6.1 Motivation for approximations

• Remember we are trying to compute the posterior distribution:

\[ f\left(\theta | y \right) = \frac{f(\theta)L(\theta | y)}{f(y)} \]

• In previous examples (conjugate priors) we were able to do this analytically

• Numerator - no issue, we specify these distributions.

• Denominator - Can be difficult or intractable to compute the denominator f(y) !

\[ f(y) = \int_{\theta_1}\int_{\theta_2} ... \int_{\theta_k}f(\theta)L(\theta | y) d\theta_k ... d\theta_1 d\theta_2 \]

• Solution? Approximate the posterior via simulation!