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Annual Review of Nutrition, 26, 413 34 Rankinen, T, Zuberi, A, Chagnon, YC et al (2006) The human obesity gene map: the 2005 update Obesity, 14, 529 644 Ravussin, E, Lillioja, S, Knowler, WC et al (1988) Reduced rate of energy expenditure as a risk factor for body-weight gain The New England Journal of Medicine, 318, 467 72 Resnicow, K, Barone, J, Engle, A et al (1991) Diet and serum lipids in vegan vegetarians: a model for risk reduction Journal of the American Dietetic Association, 91, 447 53 Riumallo, JA, Schoeller, D, Barrera, G et al (1989) Energy expenditure in underweight free-living adults: impact of energy supplementation as determined by doubly labeled water and indirect calorimetry The American Journal of Clinical Nutrition, 49, 239 46 Roberts, SB, Savage, J, Coward, WA, Chew, B.

The reweighting of samples according to equation (53) gives a skewed perspective of the PPD as it tends to have more narrow peaks and fat tails relative to a cost function that had been correctly normalized This is because correlations among constraints in the data tend to increase the signi cance of changes in the cost function Thus, stochastic sampling uninformed about the effects of these correlations will tend to sample more frequently the regions that end up being weighted down in equation (53) This represents a sampling inef ciency Along the lines suggested by Gelman et al [28], it is possible to treat S as one of the uncertain parameters within MVFSA and use principles of Bayesian inference to select candidate choices of S from a prior Gamma distribution whose mean and variance are determined by a process similar to the choice of S in Section 5.

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41 The choice of a Gamma distribution was mostly for mathematical convenience and given that a Gamma prior is conditionally conjugate to our de nition of the cost function It is a mathematical way to express uncertainties in the denominator (ie, the variance) The Gamma distribution looks alike a skewed Gaussian distribution with no probability for a value of 0, a mean value that corresponds to the choice of S from the previous section, and a tail that conveys uncertainties in de ning an appropriate S In a fully Bayesian approach, one can construct candidate choices of S to depend on a corresponding evaluation of E(m) to incorporate uncertainties in the data covariance matrix into the stochastic sampling algorithm [29] Although this revised method adds a randomly generated number S to the acceptance/rejection criterion (equation 5.

1), there can be signi cant improvements in the accuracy and ef ciency (Figure 51) to estimating the PPD In fact, samples selected in the case where S is included through a prior no longer require weighting for estimating the PPD as these samples are now assumed to be drawn directly from a distribution that is proportional to the actual PPD The effect of this choice can be dramatic, as illustrated in Figure 52 where the PPD derived from including S as a prior is a much better match to the target distribution of an idealized example than without it..

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and Lucas, A (1988) Energy-expenditure and intake in infants born to lean and overweight mothers New England Journal of medicine, 318, 461 6 Rolls, BJ, Bell, EA and Waugh, BA (2000) Increasing the volume of a food by incorporating air affects satiety in men The American Journal of Clinical Nutrition, 72, 361 8..

1 E2 2 1 4E2 10E4 4E6 E8 1 9E2 45E4 65E8 45E8 9E10 E12 1=2 p 4 14xy x2 y2 6r2 5 1 E12 = 1 E10 f 1 E2 1 36 1 E14 35 1 E12 2 = 1 E10 g1=2 p 14xy 32x4 32x2 r2 6r4 = 1 E2 E4 E6 E8 E10 E12 1=2 p 14 32x6 48x4 r2 18x2 r4 r6 = 1 E2 E4 E6 E8 E10 E12 1=2 f 1 E2 1 36 1 E14 35 1 E12 2 = 1 E10 g1=2 p 4 14 8x 8x2 r2 r4 6r2 5 1 E12 = 1 E10

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