Π, thank you for bringing up Gelman. I take his name in vain in a paper about logic in probability, which you can find on my Resume tab : look for “Broccoli”.

However, this does not answer your point, which I promise to do at a later time in a post about “randomness.”

The distinction between climate and weather is true, but it is separate entirely from this posting. Whether “trend” is useful is relevant, but only insofar as it helps us make predictions. There, the main point still stands: usual statistical methods lead to overconfidence.

I’m a little bit lost here. I thought that a stationary process (wide-sense stationarity) requires that 1st and 2nd moments do not vary with respect to time:

E[y(t]]= m(t)= m(t+Dt)

which is not the case here.

And while the trend doesn’t tell you the future values of observables, the posterior predictive distribution also obscures the trend. We always experience the combination of weather and climate, but it is of physical interest to disentangle weather from climate. Both kinds of inferences are useful.

Mike D, Amen. Measurement error just adds to the uncertainty. I have pointed this out elsewhere, but have given no examples yet.

Bob, right on. It’s not that a model cannot describe reality, it’s just that they often do a poor job.

Π 1. Yes, the posterior predictive, not the posterior for μ 2. Not quite. See today’s post on “meaning on mean means”. Red lines are just posterior predictive. The blue lines are the parameter posterior (on μ).

And I agree with you that it’s always better to show the full posterior distribution when possible. This is because showing an interval can give misleading results, as you say. However, these posteriors (for observables or μ) are all t, and are symmetric, so the intervals are reasonable summaries.

The term “random draws”, oh my. I’ll be talking about the mysticism of “random” in future posts.

And no matter if I know the exact value of the parameters, I still do not know the future value of the observables.

Briggs

“It must be stressed that the 95% interval is for the mean, which is itself an unobservable parameter. What we really want to know about is that data values themselves.”

1. This (and most of the rest of your post) implies that you’re talking about the posterior predictive distribution for the data, instead of the posterior for the model trend.

“What we have to do is to completely account for our uncertainty in the values of a and b, but also in the parameters that make up OS.”

2. On the other hand, this implies that you’re treating the distributional parameters (e.g., residual variance or autocorrelation) as uncertain with some prior, and marginalizing over them in the posterior.

Which of these represents the red lines? 1, 2, or both?

As for 1, I believe it’s useful to give both posterior trend and posterior predictive intervals. The underlying trend itself is interesting and informative, but if you want to predict what will be measured in a specific year, you need the posterior predictive distribution.

(Actually, instead of intervals, I prefer a collection of random draws from the posterior or posterior predictive distributions. Means and confidence intervals can hide structure.)

2 should always be done when computationally feasible, unless there is good reason to believe the priors on distributional parameters are very narrow.

A model of a mean is meaningless (?).

Even if you could be certain that the global mean temperature would rise by x, you would have no idea what that would me for people or polar bears or crops.

It’s easy to construct a scenario in which the global mean temp goes up by six degrees yet we all freeze to death.

I think one thing that some lose sight of when they get too caught up in the statistics and formulae is that math doesn’t explain the physical world around us, we are only trying to use math to explain the world around us as best we can. Sometimes it works, sometimes it doesn’t, but ultimately all mathematical descriptions of the physical world are subject to refutation or refinement. In other words, if there is an observable phenomenom that is proven to be true, but which is in conflict with one of our formulae with which we attempt to describe the physical world, it is the formulae that needs to modified, not the phenomenom.

Regards,
Bob North

Just to add more confusion, your model is fit to “observed” data, but in many cases (global temps being a good example) the “observed” data is squishy, or as statisticians say, loaded with measurement error.

Some phenomena can be measured with great precision but some cannot. When the “yardstick” is rubbery, the data are more like guesses than precise readings that can be replicated by unbiased observers.

Measurement error throws an entirely different uncertainty into models, called statistical bias, that is difficult to account for. When measurement error is folded in, somehow or other, it invariably boosts the model variance (a stat pun) and widens the confidence intervals.

PS Satellite data has measurement error. Don’t kid yourself about that. And terrestrial weather station data is thoroughly messed up. And anytime an analysis is based on “proxies,” you can sure the “observed” data is as squishy as mud.