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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:

enter image description here

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()
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The mean of an exponential RV is $1/\lambda$, so an increasing lifetime in data decreases your posterior $\lambda$, e.g. mean life=10 means $\lambda=0.1$. So, it's normal that your posterior belief goes towards left side of the plot.

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