贝叶斯分析——从数值积分到MCMC
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Analytical solution
Numerical integration
One simple way of numerical integration is to estimate the values on a grid of values for
One advantage of this is that the prior does not have to be conjugate (although the example below uses the same beta prior for ease of comaprsion), and so we are not restricted in our choice of an approproirate prior distribution. For example, the prior can be a mixture distribution or estimated empirically from data. For example, the prior can be a mixture distribution or estimated empirically from data. The disadvantage, of course, is that this is computationally very expenisve when we need to esitmate multiple parameters, since the number of grid points grows as
import numpy as npimport scipy.stats as stimport matplotlib.pyplot as plta, b = 10, 10n, k = 100, 61thetas = np.arange(0, 1, .001)prior = st.beta(a, b)poster = prior.pdf(thetas) * st.binom(n, thetas).pmf(k) # poster /= (poser.sum()/len(thetas)) # ∑p(θ)p(x|θ)plt.plot(thetas, prior.pdf(thetas), 'b', lw=2, label='Prior')plt.plot(thetas, n*st.binom(n, thetas).pmf(k), 'g', lw=2, label='Likelihood')plt.plot(thetas, poster, 'r', lw=2, label='Posterior')
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