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The probabilities are graphed in Figure 6 as a function of sample size. The squares mark the upper bounds, and the diamonds mark the lower bounds. The bounds shown in the graph were computed using s = 1, but s could be any positive number. If s is a very small value, the interval gets very tight very fast around the observed frequency of red marbles. If s is a very large number, the interval stays wide, close to the vacuous interval [0,1], for a long time while marbles are drawn from the bag and observed.

Figure 6 The probabilities are graphed in Figure 6 as a function of sample size. The squares mark the upper bounds, and the diamonds mark the lower bounds. The bounds shown in the graph were computed using s = 1, but s could be any positive number. If s is a very small value, the interval gets very tight very fast around the observed frequency of red marbles. If s is a very large number, the interval stays wide, close to the vacuous interval [0,1], for a long time while marbles are drawn from the bag and observed.

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so that, when we multiply these functions, expand the squares, and omit constants of proportionality, we obtain Suppose that that your prior for some parameter 0 is a normal distribution with a mean of 6 and unit variance, and that you have an observation x = 13 which is normally distributed with mean @ equal to the parameter of interest and a variance of 2. (The central limit theorem is often used as a justification for assuming that observations might be normally distributed.) Recall that a normal distribution on y with mean p1 and variance o” has probability density Numerical example: updating with sample values
so that, when we multiply these functions, expand the squares, and omit constants of proportionality, we obtain Suppose that that your prior for some parameter 0 is a normal distribution with a mean of 6 and unit variance, and that you have an observation x = 13 which is normally distributed with mean @ equal to the parameter of interest and a variance of 2. (The central limit theorem is often used as a justification for assuming that observations might be normally distributed.) Recall that a normal distribution on y with mean p1 and variance o” has probability density Numerical example: updating with sample values
Figure 2. Prior distribution (blue), likelihood function (red) and the resulting posterior distribution (black) for 0.
Figure 2. Prior distribution (blue), likelihood function (red) and the resulting posterior distribution (black) for 0.
Figure 3. Mixture (black) of f(A, B) distributions for additive model (green) and multiplicative model (orange), with Bayes factors used as weights. Both density and cumulative displays are given. incorporates model uncertainty, the relative beliefs about the two models, and the available sample data. If there had been no sample data at all available, the posteriors would be taken to be the unmodified priors for the two models, so the mixture weights would have been 0.6 and 0.4.
Figure 3. Mixture (black) of f(A, B) distributions for additive model (green) and multiplicative model (orange), with Bayes factors used as weights. Both density and cumulative displays are given. incorporates model uncertainty, the relative beliefs about the two models, and the available sample data. If there had been no sample data at all available, the posteriors would be taken to be the unmodified priors for the two models, so the mixture weights would have been 0.6 and 0.4.
Figure 4. Posterior marginal cumulative distributions (black) for the random variables width, height and area along with their possible ranges (gray).
Figure 4. Posterior marginal cumulative distributions (black) for the random variables width, height and area along with their possible ranges (gray).
Figure 5. Posterior probability of paternity as a function of its prior probability.
Figure 5. Posterior probability of paternity as a function of its prior probability.
Figure 6. Surety of the predictive probability of drawing a red marble as sample size grows.
Figure 6. Surety of the predictive probability of drawing a red marble as sample size grows.
Figure 7. Robust Bayes analysis of a prior p-box (blue), a likelihood p-box (red), and the resulting posterior p-box (black). The only thing that does matter is their intersection. This conclusion seems very unsatisfying. I is the result, of course, of saying very little about the prior and the likelihood. Although the graphs of p-boxes that characterize them would seem to suggest that we are saying a lot, the graphs are misleading in this situation because Bayes’ rule involves probability densities, not cumulative probabilities.
Figure 7. Robust Bayes analysis of a prior p-box (blue), a likelihood p-box (red), and the resulting posterior p-box (black). The only thing that does matter is their intersection. This conclusion seems very unsatisfying. I is the result, of course, of saying very little about the prior and the likelihood. Although the graphs of p-boxes that characterize them would seem to suggest that we are saying a lot, the graphs are misleading in this situation because Bayes’ rule involves probability densities, not cumulative probabilities.
Figure 8. Shouldering, by which posterior distributions can be zero anywhere. Parametric classes
Figure 8. Shouldering, by which posterior distributions can be zero anywhere. Parametric classes
Figure 9. Parametric classes of distributions, varying the mean (green) or the variance (blue). of distribution functions the analyst might think are actually possible. For instance, I could set the limits on the mean and the variance to be very wide so that the prior is very uncertain and yet still be dissatisfied because the elements of the class are all vanilla normals. If I think that skewed, multimodal, truncated or non-continuous distributions should need to change the parametric family or consider some non-parametric In a sense, the parametric approach to robust Bayes analysis has the op be in the class, I would scheme to define the class. posite problem as that of the trivial example involving p-boxes in the previous section. In that case, the classes were too large to be able to conclude anything about important about the posterior. Here, the classes are too small to be really representative of the kind of uncertainty we might have about the prior and likelihood.
Figure 9. Parametric classes of distributions, varying the mean (green) or the variance (blue). of distribution functions the analyst might think are actually possible. For instance, I could set the limits on the mean and the variance to be very wide so that the prior is very uncertain and yet still be dissatisfied because the elements of the class are all vanilla normals. If I think that skewed, multimodal, truncated or non-continuous distributions should need to change the parametric family or consider some non-parametric In a sense, the parametric approach to robust Bayes analysis has the op be in the class, I would scheme to define the class. posite problem as that of the trivial example involving p-boxes in the previous section. In that case, the classes were too large to be able to conclude anything about important about the posterior. Here, the classes are too small to be really representative of the kind of uncertainty we might have about the prior and likelihood.
Figure 10. Example of robust Bayes combination of several prior distributions and likelihood functions to obtain many possible posterior distributions.
Figure 10. Example of robust Bayes combination of several prior distributions and likelihood functions to obtain many possible posterior distributions.
Figure 11. Bounded-density classes of priors (blue), likelihoods (red) and unnormalized posteriors (gray).
Figure 11. Bounded-density classes of priors (blue), likelihoods (red) and unnormalized posteriors (gray).
Figure 12. Bounds on the posteriors (black) arising from bounds on priors (blue) and likelihoods (red).
Figure 12. Bounds on the posteriors (black) arising from bounds on priors (blue) and likelihoods (red).
Figure 13. Hypothetical distribution of distributions (metadistribution).
Figure 13. Hypothetical distribution of distributions (metadistribution).
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