Nuisance parameters are those that we don't care about but we need to estimate to estimate the parameter we do care about (e.g., variance for a normal distribution when we want the mean). Posterior distributions with nuisance parameters can be computed by marginal and conditional simulation, first drawing the nuisance parameter $\theta_2$ from its marginal posterior distribution and then drawing $\theta_1$ from its conditional posterior distribution, given the drawn value of $\theta_2$. Let $X_1 \dots, X_n \overset{iid} \sim f(x | \theta)$ where $\theta = (\theta_1, \theta_2)^T$. Let $\pi(\theta)$ be the joint prior on the parameter vector $\theta$. If we are mainly concerned with $\theta_1$ and $\theta_2$ is a nuisance parameter, then we can find the marginal posterior for $\theta_1$ given the data. $\pi(\theta_1 | x) = \int \pi(\theta_1, \theta_2|x) \ d\theta = \int \pi(\theta_1|\theta_2, x) \ \pi(\theta_2|x) \ d\theta_2$ We can randomly draw values from the marginal pdf $\pi(\theta_2|x)$ and use them to get the conditional posterior pdf $\pi(\theta_1|\theta_2, x)$ using [[Markov Chain Monte Carlo]].