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All maps of parameter estimates remain misleading

Roland Rau writes:

After many years of applying frequentist statistical methods in mortality research, I just began to learn about the application of Bayesian methods in demography. Since I also wanted to change a part of my research focus on spatial models, I discovered your 1999 paper with Phil Price, All maps of parameter estimates are misleading. As this article is already 17 years old, I wanted to ask whether you think that the last part of the final sentence of the article—“we know of no satisfactory solution to the problem of generating maps for general use”—is still valid. Or would you recommend some other technique to avoid the pitfalls of plotting observed rates or posterior means/medians?

My reply:

For the reasons discussed in our article, I think that there is inherently no way to avoid a map of parameter estimates being misleading in some way (unless variation is tiny or the data have some symmetry so that all sample sizes are identical). It’s just not possible to project the globe of multivariate uncertainty onto the plane of point estimates.

That said, there could well be new ideas in how best to map uncertainty and variation. So I expect there has been progress in mapping parameter estimates in the past twenty years, even if there are fundamental mathematical constraints that will always be with us.


  1. Rahul says:

    Dumb question: What is a “map of parameter estimates”?

    Is it the estimate vs the probability of that particular estimate?

  2. Jeff S says:

    There’s an error in note 15 in the linked paper; Tufte’s book is titled “The Visual Display of Quantitative Information.”

  3. Alex says:

    It seems like the main issue is a lack of way to depict the point estimate and variability at the same time, right? But if you’re willing to alter the map itself a bit you can do that. You could use a tile map, for example (, and use color for one dimension and size for another.

    Or, keeping the map intact and using the radon example, couldn’t you change your plotted variable from ‘radon concentration’ to ‘probability of high radon concentration’? This would allow one to keep the preferred Bayesian analysis and plot a point estimate, but one that’s inherently framed in terms of uncertainty.

  4. If it’s on line, I’d try a dynamic plot of a random tour through the multivariate space. Or, simpler, just a sequence of samples from the posterior.

  5. Will says:

    I’m curious how severe this problem is in practice. Red State, Blue State and related work, for instance, feature many (excellent) maps of parameter estimates for subsets of the population (e.g., vote share among high-income whites in Rhode Island). It seems to me that even given the caveats in the 1999 paper, the benefits of visualizing data this way probably outweigh the potential costs.

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