regression normal linear models. Hence, the first chapter of the thesis is: Bayes Factor for Nested Linear mod- els includes a statement of the problem, the de nition and presentation of the Bayes factors for all the priors we examine in the thesis, the materials, and theorems, that will be related to its applications in the studies of the consistency of the Bayes factors under the priors de ned in the chapters below. The second chapter is Approximation to the Bayes factors, we introduce the Stirling's approximation, Laplace approximation, we de ne the Stirling's approximation and Laplace approximation to the Bayes factors, we approximate our Bayes factors by using these approximations, which will be applied to study and nd simulation of the Bayes factors under the priors presented in the chapter below. In the third chapter, the chapters 4, and 5 are entitled Consistency of the Bayes Factor under Intrinsic Prior, Consistency of the Bayes Factor under Je reys Power Expected Posterior and Consistency of the Bayes Factor under Multivariate Cauchy and Smooth Cauchy respectively, we study the validity of the consistency of the Bayes factors under these priors, in each case we presented a table that summarizes the study of the consistency. In the chapter 6: Simulations and E ciency of the Bayes factors, we present the results of some simulations by using the appropriate approximations of the chapter 2. At the end of each of the chapters 3, 4, and 5 are shown a series of discussions, concerning the validity of the consistency. As the conclusion of the thesis, we present a discussion which is summarized in chapters three, four and five. Possible future investigations are also indicated and that the Ph.D. thesis ends with a set of references articles or books relevant to this thesis. The contributions of the thesis We Examine the consistency of the Bayes factors for nested normal linear models for which the number of regressors is equal to its dimension, the number of parameters of the simple model is always xed and the full model is of order O(nα ); where α ϵ [0, 1] and the dimension of the full model increases with a rate as the sample size grows to in finity. Completely New Let dim(Mi) = i = O(1) be the dimension of the reduced model Mi. Let dim(Mp) = p = O(n α ); α ϵ [0, 1] be the order of the full model Mp. We take the sample size n equal to the training sample n * and the power 1/dj is so that dj ϵ {n,n - p,p,d}, where d is a non negative natural number. We de ne and nd the Stirling's approximation to the Bayes factors for Je reys Power Expected Posterior ( J-PEP ) priors and then use it to study the consistency of the Bayes factor for J-PEP. We examine the consistency of the Bayes factor for Je reys Power Expected Posterior priors by showing that it is consistent when the power is so that dj ϵ {n,n-p}, for the case where the number of parameters of the full model is of order O(n α); where 0 ≤ α < 1, and for α = 1 it is inconsistent for a set of the alternative model. When the power is so that dj = p; the Bayes factor for Je reys Power Expected Posterior prior is inconsistent under the reduced model and consistent under the full model for the case where the number of parameters of the full model is of order O(n α ); where 0 ≤ α < 1; and for = 1 it is inconsistent for a set of the alternative model. The Bayes factor for J-PEP is inconsistent for a small set of the alternative model, when the power is so that dj = d is a large non-negative natural number, for the case where the number of parameters of the full model is of order O(nα) and α = 1. We illustrate the results by simulating the Stirling's approximation to the Bayes factors for J-PEP under the considered situations that will be de ned in the chapter 6. Partially New A new consistency proof to the Bayes factor for J-PEP when the dimension of both models is a xed number of parameters. Extension of the theorem. 2 in the page 1942 from Moreno, E. , Gir on, F. J. and Casella, G. ( August 2010) Consistency of Objectives Bayes Factors as the Model Dimen- sion Grows, Annals of Statistics. Vol 38, No 4, pp 1937-1952 , by adding the case where the limit of the distance between the both models is δ(r) = r-1/(1+r)[r-1/r]-1 for large value of r and show that the Bayes factor for intrinsic prior is also consistent under the full model, when the models increase their number of parameters with rate i = O(nα),p = O(n); where 0 ≤ a < 1; and n ≈ rp: Generalization of the theorem. 3.2 in the pages 245-246 by jumping the dimension 1 to the dimension i = O(1) of the reduced model from Berger, J. O. , Ghosh, J. K. and Mukhopadhyay, N. (2003) Approximations and Consistency of Bayes Factors as Model Dimension Grows, Journal of Statistical Planning and Inference 122, 241-258. MR1961733. New Presentation of Results Use of Laplace approximation to the Bayes factors for Intrinsic priors. Use of Laplace approximation to the Bayes factors for Multivariate Cauchy priors and Smooth Cauchy priors instead of using either the BIC or High Dimension Approximation from Berger, J. O. , Ghosh, J. K. and Mukhopadhyay, N. (2003). Use of Stirling's approximation to the Bayes factor for J-PEP, even when Laplace approximation is valid for p fixed.