statisticalrethinkin..

48 2. SMALL WORLDS AND LARGE WORLDS2.4.1. Grid approximation. One of the simplest techniques is grid approximation. Whilemost parameters are continuous, capable of taking on an infinite number of values, it turnsout that we can achieve an excellent approximation of the continuous posterior distributionby considering only a finite grid of parameter values. At any particular value of a parameter,p ′ , it’s a simple matter to compute the posterior probability: just multiply the prior probabilityof p ′ by the likelihood at p ′ . Repeating this procedure for each value in the grid generatesan approximate picture of the exact posterior distribution. In this section, you’ll see how toaccomplish this, using simple bits of R code.In the context of the globe tossing problem, grid approximation works extremely well.So let’s build a grid approximation for the model we’ve constructed so far. Here is the recipe:(1) Define the grid. is means you decide how many points to use in estimating theposterior, and then you make a list of the parameter values on the grid.(2) Compute the value of the prior at each parameter value on the grid.(3) Compute the likelihood at each parameter value.(4) Compute the unstandardized posterior at each parameter value, by multiplying theprior by the likelihood.(5) Finally, standardize the posterior, by dividing each value by the sum of all values.In the globe tossing context, here’s the code to complete all five of these steps:R code2.9# define gridp_grid