# Monte Carlo integration for Bayesian parameter estimation

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.

1. Draw a large number of samples of $$\boldsymbol \theta$$ from a uniform random distribution. (Monte Carlo step)

2. Calculate the posterior $$p = \mathcal{L}(x|\theta)\cdot\pi(\theta)$$ for each sample. (Bayes step)

3. 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.

• The uniform distribution only makes sense as an importance function if the support of the posterior is inside the support of the uniform distribution. – Xi'an Apr 15 at 14:24
• Simulating from an unrelated importance distribution like the Uniform is quite likely to be more costly than MCMC. Even quasi-Monte Carlo is an issue when the posterior is well-concentrated. – Xi'an Apr 15 at 14:25