It also occurred to me while writing the post that the figure could easily have been someone else’s idea entirely. The prose I quote appears directly above the graph, which makes it seem extremely silly in situ.

As for whether a figure is necessary, I completely agree with you; it’s not. I imagine that an editor may have decided, just as many do, that too much text without figures is inherently boring. However, it seems that the editor underestimates the audience here. Those who are reading Nate Silver’s book probably don’t need pictures every third page to hold their attention.

If one is not careful, Power Point can rock one’s socks off.

The fact that the random effects are estimated differently than are the parameters governing them doesn’t make the former something other than parameters. The key difference between fixed effects and random effects here is that random effects are modeled (i.e., there are variances and covariances governing them [as well as means, namely the associated fixed effects]) whereas the fixed effects aren’t. Yet there is still uncertainty about the fixed effects, else why have standard errors and confidence intervals and all the associated statistical testing machinery?

Bates says (on p. 2 of lrgprt.pdf) that “random effects are unobserved random variables.” Well, the fixed effects aren’t observed either, and the fact that nobody disputes that there is uncertainty about them (again, CIs, SEs, etc…) suggests that, whether we’re modeling them or not (i.e., whether we are willing to make particular distributional assumptions about them or not), there is an element of randomness smack dab in the middle of fixed effect estimation, too.

Ultimately, I can’t figure out what exactly Bates and others mean by ‘parameter’ that includes fixed effects and variances/covariances while excluding random effects.

This distinction, whatever it is, seems at odds with a lot of other statistical modeling, too. For example, in CFA and SEM, there are potentially huge numbers of latent variables, all of which are parameters in those models. Pretty much all of the cognitive/math psych modeling I know about has analogous structure – latent variables like the spatial location, dimension weights, and bias parameters in Generalized Context Model or the means and correlations in the perceptual distributions in GRT. I link to those two papers specifically because they illustrate maximum likelihood estimation and not Bayesian estimation (i.e., the unobserved parameters aren’t explicitly probabilistically modeled).

The fact that the variance of the estimated random effects can be different from the estimated variance doesn’t do much work here either. For any given step in an MCMC chain in a Bayesianly-estimated model, the actual variance of a set of ‘random effects’ parameters may or may not match the variance parameter governing them exactly. In fact, given that there’s exactly one way for the two numbers to match exactly and approximately infinitely many ways for them to fail to match, I would expect the latter to occur far more often.

Maybe the best place for you to look (and I mean that, because you will understand the math better than I do) is one of the drafts of Bates’ long-in-progress book on lme4. If you go to this site:
http://lme4.r-forge.r-project.org/
And find a doc called “lrgprt.pdf”, you will find a relevant discussion in section 1.6 (at least, in the version at the time of this posting). Here’s a paragraph of it:

These values [talking about BLUPs] are often considered as some sort of “estimates” of the random effects. It can be helpful to think of them this way but it can also be misleading. As we have stated, the random effects are not, strictly speaking, parameters—they are unobserved random variables. We don’t estimate the random effects in the same sense that we estimate parameters. Instead, we consider the conditional distribution of B given the observed data, (B|Y = y).