Posterior Inference

Loss Function

Def Loss Function For a decision , a loss function defines the penalty in taking decision given parameter value . In Bayesian scenario, we have to be the point estimate.

Def Quadratic Loss We define the quadratic loss asProp Quadratic loss is minimized when .

Def Linear Loss We define the linear loss:For given scalars and . Prop Linear loss is minimized at , where is the quantile of the posterior.

Def Absolute Error Loss Absolute error loss is a special case of the linear loss when :Prop The posterior median minimizes absolute error loss.

Def 0-1 Loss The 0-1 loss is defined as follows:Prop Prop The posterior mode minimizes 0-1 loss when choosing arbitrarily small.

Predictive Inference

Def Predictive Density Function Predictive density function of a future observation ise.g. Suppose we have with conjugate prior . Then we know that . Now suppose we intend to make further observations, let be the number of successes, . HenceSo for ,e.g. Suppose and , then we can derive that .

Algorithm Sampling from Posterior Predictive Distributions

1. Obtain posterior samples
2. For each we can generate
3. This gives us joint samples
4. To obtain samples from , integrate out (discard the values), leave only.