Bayesian inference for models with recursively defined likelihoods is computationally demanding, limiting scalability to large datasets. We propose a stabilised weighted subsampling methodology for accelerated Bayesian inference based on an unbiased estimator of the log-likelihood. By assigning higher sampling probabilities to early observations, the method reduces the effective depth of recursive likelihood evaluations and hence computational cost. This creates a challenging variance-cost trade-off: sampling probabilities that decay too slowly yield limited computational savings, while overly aggressive decay can substantially inflate estimator variance. We develop a stabilisation framework, supported by theory, that controls this trade-off and leads to a practical, theory-informed tuning strategy. We also derive an unbiased subsampling estimator of the log-likelihood gradient, enabling gradient-based Bayesian inference. The methodology can be embedded within a range of Bayesian computational algorithms. We illustrate its use in variational Bayes and subsampling Markov chain Monte Carlo for conditional volatility models, including leverage effects. Empirical results show substantial computational speed-ups relative to full-data methods while maintaining inferential accuracy. We also compare with recent stochastic gradient MCMC and divide-and-conquer MCMC methods for temporally dependent data, observing favourable empirical performance.

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