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This paper develops a generalization of the balanced truncation algorithm applicable to a class of discrete-time stochastic jump linear systems. The approximation error, which is captured by means of the stochastic L2 gain, is bounded from above by twice the sum of singular numbers associated to the truncated states, similar to the case of linear time-invariant systems. A two step model reduction algorithm for hidden Markov models is also developed. The first step relies on the aforementioned balanced truncation algorithm due to a topological equivalence established between hidden Markov models and a subclass of stochastic jump linear systems. In a second step the positivity constraints, which reflect the hidden Markov model structure, are enforced by solving a low dimensional optimization problem. [ABSTRACT FROM AUTHOR]