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To sample the posterior distribution, there are a few MCMC algorithms (pyMC uses the NUTS algorithm), but here I will focus on the Metropolis algorithm which I have used before to solve the Ising spin model. The algorithm starts from some point in parameter space θ0\theta_0θ0. Then at every time step ttt, the algorithm proposes a new point θt+1\theta_{t+1}θt+1 which is accepted with probability min(1,P(θt+1∣X)P(θt∣X))\min\left(1, \frac{P(\theta_{t+1}|X)}{P(\theta_t|X)}\right)min(1,P(θt∣X)P(θt+1∣X)). Because this probability only depends on the ratio of posterior distributions, it is independent on the normalization term P(X)P(X)P(X) and instead only depends on the likelihood and the prior distributions. This is a huge advantage since both of them are usually well-known and easy to compute. The algorithm continues for some time, until the chain converges to the posterior distribution, and the observed data points show the shape of the posterior distribution.。whatsapp网页版对此有专业解读
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Optionally, install markdown-mode to improve documentation buffer display.
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