Statistical methods should do better with more data. That’s essentially what the technical term “consistency” means. But with improper numerical techniques, the numerical error can increase with more data, overshadowing the decreasing statistical error.

There are three ways Bayesian posterior probability calculations can degrade with more data:

- Polynomial approximation
- Missing the spike
- Underflow

Elementary numerical integration algorithms, such as Gaussian quadrature, are based on polynomial approximations. The method aims to exactly integrate a polynomial that approximates the integrand. But likelihood functions are not approximately polynomial, and they become less like polynomials when they contain more data. They become more like a normal density, asymptotically flat in the tails, something no polynomial can do. With better integration techniques, the integration accuracy will *improve* with more data rather than degrade.

With more data, the posterior distribution becomes more concentrated. This means that a naive approach to integration might entirely miss the part of the integrand where nearly all the mass is concentrated. You need to make sure your integration method is putting its effort where the action is. Fortunately, it’s easy to estimate where the mode should be.

The third problem is that software calculating the likelihood function can underflow with even a moderate amount of data. The usual solution is to work with the logarithm of the likelihood function, but with numerical integration the solution isn’t quite that simple. You need to integrate the likelihood function itself, not its logarithm. I describe how to deal with this situation in Avoiding underflow in Bayesian computations.

When working on my dissertation I had exactly the kind of underfoot problem you describe here. I ended up using a very high precision library for that particular piece of the computation which I’m sure was terribly inefficient, but did the trick.

John:

Here’s one of my favorite examples, where the maximum likelihood estimate is reasonable but the full Bayesian posterior is not: see section 3 of this paper: http://www.stat.columbia.edu/~gelman/research/published/deep.pdf