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In Bayesian analysis, we seek a balance between prior information in a form of expert knowledge
or belief and evidence from data at hand. Achieving the right balance is one of the difficulties in
Bayesian modeling and inference. In general, we should not allow the prior information to overwhelm
the evidence from the data, especially when we have a large data sample. A famous theoretical
result, the Bernstein–von Mises theorem, states that in large data samples, the posterior distribution is
independent of the prior distribution and, therefore, Bayesian and likelihood-based inferences should
yield essentially the same results. On the other hand, we need a strong enough prior to support weak
evidence that usually comes from insufficient data. It is always good practice to perform sensitivity
analysis to check the dependence of the results on the choice of a prior.

Finally, as we briefly mentioned earlier, the estimation precision in Bayesian analysis is not limited
by the sample size—Bayesian simulation methods may provide an arbitrary degree of precision.
Despite the conceptual and methodological advantages of the Bayesian approach, its application in
practice is still considered controversial sometimes. There are two main reasons for this—the presumed
subjectivity in specifying prior information and the computational challenges in implementing Bayesian
methods. Along with the objectivity that comes from the data, the Bayesian approach uses potentially
subjective prior distribution. That is, different individuals may specify different prior distributions.
Proponents of frequentist statistics argue that for this reason, Bayesian methods lack objectivity and
should be avoided. Indeed, there are settings such as clinical trial cases when the researchers want to
minimize a potential bias coming from preexisting beliefs and achieve more objective conclusions.
Even in such cases, however, a balanced and reliable Bayesian approach is possible. The trend in
using noninformative priors in Bayesian models is an attempt to address the issue of subjectivity. On
the other hand, some Bayesian proponents argue that the classical methods of statistical inference
have built-in subjectivity such as a choice for a sampling procedure, whereas the subjectivity is made
explicit in Bayesian analysis.
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