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mean and posterior standard deviation, involve integration. If the integration cannot be performed
analytically to obtain a closed-form solution, sampling techniques such as Monte Carlo integration
and MCMC and numerical integration are commonly used.
Bayesian hypothesis testing can take two forms, which we refer to as interval-hypothesis testing
and model-hypothesis testing. In an interval-hypothesis testing, the probability that a parameter or
a set of parameters belongs to a particular interval or intervals is computed. In model hypothesis
testing, the probability of a Bayesian model of interest given the observed data is computed.
Model comparison is another common step of Bayesian analysis. The Bayesian framework provides
a systematic and consistent approach to model comparison using the notion of posterior odds and
related to them Bayes factors. See [BAYES] bayesstats ic for details.
Finally, prediction of some future unobserved data may also be of interest in Bayesian analysis.
The prediction of a new data point is performed conditional on the observed data using the so-called
posterior predictive distribution, which involves integrating out all parameters from the model with
respect to their posterior distribution. Again, Monte Carlo integration is often the only feasible option
for obtaining predictions. Prediction can also be helpful in estimating the goodness of fit of a model.

How to do Bayesian analysis
Bayesian analysis starts with the specification of a posterior model. The posterior model describes
the probability distribution of all model parameters conditional on the observed data and some prior
knowledge. The posterior distribution has two components: a likelihood, which includes information
about model parameters based on the observed data, and a prior, which includes prior information
(before observing the data) about model parameters. The likelihood and prior models are combined
using the Bayes rule to produce the posterior distribution

Bayesian inference provides a straightforward and more intuitive interpretation of the results in
terms of probabilities. For example, credible intervals are interpreted as intervals to which parameters
belong with a certain probability, unlike the less straightforward repeated-sampling interpretation of
the confidence intervals.
Bayesian models satisfy the likelihood principle (Berger and Wolpert 1988) that the information in
a sample is fully represented by the likelihood function. This principle requires that if the likelihood
function of one model is proportional to the likelihood function of another model, then inferences
from the two models should give the same results. Some researchers argue that frequentist methods
that depend on the experimental design may violate the likelihood principle.

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