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Is it possible to plot the posterior distribution?
In practice, we must also present the posterior distribution somehow. If the examined parameter θ θ is one- or two dimensional, we can simply plot the posterior distribution. Or when we use simulation to obtain values from the posterior, we can draw a histogram or scatterplot of the simulated values from the posterior distribution.
When to use an ad hoc posterior distribution?
We may for example have an ad hoc estimate of the region of the parameter space where the true parameter value lies with 95% certainty. Then we just have to find a prior distribution whose 95% credible interval agrees with this estimate. But usually credible intervals are examined after observing the data.
Which is stronger the gamma or the posterior distribution?
But its probability mass is concentrated on much smaller area compared to the relatively flat Gamma (1,1) ( 1, 1) -prior, so it has a much stronger effect on the posterior inferences:
Which is the smallest region of the posterior distribution?
This means that a (1 −α) ( 1 − α) -highest density posterior region is a smallest possible (1 −α) ( 1 − α) -credible region.
What kind of Statistics are used to summarize posterior distributions?
The usual summary statistics, such as the mean, median, mode, variance, standard devation and different quantiles, that are used to summarize probability distributions, can be used. These summary statistics are often also easier to present and interpret than the full posterior distribution.
How to calculate the posterior mean of the Poisson-gamma model?
The formula for the posterior mean of the Poisson-gamma model given in Equation (3.2) also gives us a hint why increasing the rate parameter β β of the prior gamma distribution increased the effect of the prior of the posterior distribution: The location parameter α α is added to the sum of the observations, and β β is added to the sample size.