Weighted Prior as an application of IS in BP simulation
In this section, we show how IS plays a role in BP simulation. As Bayes
rule, posterior distribution is proportional to the likelihood of
observed data and prior distribution of parameter (Supplementary
Formula-3). It emphasizes that likelihood can play the role of
importance function for priors or for a prior. Using this approach, the
expectation of a posterior distribution function can be calculated. In
addition, to paint a clearer picture of the posterior density and its
properties- such as percentiles-it suffices to generate weighted samples
from the prior.
This follows a straightforward algorithm: 1) Generate a random sample
from a prior distribution. 2) Consider the values of the likelihood of
the data in this generated sample as the weight and continue this
process, 3) Normalize the weights. Consequently, each point has its own
probability of occurrence by the likelihood of data. This naïve sampling
method which we referred to as Weighted Prior (WP) is equivalent to IS.
As such IS method provides an alternative way to exact simulation from
posterior (Supplementary Formula-4). It is worth mentioning that, bias
estimated through this approach is of an order of\(\frac{\mathbf{1}}{\mathbf{n}}\) so in large samples it provides
unbiased estimates. It should be considered that, the contribution of
generated samples to the estimation of the final posterior depends on
how much it is supported by data. This means that, prior distributions
should not be that much far off from the likelihood if so, most
generated points receive very small importance weights. We depicted a
case when this method fails (Figure 2).