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.. ;. {0, M ? 1} in the example of a logistic regression. The mean of? i (g i ) is displayed in black and is spaced apart from the other two curves by the standard deviation of? i (g i ), p.20

, 46 3.2 Computation of the volume of the cube with MYULA and hit-and-run algorithm

, Boxplots of ? 1 , ? 2 , ? 3 for the truncated Gaussian variable in dimension 10, p.53

, Boxplots of ? 1 , ? 2 , ? 3 for the truncated Gaussian variable in dimension 100. 54 3.5 Lasso path for the Gibbs sampler, Wall HMC and MYULA algorithms, p.55

, Boxplots of the error on the first moment for the multivariate Gaussian (first coordinate) in dimension 1000 starting at 0 for different step sizes, p.74

, Boxplots of the error on the first moment for the double well in dimension 100 starting at (100, 0 ?99 ) for different step sizes

, Boxplots of the error on the second moment for the double well in dimension 100 starting at 0 for different step sizes

, Boxplots of the error on the first moment for the Ginzburg-Landau model in dimension 1000 starting at (100, 0 ?999 ) for different step sizes, p.77

, Boxplots of the error for TULAc on the first and second moments for the badly conditioned Gaussian variable in dimension 100 starting at 0 for different step sizes

. .. , Boxplots of the logarithm of the normalizing constants of a multivariate Gaussian distribution in dimension d ? {10, 25, 50}, p.113

, 2 Boxplots of the log evidence for the two models on the Gaussian regression, vol.115

.. {0, The mean of? i (g i ) is displayed in black and is spaced apart from the other two curves by the standard deviation of? i, M ? 1} in the example of the Gaussian regression (model M 1 ), p.116

, Boxplots of the log evidence for the two models on the logistic regression. The methods are the Laplace method (L)

.. {0, The mean of? i (g i ) is displayed in black and is spaced apart from the other two curves by the standard deviation of? i, M ? 1} in the example of the logistic regression (model M 1 ), p.118

, Boxplot of the log evidence for the mixture of Gaussian distributions, p.119

, MALA and RWM algorithms for the logistic regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zero-variance estimators of [PMG14] using a first (ZV-1) or second (ZV-2) order polynomial bases. The plots in the second column are

, 177 the ULA, MALA and RWM algorithms for the logistic regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zerovariance estimator of [PMG14] using a first (ZV-1) or second (ZV-2) order polynomial basis

, x 3 using the ULA, MALA and RWM algorithms for the probit regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zero-variance estimator of, p.179

, 3 using the ULA, MALA and RWM algorithms for the probit regression. The compared estimators are the ordinary empirical average (O), our estimator with a control variate (6.17) using first (CV-1) or second (CV-2) order polynomials for ?, and the zero-variance estimator of, p.180

, 7 in the asymptotic N ? +?.?,? SGD ,? LMC ,? FP and? SGLD are the means under the stationary distributions ?, ? SGD , ? LMC , ? FP and ? SGLD , respectively. The associated circles indicate the order of magnitude of the covariance matrix. While LMC and SGLDFP concentrate to the posterior mean ? with a covariance matrix of the order 1/N , SGLD and SGD are at a distance of order ? 1 of? and do not concentrate as N ? +?, Figure illustrating the definitions of cone(0, ? ? ), b(z ?1 ), c(z ?1 ) and ?(z ?1 ). 186 xx 7.1 Illustration of Proposition 7.5, Theorem 7.6 and Theorem 7

S. Lmc and .. .. ,

, Variance of the stochastic gradients of SGLD, SGLDFP and SGD function of N , in logarithmic scale

, Negative loglikelihood on the test dataset for SGLD, SGLDFP and SGD function of the number of iterations for different values of N ? 10 3, vol.10