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We propose a new type of variable-parameter hidden Markov model (VPHMM) whose mean and variance parameters vary each as a continuous function of additional environment-dependent parameters. Different from the polynomial-function-based VPHMM proposed by Cui and Gong (2007), the new VPHMM uses cubic splines to represent the dependency of the means and variances of Gaussian mixtures on the environment parameters. Importantly, the new model no longer requires quantization in estimating the model parameters and it supports parameter sharing and instantaneous conditioning parameters directly. We develop and describe a growth-transformation algorithm that discriminatively learns the parameters in our cubic-spline-based VPHMM (CS-VPHMM), and evaluate the model on the Aurora-3 corpus with our recently developed MFCC-MMSE noise suppressor applied. Our experiments show that the proposed CS-VPHMM outperforms the discriminatively trained and maximum-likelihood trained conventional HMMs with relative word error rate (WER) reduction of 14% and 20% respectively under the well-matched conditions when both mean and variances are updated.