Two Moons

Mathematical Set-Up

Given $\theta \in {\mathbb{R}}^2$, produces data $y\in {\mathbb{R}}^2$ as follows $$\begin{array}{rl}a& \sim \mathcal{U}(-\frac{\pi }{2},\frac{\pi }{2})\\ r& \sim \mathcal{N}(0.1,{0.01}^2)\\ p& =(r\cos (a)+0.25,r\sin (a){)}^{\mathrm{\top }}\\ y& =p+(-\frac{|{\theta }_1+{\theta }_2|}{\sqrt{2}},\frac{-{\theta }_1+{\theta }_2}{\sqrt{2}})\end{array}$$

Coding

def TM_simulator(theta):
    """Two Moons simulator for ABC."""
    t0, t1 = theta[0], theta[1]
    a = uniform(low=-np.pi/2, high=np.pi/2)
    r = normal(loc=0.1, scale=0.01)
    p = np.array([r * np.cos(a) + 0.25, r * np.sin(a)])
    return p + np.array([-np.abs(t0 + t1), (-t0 + t1)]) / sqrt(2)