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If the posterior distribution \(p(\theta|y)\) is unimodal and roughly symmetric(very stringent requirements), it can be convenient to approximate it by a normal distribution; that is, the logarithm of the posterior density is approximated by a quadratic function of \(\theta\).
A Taylor series expansion of \(p(\theta|y)\) around its posterior mode \(\hat\theta\) is \[\log p(\theta | y)=\log p(\hat{\theta} | y)+\frac{1}{2}(\theta-\hat{\theta})^{T}\left[\frac{d^{2}}{d \theta^{2}} \log p(\theta | y)\right]_{\theta=\hat{\theta}}(\theta-\hat{\theta}).\]
The remainder terms of higher order fade in importance relative to the quadratic term when \(\theta\) is close to \(\hat\theta\) and \(n\) is large. The normal approximations to the posterior distribution is then \[p(\theta | y) \approx \mathrm{N}\left(\hat{\theta},[I(\hat{\theta})]^{-1}\right),\] where \(I(\theta)\) is information matrix \(I(\theta) = -\frac{d^2}{d\theta}\log p(\theta|y)\). If the mode is in the interior of parameter space, then the matrix \(I(\hat\theta)\) is positive definite.
We can rewrite the coefficient of the quadratic term as
\[\left[\frac{d^{2}}{d \theta^{2}} \log p(\theta | y)\right]_{\theta=\hat{\theta}}=\left[\frac{d^{2}}{d \theta^{2}} \log p(\theta)\right]_{\theta=\hat{\theta}}+\sum_{i=1}^{n}\left[\frac{d^{2}}{d \theta^{2}} \log p\left(y_{i} | \theta\right)\right]_{\theta=\hat{\theta}}\] The importance of the prior distribution diminishes as the sample size increases.
When theory fails: 1. the likelihood is flat; 2. Number of parameters increasing with sample size; 3. …