I'm following these slides. I have some lamps, which I expect to die at a time $t$ where $p(t) = \lambda e ^{-\lambda t}$ for some $\lambda$. My prior for $\lambda$ is given by $p(\lambda) = \Gamma(\lambda, 2, 2)$. I observe one lamp die at time 10. According to those slides, my posterior is $$k' = 2 + 1 = 3$$ $$ \theta' = 2 / (1 + 2 * 10) = 0.095$$ so
$$p(\lambda) =\Gamma(\lambda, 3, 0.095)$$ When I plot this though, it doesn't look right:
My posterior has shifted far to the left, even though my observation was to the right of the expectation of my prior. What's going on?
Code:
import matplotlib.pyplot as plt
import numpy as np
from scipy.special import gamma as gamma_function
def gamma(x, k, theta):
return np.power(x, k-1) * np.exp(-x/theta) / (gamma_function(k) * np.power(theta, k))
k = 2
theta = 2
data = [10]
new_k = k + len(data)
new_theta = theta / (1 + theta * sum(data))
x = np.linspace(0, 10, 100)
plt.plot(x, gamma(x, 2, 2), label = 'prior')
plt.plot(x, gamma(x, new_k, new_theta), 'r', label = 'posterior')
plt.legend()
