There’s an innumerable amount of plausible priors, and a larger set if implausible nonsensical priors. This is a perfect case of the latter. Whether it’s Jeffreys’ prior or not is irrelevant for that consideration.

]]>Yes indeed, I deliberately chose a strictly monotonic function – I said it was smooth and monotonic, but perhaps not everyone would have realised that the sum of sigma functions is strictly monotonic.

As you say, I didn’t really need to do an MC analysis, at least in the middle segments. Nearer the ends of the calibration curve there is a slight difference between Bayesian and frequentist results. But I’ve found subjective Bayesians often impervious to logical arguments. Demonstrating that Bayesian inference using Jeffreys prior and inference using likelihood ratios gives exact probability matching and hence accurate CIs, whereas subjective Bayesian methods don’t except in a special, unrealistic, case, shakes them up a bit and hopefully makes them think again.

]]>“They are not defined to tell that every point outside of the confidence interval has a lower value of PDF than every point inside the PDF.”

Just to be clear, I have never suggested any such thing.

Your example of CI’s on a uniform distribution gives me some clue as to why we may be talking at cross purposes. I think that you are working under a misapprehension.

You may be under the impression that Nic’s flatspots on the calibration curve have a zero gradient segment somewhere along their length, rather than something getting “very close to” a zero gradient. (?)

If my own analysis is correct, Nic needs a STRICTLY monotonic function for his probability matching to work across the various intervals he has chosen. In other words, he has tied sigmoid functions together to generate a synthetic curve which gets very close to zero-gradient segments, but which in reality remains invertible to within numerical limits.

For this strictly monotonic function, Nic did not actually need to run a MC analysis to show probability matching. This can be demonstrated analytically directly from the definition of the Fisher Information for a single parameter space if a constant variance is assumed for the distribution of RC age given a calendar date, and the “true” calendar dates for testing are selected from a uniform distribution. The MC analysis does however do two things very effectively. It highlights the lack of credibility of confidence intervals obtained from the assumption of a uniform prior by segment. Secondly, it highlights the mathematical properties of the posterior obtained from the Jeffrey’s prior with respect to credible interval analysis, despite the apparent absurdity of the pdf.

If Nic had not retained strict monotonicity, then his exceedance test results would have had poor tail characteristics when the flat spot was occurring at the beginning or end of his selected calendar date segments. And when the flatspot was occurring in the middle of the segment, the central portion of his probability matching would have gone slightly sigmoid, I believe, because of over/underweighting at each end of the flat interval, but the tail probabilities would still have been usable to provide credible interval analysis for high probability levels.

I would invite you to think about whether your intellectual outrage at the “nonsensical” nature of the posterior pdf from the Jeffrey’s prior may just be blinding you to its mathematical properties for this hypothetical problem.

]]>that will be the time I go and WALK

]]>whereby you would demand to put a higher weight on certain years.

So it certainly is not somehting to attack the method on.

But this would be the customer(the historian providing the sample)’s wish.

Most likely customer wants to see the blunt experimental data with propagated errors indicated and do weighting or further speculation on the sample’s contamination etc by themselves and not done by some mishmash tool which should not be used.

A good start is CGSE undergraduate level Taylor’s “An Introduction to error analysis”, chapter 3 , “propagation of uncertainties. It mentions curves and INdependies of uncertainties. He refers to more scholarly work.

But the scientific take home is that a correct error budget can only be made following measure theoretic principles.

Now Keenan’s solution is discretized and measure theory is dead simple then: you need only keep track of the probabilities each discrete bit carries in your “propagation”.

He lays out a proper discretized sample space for the calibration curve which allows him to use Bayes formula, there. Note this is mathematical formula relating 2 events in a completely defined sample space. It does not use anything “prior” or anything coming from prior art or distributions or subjective choices etc etc. zilch nope nada. The method is a close relation of the “rule of three”, which some statistics experts here must have heard about when they were young.

The layout for this sample space uses the provided calibration curve input which is the YEARLY (interpret this as uniform if you like but it certainly is no uniformity in the context of statiscal estimation techniques prior distributions and the like) normal pdf’s. As he is using only probability respecting formulas, all the issues of flat curves , non monotonuous curves is been taken account of , as there simply IS no curve at all. There is only a discretized sample space with atomic element P(carbonyear,sampleageyear).

The “summation” of the uncertainties , for one carbon age measurement with the curve, eventually corresponds to the convolution of 2 independent random variables, discretized.

There is no subjective choice WHATSOEVER taken here, for the whole solution.

The mish mash being discussed here for the “other techniques” ALSO uses the “uniformly” provided calibration curve. No difference there.

]]>This is a context whereby the statistical work is DONE:

1) The sample’s measured; a carbon age is obtained with an error and a reference to gaussian form of the pdf. The pdf is hereby DEFINED. DONE.

2) the calibration curve has uncertainty with it associated. In principle this has all been measured, with the proper statistics sampling techniques, and the result of all that is a “uniform”(yearly) presented format of the uncertainties.

This curve defines a probability distribution for all possible carbon ages vs real ages. That work is DONE. The PDFs are defined.

What the discussion is about is how to translate the sample’s PDF into a “Real Age”, for that specific sample.

That is question around how probabilities are transferred/combined when you use such a calibration curve. It is an exercise in probability theory, and NOT statistics.

No heuristic statistical methods for parameter estimation should be used AT ALL for that last work and discussion.

statistics is an ENGINEERING tool. It uses a lot of math but it is engineering.

You use it, then close the tool box and proceed.

probability is a MATHS tool used in algortihms where you try to derive solutions, given certain inputs.

It is not the first time engineers or a club of bien pensants do not know when to use what tool and foremost when NOT to use certain tools.

the whole discussion is one which makes me think of someoen who says he has a headache in a company of brain surgeons. Of course he will end up on an operation table with his skull open, and half his brains sliced up.

Even if he only needed a paracetamol for the heavy drink of last night.

I agree that determining the PDF from confidence intervals requires the knowledge of those intervals for all levels of certainty. In this case you are proposing a method that can be used to determine all those confidence intervals. Thus we have an example, where the the questions are equivalent. Deciding to apply the method only to a single value of certainty does not make the calculated value any more correct than it is when the other values are calculated as well.

When PDF is constant, CDF is, indeed, linear, and so it must be, nothing else is correct. Range of values of zero for PDF from Jeffreys’ prior leads to **a seriously non-sensical outcome**. If that’s not immediately obvious, it should become so through looking at further examples. I wrote the first two sentences of my previous comment as I wrote precisely to refer to this state of matter. I had not written such sentences without full knowledge of these issues.

As a familiar example. Take a PDF constant over a finite range and zero outside. It’s possible to calculate upper and lower limits that leave 2.5% outside at both edges. The values outside those limits are equally likely than those in the central 95%, but the confidence interval is correct. If we have a peak that includes 95% and a long flat tail that includes 5%, then the 97.5% edge of the confidence interval is in the middle of the tail. That’s how confidence intervals are normally defined. They are not defined to tell that every point outside of the confidence interval has a lower value of PDF than every point inside the PDF.

When the empirical results allow with equal likelihood all values higher than some limiting value, the measurements cannot give any upper limit. Claiming an upper limit or cutoff based on Jeffreys’ prior is, again seriously wrong. We need some additional argument to justify a prior that leads to a cutoff, the reason cannot be derived from the incapability of the method to differentiate between large values, as it would be, when justified by Jeffreys’ prior.

The same problem enters also the method of Nic Lewis to determine the climate sensitivity using Jeffreys’ prior. He used the weak differentiating power of the method for high values of climate sensitivity as evidence against such high values. That’s wrong. His choice of method is not a legitimate reason for a cutoff. His method could not differentiate effectively between high values. The power of the method stops there. (There are better reasons for some cutoff in the prior of climate sensitivity, but that’s another matter. Using Jeffreys’ prior is not a legitimate argument.)

]]>“…then the problem of determining the confidence limits is equivalent to the problem of determining the pdf.”

No, it is not exactly equivalent. If the aim is to establish confidence limits over the entire range of probability space, then I would agree that you need the pdf to be correct over the entire interval of calendar dates, something which it is not possible to obtain unambiguously whatever method is applied. However, if you are interested in, say, 90% or 95% CI’s, then you are (only) looking for the correct calendar dates which corresponds to 5% or 2.5% probability in the tails. A SUFFICIENT condition for a correct answer then is that the estimated CDF asymptotes towards the true unknown CDF in the lower and upper tails.

If, from the sample measurement, a flat calendar date interval appears in the set of possible calendar dates, the posterior CDF for the uniform prior will show a straight line segment over that interval; the CDF using Jeffrey’s prior will show a flat segment over the same interval. The tails in the two posterior distributions are radically different. In particular, the tails in the case of the uniform prior are arbitrarily determined by the length of the length of the flatspot. The exceedance tests carried out by Nic suggest that Jeffrey’s prior yields a CDF which captures likelihood of occurrence of these tails more accurately in a frequentist sense than does a uniform prior.

I agree that all answers are subjective, but some are more subjective than others.

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