Probit Approximation

• avoid long computation time
• assume likelihood is also normally distributed

$$p(w|D) = \text{N}(w|\mu, \Sigma)$$

Approximation of Posterior

• sigmoid is similar to normal CDF

$$\sigma(a) \approx \Phi\left(\frac{a\sqrt{\pi}}{\sqrt{8}}\right)$$

• approx posterior via sigmoid

$$\begin{array}{rcccl} p(y=1|x,D) & = & \sigma\left(\frac{m}{\sqrt{1+\frac{\pi v}{8}}}\right) \\ m & = & \text{E}[a] & = & x^{T}\mu \\ v & = & \text{V}[a] & = & x^{T}\Sigma x \\ a & = & x^{T}w \end{array}$$

• produces estimates that are closer to the decision boundary