... universally11.1
In this chapter, the denomination universal is used in the sense of uniformly over all distributions. 
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... varies11.2
To impose the stationarity constraint when the order of the $ AR(p)$ model varies, it is necessary to reparameterise this model in terms of either the partial autocorrelations or of the roots of the associated lag polynomial. (See, e.g., [35], Sect. 4.5.) 
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... integrals.11.3
In this presentation of Bayes factors, we completely bypass the methodological difficulty of defining $ \pi(\theta \in \Theta_0)$ when $ \Theta_0$ is of measure 0 for the original prior $ \pi $ and refer the reader to Robert (2001, Section 5.2.3) for proper coverage of this issue. 
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... density11.4
The prior distribution can be used for importance sampling only if it is a proper prior and not a $ \sigma $-finite measure. 
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... methods.11.5
The constant order of the Monte Carlo error does not imply that the computational effort remains the same as the dimension increases, most obviously, but rather that the decrease (with $ m$) in variation has the rate $ 1/\sqrt m$. 
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... simulations.11.6
The empirical (Monte Carlo) confidence interval is not to be confused with the asymptotic confidence interval derived from the normal approximation. As discussed in Robert and Casella (2004, Chap. 4), these two intervals may differ considerably in width, with the interval derived from the CLT being much more optimistic! 
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... quickly.11.7
An alternative to the simulation from one $ \mathcal{T}(\nu,x_i,1)$ distribution that does not require an extensive study on the most appropriate $ x_i$ is to use a mixture of the $ \mathcal{T}(\nu,x_i,1)$ distributions. As seen in Sect. 11.5.2, the weights of this mixture can even be optimised automatically. 
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... Sect. 10.3.2).11.8
Even in the simple case of the probit model, MCMC algorithms do not always converge very quickly, as shown in [37] (2004, Chap. 14). 
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... occur.11.9
It is quite interesting to see that the mixture Gibbs sampler suffers from the same pathology as the EM algorithm, although this is not surprising given that it is based on the same completion scheme. 
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... moves.11.10
This wealth of possible alternatives to the completion Gibbs sampler is a mixed blessing in that their range, for instance the scale of the random walk proposals, needs to be scaled properly to avoid inefficiencies. 
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... distribution?11.11
Early proposals to solve the varying dimension problem involved saturation schemes where all the parameters for all models were updated deterministically ([9]), but they do not apply for an infinite collection of models and they need to be precisely calibrated to achieve a sufficient amount of moves between models. 
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....11.12
For a simple proof that the acceptance probability guarantees that the stationary distribution is $ \pi(k,\theta^{(k)})$, see Robert and Casella (2004, Sect. 11.2.2). 
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....11.13
In the birth acceptance probability, the factorials $ k!$ and $ (k+1)!$ appear as the numbers of ways of ordering the $ k$ and $ k+1$ components of the mixtures. The ratio cancels with $ 1/(k+1)$, which is the probability of selecting a particular component for the death step. 
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... distributions.11.14
The ''sequential'' denomination in the sequential Monte Carlo methods thus refers to the algorithmic part, not to the statistical part. 
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... a proposal.11.15
Using a Gaussian non-parametric kernel estimator amounts to (a) sampling from the $ x_i^{(t)}$'s with equal weights and (b) using a normal random walk move from the selected $ x_i^{(t)}$, with standard deviation equal to the bandwidth of the kernel. 
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... resampling.11.16
When the survival rate of a proposal distribution is null, in order to avoid the complete removal of a given scale $ v_k$, the corresponding number $ r_k$ of proposals with that scale is set to a positive value, like $ 1\,{\%}$ of the sample size. 
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