Figure 1.12: The Laplace approximation to distributions with skew and kurtosis. These two plots show the Laplace approximation to (a) a sub-Gaussian distribution p(x) ∝ exp(−x4 − 0.1x2) and (b) a skewed distribution p(x) ∝ exp(−0.7x4 + 1.5x3 − 0.1x2). In (a) the approximation over-estimates the tails of the distribution. In (b) it is unaffected by the region of probability to the left of the mode. The distributions are all shown subject to a constants of proportionality.