Abstract
If, in the mid 1980s, one had asked the average statistician about the difficulties of using Bayesian Statistics, the most likely answer would have been “Well, there is this problem of selecting a prior distribution and then, even if one agrees on the prior, the whole Bayesian inference is simply impossible to implement in practice!” The same question asked in the Twenty first Century does not produce the same reply, but rather a much less aggressive complaint about the lack of generic software (besides winBUGS), along with the renewed worry of subjectively selecting a prior! The last 20 years have indeed witnessed a tremendous change in the way Bayesian Statistics are perceived, both by mathematical statisticians and by applied statisticians and the impetus behind this change has been a prodigious leap-forward in the computational abilities. The availability of very powerful approximation methods has correlatively freed Bayesian modelling, in terms of both model scope and prior modelling. This opening has induced many more scientists from outside the statistics community to opt for a Bayesian perspective as they can now handle those tools on their own. As discussed below, a most successful illustration of this gained freedom can be seen in Bayesian model choice, which was only emerging at the beginning of the MCMC era, for lack of appropriate computational tools.
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Notes
- 1.
In this chapter, the denomination universal is used in the sense of uniformly over all distributions.
- 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., Robert 2007, Sect. 4.5.)
- 3.
In this presentation of Bayes factors, we completely bypass the methodological difficulty of defining π(θ ∈ Θ 0) when Θ 0 is of measure 0 for the original prior π and refer the reader to Robert (2007, Sect. 5.2.3) and Marin and Robert (2007, Sect. 2.3.2) for proper coverage of this issue.
- 4.
The prior distribution can be used for importance sampling only if it is a proper prior and not a σ-finite measure.
- 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 \(1/\sqrt{m}\).
- 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!
- 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. 26.5.2, the weights of this mixture can even be optimised automatically.
- 8.
We stress the point that this is mostly an academic exercise as, in regular settings, it is rarely the case that independent components are used for the importance function.
- 9.
Sect. 26.4.3 covers in greater details the setting of varying dimension problems, with the same theme that completion distributions and parameters are necessary but influential for the performances of the approximation.
- 10.
Even in the simple case of the probit model, MCMC algorithms do not always converge very quickly, as shown in Robert and Casella (2004, Chap. 14).
- 11.
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.
- 12.
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.
- 13.
The difficulty with the infinite part of the problem is easily solved in that the setting is identical to simulation problems in (countable or uncountable) infinite spaces. When running simulations in those spaces, some values are never visited by the simulated Markov chain and the chances a value is visited is related with the probability of this value under the target distribution.
- 14.
Early proposals to solve the varying dimension problem involved saturation schemes where all the parameters for all models were updated deterministically (Carlin and Chib 1995), 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.
- 15.
For a simple proof that the acceptance probability guarantees that the stationary distribution is π(k, θ(k)), see Robert and Casella (2004, Sect. 11.2.2).
- 16.
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.
- 17.
The “sequential” denomination in the sequential Monte Carlo methods thus refers to the algorithmic part, not to the statistical part.
- 18.
- 19.
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.
- 20.
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.
- 21.
An R package called mcsm has been developed in association with Robert and Casella (2009) for training about Monte Carlo methods.
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This work had been partly supported by the Agence Nationale de la Recherche (ANR, 212, rue de Bercy 75,012 Paris) through the 2009–2012 projects Big’MC and EMILE.
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Robert, C.P. (2012). Bayesian Computational Methods. In: Gentle, J., Härdle, W., Mori, Y. (eds) Handbook of Computational Statistics. Springer Handbooks of Computational Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21551-3_26
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