I am interested in an expression of the density of the distribution from which this is sampled:
The process is the following: along the semi-circle the variable $Z$ is uniformly distributed (equally spaced, not random). Then, a bivariate gaussian with mean $z$ and variance $\Sigma=I_2$ is sampled.
What is the density of this distribution?
Let $X=[x_1,x_2]$, then
$$p(X)=\int p(X|z)p(z)dz=\int N(X;\mu(z),\Sigma)p(z)dz$$
but I do not know how to proceed from here.
Code to reproduce plot:
theta = np.linspace(0, np.pi, 1000)
x1 = np.cos(theta)
x2 = np.sin(theta)
x = np.vstack((x1, x2))
x += np.random.normal(0, 0.1, x.shape)
sns.set_style('whitegrid')
plt.figure(figsize=(5, 5))
plt.plot(x[0], x[1], 'o', alpha=0.5)
plt.axis('equal')
plt.show()