I want to determine the credible interval of a quantity $\theta_1$. I want to make this estimate using observed data by assuming a certain model which depends on $\theta_1$ as well as about n=15 nuisance parameters $\theta_2, \ldots, \theta_n$.
I have a likelihood function $\mathcal L(x|\boldsymbol\theta)$ where $x$ are my observations which I can calculate for any $\boldsymbol \theta$.
I also have priors $\pi(\boldsymbol \theta)$ which I can similarly calculate for any $\boldsymbol \theta$.
My plan is as follows.
Draw a large number of samples of $\boldsymbol \theta$ from a uniform random distribution. (Monte Carlo step)
Calculate the posterior $p = \mathcal{L}(x|\theta)\cdot\pi(\theta)$ for each sample. (Bayes step)
Calculate the weighted mean and standard deviation of $\theta_1$, where the weights are $p$. (Integration step)
Does this make sense? If not, can someone please point me in the right direction? If so, is there a way to estimate how many samples I will need?
(In practice I think I will actually use quasi-Monte Carlo sampling by making use of Sobol numbers, but I think this detail is not important for the question.)
I know how to solve this problem using MCMC, but my likelihood function is very expensive to calculate and so I prefer a simpler way to make this estimate.