## Abstract

A multimove sampling scheme for the state parameters of non-Gaussian and nonlinear dynamic models for univariate time series is proposed. This procedure follows the Bayesian framework, within a Gibbs sampling algorithm with steps of the Metropolis–Hastings algorithm. This sampling scheme combines the conjugate updating approach for generalized dynamic linear models, with the backward sampling of the state parameters used in normal dynamic linear models. A quite extensive Monte Carlo study is conducted in order to compare the results obtained using our proposed method, conjugate updating backward sampling (CUBS), with those obtained using some algorithms previously proposed in the Bayesian literature. We compare the performance of CUBS with other sampling schemes using two real datasets. Then we apply our algorithm in a stochastic volatility model. CUBS significantly reduces the computing time needed to attain convergence of the chains, and is relatively simple to implement.

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## Acknowledgments

This work was part of Romy E. R. Ravines’ PhD program under the supervision of Helio S. Migon and Alexandra M. Schmidt. João B. M. Pereira contributed with some applications and simulations. The work of Romy R. Ravines was supported by a grant from *Coordenação de Aperfeiçoamento de Pessoal de Nível Superior* (CAPES), Brazil. Helio S. Migon, Alexandra M. Schmidt and João B. M. Pereira were supported by *Conselho Nacional de Desenvolvimento Científico e Tecnológico* (CNPq), Brazil. The authors thank Mike West and Yuhong Wu, the editor and two anonymous reviewers’, whose comments greatly improved the presentation of the paper.

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## Appendix: Some equations for prior and posterior parameters

### Appendix: Some equations for prior and posterior parameters

Let \(r_t\) and \(s_t\) be the parameters of the conjugate prior of \(\eta _t\), that is \(\eta _t \sim CP[r_t,s_t]\). Consider \(f_t=E[g(\eta _t)|Y^{t-1},\Phi ]\) and \(q_t=var[g(\eta _t)|Y^{t-1},\Phi ]\), the prior moments obtained from the linear predictor in (1b), and let \(f_t^*\) and \(q_t^*\) be the resultant posterior moments. Assuming the most used distributions in practice, Table 6 shows the equations to be solved in order to obtain \(r_t\) and \(s_t\) as a function of \(f_t\) and \(q_t\); and \(f^*_t\), \(q^*_t\) as a function of \(r^*_t=r_t +\phi y_t\) and \(s^*_t = s_t + \phi \). In Table 6, \(\gamma (\cdot )=\frac{d}{dz} \Gamma (z)\) and \(\gamma ^{\prime }(\cdot )= \frac{d}{dz} \gamma (z)\) denote the digamma and trigamma functions, respectively. Useful recurrence relationships and approximations are: for the digamma function, \( \gamma (z) = \gamma (z+1) - \frac{1}{z}\) and \(\gamma (z) \simeq log(z) - \frac{1}{2z}\); for the trigama function, \( \gamma ^{\prime }(z) = \gamma ^{\prime }(z+1) + \frac{1}{z^2}\) and \(\gamma ^{\prime } (z) \simeq \frac{1}{z} + \frac{1}{2z^2}\). When \(z>3\) the following approximation is better: \(\gamma (z) \simeq log(z-\frac{1}{2})\). In the third column of the table we find the solution in an implicit form which needs to be solved numerically, and the following approximations for the gamma and digamma functions are used: \(\gamma (x) \approx \log (x)\) and \(\gamma ^{\prime }(x) \approx 1/x\) (see Abramovitch and Stegun 1965, pp. 258–259, for details).

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Migon, H.S., Schmidt, A.M., Ravines, R.E.R. *et al.* An efficient sampling scheme for dynamic generalized models.
*Comput Stat* **28**, 2267–2293 (2013). https://doi.org/10.1007/s00180-013-0406-9

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DOI: https://doi.org/10.1007/s00180-013-0406-9

### Keywords

- Bayesian paradigm
- Linear Bayes
- Nonlinear models
- Markov chain Monte Carlo