MCMC - how to compute prior(𝜃)

Question

Question

How to compute the prior $P(𝜃)$ and $P(𝜃')$ in MCMC when calculating the posteriors?

Prior

I thought prior keeps updated with the accepted θ'. However, the way it is computed in the articles referenced seem to do different ways.

From Scratch: Bayesian Inference, Markov Chain Monte Carlo and Metropolis Hastings, in python is using 1.

def prior(x):
    #x[0] = mu, x[1]=sigma (new or current)
    #returns 1 for all valid values of sigma. Log(1) =0, so it does not affect the summation.
    #returns 0 for all invalid values of sigma (<=0). Log(0)=-infinity, and Log(negative number) is undefined.
    #It makes the new sigma infinitely unlikely.
    if(x[1] <=0):
        return 0
    return 1

MCMC sampling for dummies keeps using the initial prior, mu_prior_mu=0, mu_prior_sd=1. (In this article, log is applied to the P, hence using norm.pdf directly).

def sampler(data, samples=4, mu_init=.5, proposal_width=.5, plot=False, mu_prior_mu=0, mu_prior_sd=1.):
    ...
    # Compute prior probability of current and proposed mu        
    prior_current = norm(mu_prior_mu, mu_prior_sd).pdf(mu_current)
    prior_proposal = norm(mu_prior_mu, mu_prior_sd).pdf(mu_proposal)

Metropolis-Hastings sample also keep using the initial prior. In this article, the distribution is binomial so using beta function.

def target(lik, prior, n, h, theta):
    if theta < 0 or theta > 1:
        return 0
    else:
        return lik(n, theta).pmf(h)*prior.pdf(theta) # <--- prior = st.beta(a, b) where a = 10, b = 10 

n = 100
h = 61
a = 10
b = 10
lik = st.binom
prior = st.beta(a, b)
sigma = 0.3

for i in range(niters):
    theta_p = theta + st.norm(0, sigma).rvs()
    rho = min(1, target(lik, prior, n, h, theta_p)/target(lik, prior, n, h, theta ))
    u = np.random.uniform()
    if u < rho:
        naccept += 1
        theta = theta_p

Are these correct ways? If so, kindly explain why no need to use the updated 𝜃.

Answer 1

Whenever we have a prior on a parameter $p(\theta)$, and some likelihood function $p(\mathcal{D} | \theta)$, the posterior is proportional to

$$\pi(\theta) \propto p(\mathcal{D} | \theta) p(\theta),$$

since the evidence does not depend on $\theta$ - it has been averaged out. Now, if you want to run a MCMC sampler that targets the density $\pi(\theta)$, you define a proposal mechanism $\theta' \sim q(\theta'|\theta)$. Then your Metropolis-Hastings ratio is

$$\frac{\pi(\theta')q(\theta|\theta')} {\pi(\theta)q(\theta'|\theta)} = \frac{p(\mathcal{D} | \theta') p(\theta')q(\theta|\theta')} {p(\mathcal{D} | \theta) p(\theta)q(\theta'|\theta)},$$

so as you can see the prior does enter into the acceptance ratio. This also illustrates a nice principle if you choose your prior as your proposal mechanism, such that $\theta' \sim p(\theta')$. In this particular case the priors cancel out with the proposal ratio, and you are now just comparing the likelihood of the relative points.