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.

  • $\begingroup$ The uniform distribution only makes sense as an importance function if the support of the posterior is inside the support of the uniform distribution. $\endgroup$ – Xi'an Apr 15 at 14:24
  • $\begingroup$ 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. $\endgroup$ – Xi'an Apr 15 at 14:25

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