Two Moons

$\newcommand{\ystar}{y^{*}} \newcommand{\Ycal}{\mathcal{Y}} \newcommand{\isample}{^{(i)}} \newcommand{\kernel}{p_{\epsilon}(\ystar \mid y)} \newcommand{\tkernel}{\tilde{p}_{\epsilon}(\ystar \mid y)} \newcommand{\jointABCpost}{p_{\epsilon}(\theta, y \mid \ystar)} \newcommand{\like}{p(y \mid \theta)} \newcommand{\prior}{p(\theta)} \newcommand{\truepost}{p(\theta \mid \ystar)} \newcommand{\ABCpost}{p_{\epsilon}(\theta \mid \ystar)} \newcommand{\ABClike}{p_{\epsilon}(\ystar \mid \theta)} \newcommand{\kerneltilde}{\tilde{p}_{\epsilon}(\ystar \mid y)} \newcommand{\zkernel}{Z_{\epsilon}} \newcommand{\truelike}{p(\ystar \mid \theta)}$

Mathematical Set-Up

Given $\theta\in\mathbb{R}^2$, produces data $y\in\mathbb{R}^2$ as follows $$ \begin{align} a &\sim \mathcal{U}\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \newline r &\sim \mathcal{N}(0.1, 0.01^2) \newline p &= (r\cos(a) + 0.25, r\sin(a))^\top \newline y &= p + \left(-\frac{|\theta_1 + \theta_2|}{\sqrt{2}}, \frac{-\theta_1 + \theta_2}{\sqrt{2}}\right) \end{align} $$

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)