On Monte Carlo computation of posterior expectations with uncertainty

ABSTRACT

In this paper, we study computation of the range of posterior expectations that arise from robust Bayesian statistics. We compute supremum and infimum of the posterior expectations, when allowing uncertainty for the choice of the likelihood function, or uncertainty for the choice of the prior distribution. In the standard approach of sensitivity analysis, posterior statistics is computed a multiple number of times for each choice of the uncertainty scenarios, which might involve heavy computation due to running Monte Carlo sampling many times. Our paper proposes a more efficient computational method that only requires one Monte Carlo sample for all possible choices of the uncertainty scenarios. The proposed computational method involves three steps (with the mnemonic PSI):

1. (Prior step.) Introduce an auxiliary hyperprior distribution on a parameter λ that indexes the uncertainty.

2. (Sampling step.) For any parameter of interest h, we derive a sample of $(h,\lambda )$ from the joint posterior distribution given the observed data, using a Monte Carlo method. ¹

3. (Inference step.) Based on this posterior sample of $(h,\lambda )$, we estimate the range of posterior expectations $\left\{\underset{\lambda }{inf}E\right(h|\lambda ,data),\underset{\lambda }{sup}E(h|\lambda ,data\left)\right\}$ (and similarly the range for any posterior quantile).

KEYWORDS:

• Expectation
• Bayes
• Monte Carlo
• prior
• posterior
• uncertainty

Acknowledgments

We thank the anonymous referees for many comments that have improved the presentation of this paper, and for providing some useful references.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Details: 2.1: Use a Monte Carlo method to sample from the joint posterior distribution of $(\theta ,\lambda )$, where θ is the structural parameter of the data generation process after fixing the uncertainty.

2.2: For any parameter of interest $h=F(\theta ,\lambda )$, we derive the corresponding sample of $(h,\lambda )$ from the posterior distribution.

2 Note that for the reparameterized uncertainty parameter λ, the corresponding uncertainty range is $\mathrm{\Lambda }=[0.09975062,8]$.

Additional information

Funding

The first author was supported by the State Scholarship Fund of China during her visit in the US, when part of this work was completed. The second author thanks Qilu Securities Institute for Financial Studies, Shandong University, for the hospitality during his visit, where part of this work was done. He was partially supported by the Taishan Scholar Program of the Shandong Province, China.