An i-vector has become the state-of-the-art algorithm for text-independent recognition. Most of related works take the extraction of the i-vector as a black-box by using some open software (e.g. Kaldi, Alize) and focus on the vector-based back-end algorithms, such as length normalization, WCCN, or PLDA. In this paper, we study the variational method and present a concise derivation for the i-vector. Based on our proposed methods, three criteria for derivation are compared. There are maximum likelihood (ML), maximum a posteriori (MAP) and maximum
marginal likelihood (MML) criterion respectively. Experimental results on the NIST SRE08 tel-tel-English condition task proved our works.