# If we take estimated parameters from an MCMC and plug it back into the likelihood to draw new observations, what does the histogram approximate?

Suppose we have the following set-up and we conduct an MCMC on it.

# Likelihood:

$$X\sim ~ Gamma(\alpha,\beta)$$

# Prior:

$$\alpha \sim Unif(0,10)$$ $$\beta \sim Gamma(0.5,0.5)$$

Assume the MCMC over $$M$$ iterations gives us estimates of $$\alpha$$ and $$\beta$$ where $$\widehat{\alpha} = (\alpha_1, \ldots, \alpha_M)$$ and $$\widehat{\beta} = (\beta_1, \ldots, \beta_M)$$.

Assume we iteratively for $$i = \{1, \ldots, M\}$$:

• Take $$\alpha_i$$ and $$\beta_i$$ and sample a value of $$x_i \sim Gamma(\alpha_i,\beta_i)$$

If we plot the density of these $$(x_1, \ldots, x_M)$$, what is it approximating? Is it the posterior predictive?

$$p(\tilde{y} \vert y) = \int p(\tilde{y} \vert \theta) p(\theta \vert y) \, d\theta$$
You'll note the first part of the integrand is the likelihood and the second is the posterior. This integral is integrating over all $$\theta$$ drawn from the posterior and plugged into the likelihood.
• Thanks. For my procedure above, the $\alpha$ and $\beta$ to be plugged into the likelihood is taken iteratively. Is it necessary for me to sample $\alpha$ and $\beta$ randomly before plugging in? Is there a difference between randomly sampling the posterior parameters vs. taking them one by one iteratively up to $M$? Thanks! Feb 20, 2020 at 5:47