Updating a view

This has the disadvantage that it does not account for any uncertainty in the value of the parameter, and hence will underestimate the variance of the predictive distribution.

(In some instances, frequentist statistics can work around this problem.

In Bayesian statistics, however, the posterior predictive distribution can always be determined exactly—or at least, to an arbitrary level of precision, when numerical methods are used.) Note that both types of predictive distributions have the form of a compound probability distribution (as does the marginal likelihood).

In fact, if the prior distribution is a conjugate prior, and hence the prior and posterior distributions come from the same family, it can easily be seen that both prior and posterior predictive distributions also come from the same family of compound distributions.

Hacking wrote "And neither the Dutch book argument, nor any other in the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. So the personalist requires the dynamic assumption to be Bayesian.

It is true that in consistency a personalist could abandon the Bayesian model of learning from experience.

Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available.

Bayesian inference is an important technique in statistics, and especially in mathematical statistics.

The figures denote the cells of the table involved in each metric, the probability being the fraction of each figure that is shaded. P(A|B) = Bayesian inference derives the posterior probability as a consequence of two antecedents, a prior probability and a "likelihood function" derived from a statistical model for the observed data.

However, it is not the only updating rule that might be considered rational.

Ian Hacking noted that traditional "Dutch book" arguments did not specify Bayesian updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch books.

For example, if one does not know whether the newborn baby next door is a boy or a girl, the color of decorations on the crib in front of the door may support the hypothesis of one gender or the other; but if in front of that door, instead of the crib, a dog kennel is found, the posterior probability that the family next door gave birth to a dog remains small in spite of the "evidence", since one's prior belief in such a hypothesis was already extremely small.

The critical point about Bayesian inference, then, is that it provides a principled way of combining new evidence with prior beliefs, through the application of Bayes' rule.

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By parameterizing the space of models, the belief in all models may be updated in a single step.

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