This paper addresses a speech recognition problem in non-stationary noise environments: the estimation of noise sequences. To solve this problem, we present a particle filter-based sequential noise estimation method for front-end processing of speech recognition in noise. In the proposed method, a noise sequence is estimated in three stages: a sequential importance sampling step, a residual resampling step, and finally a Markov chain Monte Carlo step with Metropolis-Hastings sampling. The estimated noise sequence is used in the MMSE-based clean speech estimation. We also introduce Polyak averaging and feedback into a state transition process for particle filtering. In the evaluation results, we observed that the proposed method improves speech recognition accuracy in the results of non-stationary noise environments a noise compensation method with stationary noise assumptions.