¹ March 28, 2008

Abstract:

A month does not go by without some new study appearing in a peer-reviewed journal which purports to demonstrate some ill effect that will be caused by global warming. The effects are conditional on global warming being true, which is itself not certain, and which must be categorized and bounded. Evidence for global warming is in two parts: observations and explanations of those observations, both of which must be faithful, accurate, and useful in predicting new observations. To be such, the observations have to be of the right kind, the locations and timing where and when they were taken should be ideal, and the measurement error should be negligible. The physics of our explanations, both of motion and e.g. heat, must be accurate, the algorithms used to solve and approximate the physics inside software must be good, chaos on the time scale of predictions must be unimportant, and there must be no experimenter effect. None of these categories is certain. As an exercise, bounds are estimated for their certainty and for the unconditional certainty in ill effects. Doing so shows that we are more certain than we should be.

Robustness and uncertainties of climate change predictions

William M Briggs
300 E. 71st Apt 3R
New York, NY
HTTP://WMBRIGGS.COM
matt@wmbriggs.com

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Here is a Syndney Morning Herald newspaper headline Bibby (2008) , which was derived, as these things frequently are, from a scholarly article, whose purpose is to state what will happen because of global warming

• Global warming to impact health: study

The article by Bi and Parton (2008) which gave rise to the headline is ``Effect of climate change on Australian rural and remote regions: What do we know and what do we need to know?" Among the effects are several notable ``impacts" to health such as increases in people ``admitted to hospital with kidney disease, heart disease and mental illness." There will also be an increase in ``ambulance trips."

It is important to examine these kinds of findings because global warming is not important by itself: it becomes significant only when its effects are consequential to humans. The distinction between questions like ``Will it warm?" and ``What will happen if it warms" is under-appreciated or conflated. For example, when asking how likely are the results of Bi and Parton's study, we are apt to confuse the likelihood of global warming as a phenomenon with ``more kidney disease etc." happening because of global warming. When of course the two kinds of questions and likelihoods are entirely separate.

Because of the frequency of confusion, I want to follow the path to Bi and Patron's conclusions starting from first principles, and untangle and carefully focus on the chain of causation leading up their central claims, and to quantify the uncertainty of the steps along the way. In doing this, I will point out how it is easy to muddle what is being claimed and the uncertainties in those claims.

Let us, for ease, describe Bi and Patron's (B&P's) results by the shorthand A:

$$=$$ (1)

There is no shortage of studies making claims like A; producing them has become a minor scholarly industry. John Brignell (2008) runs a website which has a running list of hundreds of effects supposed due to global warming that have appeared in scholarly literature and in the press. Some examples: a decline in circumcisions in Africa (Flannery, 2006); decreases in human brain size (Ash and Gallup, 2007); less milk in cows (Abeni1 et al., 2007); a decrease in human mortality (Laaidi et al., 2006); an increase in human mortality (Gosling et al., 2007). Biological statements--these are mostly found in the press--about particular species usually find that the population of that species will deline due to global warming: polar bears starve (Lunn and Iacozza, 2000); birds, butterflies and alpine herbs suufer because of habitat changes of every kind (Parmesan and Yohe, 2003). Unless that species is one which is considered a pest, in which case the population of that species will increase: the brain-eating ameoba Naegleria fowleri (McClure, 2007); the false black widow spider (Narain, 2007); mosquitoes, ticks, and mice (Struck, 2006). After reading Brignell's list, one becomes suspicious that there is no ill which global warming will not cause.

Let us stay with Bi and Parton. If A is true, then we will certainly see more kidney disease etc. because of global warming. B&P were careful to use words like ``could" and ``might" when asserting A, meaning that they are aware that A is not certain. We obviously do not know that A is exactly true and so we want to estimate its probability. We can write the uncertainty about A like this:

$$\Pr\{ \vert \}$$ (2)

which describe the probability A is true given that AGW is true, where

$$= 2\times CO_2.$$ (3)

I leave the term `significant' undefined for now, but the `` $2\times CO_2$ ", which means two times the pre-industrial level of atmospheric carbon dioxide, calls for comment. First, it is not a necessary component for the AGW statement; however, so many studies use this criterion, it is not out of place. Second, mankind can of course cause significant warming before this threshold is reached, or significant warming might not take place until well after this threshold. In other words, the threshold is arbitrary, but necessary. Some guidance may be had by examining forecasts of global temperature given by the IPCC (Meehl and Stocker, 2007), shown in Fig. 1. Each forecast, or ``scenario", is for increasing temperature, which is said to be mainly caused (directly and indirectly) by $CO_2$ . Any of these increases is significant.

Figure 1: Reproduction of Figure 10.4 from Chapter 10, Global Climate Projections, from the 2007 IPCC report. We are most interested in scenario A2 (red line), which is the ``worst" case scenario.

When B&P say ``more ambulance trips could happen because of AGW" they are, in the context of their paper, implying at least that $\Pr\{$A$\vert$   AGW$\}>0.5$ (though a stricter interpretation of the word ``could" reduces (2) to the trivial statement that $\Pr\{$A$\vert$   AGW$\}>0$ .)

However, the probability (2) is only tangential; the actual probability we want is of the chance that A is true unconditionally, which is necessary for performing any sort of decision analysis. For example, for ``more ambulance trips", we need the probability this statment is true and the cost of it given it is true. I do not address costs in this paper. The probability of A is

$$\Pr\{$$A$$\} = \Pr\{$$A$$\vert$$   AGW$$\}\Pr\{$$AGW$$\}+\Pr\{$$A$$\vert$$   AGW False$$\}\Pr\{$$AGW False$$\}$$

however, $\Pr\{$A$\vert$   AGW False$\} = 0$ (or close to that) because presumably there will be no increase in kidney disease and ambulance trips if AGW is false, hence

$$\Pr\{ \} = \Pr\{ \vert \}\Pr\{ \}.$$ (4)

It is often tempting for people--particularly reporters, politicians, and ``activists" --to substitute $\Pr\{$A$\vert$   AGW$\}$ for $\Pr\{$A$\}$ , which gives an inflated sense of certainty for the effects of global warming. This is because a simple rule of probability says that, no matter what,

$$\Pr\{$$A$$\vert$$   AGW$$\}> \Pr\{$$A$$\vert$$   AGW$$\}\Pr\{$$AGW$$\}$$

because $\Pr\{$AGW$\}<1$ .

I have also noticed that, particularly in the public, somewhat oddly and incorrectly, if $\Pr\{$A$\vert$   AGW$\}$ is high for some study A of future effects, it is taken as evidence that $\Pr\{$AGW$\}$ is high (it is less unusual, but still wrong, to equate $\Pr\{$A$\vert$   AGW$\}$ with $\Pr\{$   AGW$\vert$   A$\}$ ). That is, the more studies that say some harmful or worrisome thing will happen given AGW are published, the more likely people are to believe that AGW is true. It is a logical fact, however, that no matter how high any $\Pr\{$A$\vert$   AGW$\}$ is, nor no matter how many claims like A there are, they are absolutely no evidence for AGW (I leave aside results that claim events that have already occurred because of AGW). All of these $\Pr\{$A$\vert$   AGW$\}$ can be as close to 1 as you like and AGW can still be absolutely false. But it is usually the multitude of such studies, perversely, that convince people that $\Pr\{$AGW$\}$ is high. We should try and correct this misinterpretation whenever we meet it.

Now, so far I have presented what are, in a sense, cartoon equations. They are formally correct, but only attempts to illustrate what is a vast and complex system. It would be better if we were to eschew numbers when estimating our probabilities, because using numbers implies a level of precision which will obviously be lacking. In other words, it would be best if our results were stated in the form

$$\Pr\{$$AGW$$\} =$$   Somewhat likely

instead of, say,

$$\Pr\{$$AGW$$\} = 0.72.$$

(Really? 0.72 and not 0.73 or 0.71?) Nevertheless, the allure of precision is too strong, so I give in to temptation by erring on the side of doom: by which I mean that I will estimate the highest numbers for the worst case scenarios.

In the following section, I try to estimate $\Pr\{$AGW$\}$ ; in Section 3, I estimate $\Pr\{$A$\vert$   AGW$\}$ and $\Pr\{$A$\}$ , our main item of interest. These estimates do not pretend to be perfect, but it is the hope that they provide a framework which can be used to focus on the different aspects to uncertainty. Of course, this is an extremely sensitive topic where even questioning the probability of AGW might lead to charges of being in sympathy with slave holders (Marc D, 2008); nevertheless, I give it a try.

To help us put AGW in perspective, we reexamine the climate forecast A2 from the 2007 IPCC report, reprinted here at the end of the Vostok ice core temperature reconstruction (et al., 2001) in Fig. 1. This plot gives us a fuller understanding of climate variabilty over the course of human history. The IPCC 2007 A2 Scenario ``worst case" climate forecast for 100 years into the future is also plotted.

Figure 2: The Vostok ice core temperature reconstruction along with the A2 scenario climate foreacst (thick dot-dashed line) 100 years into the future from the 2007 IPCC report. Also, by dotted vertical lines, the approximate genesis of the human species, the end of the stone age, the year 1880, which is about when large-scael industrialization began.

Depending on your persepctive, this forecast is either extremely worrying, or unrealistic.

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It is trivially true that man--and every other organism--influences his environment and therefore his climate. It cannot be other than this. Since this is true, it is only a question of how much and to what extent, if any, it is harmful or beneficial, and to what extent its harmful effects can be mitigated, or its benefits exploited. Which is why we stated one of our hypotheses

   AGW$$=$$   Man-made global warming is significant.

Since man-made climate change is true, we have to clarify that we are not interested in trifles: AGW means discernable, large-scale important effects on climate, as in the forecasts pictured in Fig. 1.

To quantify the probability that AGW is true, we must have two necessary items: observations and explanations of those observations. Taken together, these two lead to predictions of future observations.

Let

$$=$$ (5)

$$=$$ (6)

where by ``explanations" I mean the hypotheses and models that conjecture significant man-made warming. Explanations are more likely to be correct if they lead to skillful predictions on observations completely independent of those used to create the explanations.

This means that AGW is true only if both OBS and EXP are true. Since neither our observations nor explanations are perfect, OBS and EXP are not certainly true, so we need to estimate their probability:

$$\Pr\{ \} = \Pr\{ \& \}$$

$$= \Pr\{ \vert \}\Pr\{ \}$$

$$= \Pr\{ \vert \}\Pr\{ \}.$$

It should be the case that $\Pr\{$EXP$\vert$   OBS$\}=\Pr\{$EXP$\}$ , and likewise that $\Pr\{$OBS$\vert$   EXP$\}=\Pr\{$OBS$\}$ . Which means that the probability that EXP is true given that OBS is true is just the same as the probability that EXP is true. In other words, the goodness of the explanations and the goodness of observations should ideally be independent. They are probability not, however, as it is easy to see how bad observations can lead to bad explanations, just as how bad explanations might lead to biases in observations.

For example, it is often unclear when somebody talks about global warming what time scale they are discussing: is it months?, years?, decades?, millenia?, or longer periods? It cannot be that we do as good a job explaining decades into the future as we do explaining months, and it is not true that the observations we have made are equally useful on all time scales. This is actually a more serious problem than commonly thought: the scale for each A of interest must be accounted for. Nevertheless, I will ignore these complications and assume

$$\Pr\{ \} = \Pr\{ \}\times\Pr\{ \}$$ (7)

which will lead to a biased estimate of whetther $\Pr\{$AGW$\}$ is true, but at least the approximation will allow us to carry out some elementary calculations easily, and lead to important insights.

Observations

There are several layers of uncertainty about observations. The most important are: the kinds, the locations, the timing, and measurement error of observations. Each of these categories is obviously not atomic and may themselves be broken down into subcategories. But these are roughly complete and will serve as a reasonable guideline. If OBS is true, then each of its components must be true, or

$$\Pr\{ \} = \Pr\{ \}\times\Pr\{ \}\times\Pr\{ \}\times\Pr\{ \}$$ (8)

where by ``Kinds" I mean the kinds of observations we make are relevant and discriminatory and so forth: all these terms are explained in a moment. Writing the equation in this way makes two approximations, neither of which are strictly true. The first is that the categories are independent. It means, for example, that the location of an observation gives no information about its measurement error. This is obviously false. The measurement error of coral reef temperature reconstructions, among other things, is different than that of reconstructions using oxygen isotopes found in air bubbles trapped in ice, e.g. Huang (2004). Nevertheless, this approximation won't be atrocious if we keep this limitation in mind and adjust our estimates of the individual probabilities accordingly.

The second limitation, created as a result of the first, is more problematic and is why these equations should be considered more to be schemata than exact representations of the actual state of affairs. The approximation presumes each of these categories are equally probative on the question of whether OBS is true. To understand this, suppose the only revelant categories were ``Locations" (meaning we have picked ideal locations for measurements) and ``The measurements of observations are reliable to the thirty-second significant digit". Then let the probability of ``Locations" be as close to 1 as you like (it can never equal 1). The probability of ``Thirty-second significant digits" is evidently near 0, so when these two probabilities are multiplied together, the probabilty of OBS is going to be near 0, which is absurd. This odd result comes about because the category ``Thirty-second significant digits", while not logically irrelevant to OBS, gives very little information about it.

It is an impossible task to create categories that all would agree are equally relevant, though I believe the set above to be close to it. However likely that is, we have to keep in mind the second limitation when we finally assign numerical values to each of the categories, inflating some if we believe the category itself to be less probabtive than the others, and deflating others if we believe the category provides more information than the others. At the least, we should keep in mind that we are likely to be too certain of whatever answer we arive at.

We are nearly certain of the set of relevant variables that should be measured, i.e. the ``Kinds". By relevant, I mean those atmospheric, oceanic, terrestrial, aeronomic, and solar components that are part of or influence the climate, e.g. Trenberth (1993); Washington and Parkinson (2005). Historically, we do not have contiguous records on them all. Some variables we historically measured and some we did not or we have large gaps in the measurements, e.g. North and Others (2006). Temperature is of course measured, precipitation was not always.

Ideally, for ``Locations" we would take observations at every three-dimensional spatial location. This is obviously impossible, so observations are taken where and when they can. Historically, this was by convenience and sometimes serendipity (e.g. in the case of hurricanes or tornadoes) or by proxy. We cannot be certain that the locations where the observations were taken are ideal for explaining future observations, especially at proxy locations, because these typically blend together actual (unmeasured) observations over different spatial extents. There is some chance that the locations of observations, particularly historically, may mislead us in the sense that we accept the limited observations we have as being representative for what we do not have. This is especially worrying for a large class of variables such as precipitation, gases like CO$_2$ and CH$_4$ , and radiative parameters like aerosol load and albedo. Even for temperature, there have been changes in the surface observation network, and of course changes due to the introduction of satellites.

The time the observations are ideally taken is at every moment at least at the same or faster rate then the time steps of the climate models. This obvioulsy cannot be practically accomplished, and so the timings of observations we have are also not necessarily ideal for explaining future observations, again, particularly historically. This is more properly a question of measurement error. It is also not certain that the times (usually years) assigned to proxy records are accurate.

Measurement error of observations is typically neglected when incorporating observations into explanatory models of climate: measurement error is usually accounted for when producing the ``product", but the downstream use of this product is proceeds as if there is not measurement error. Unfortunately, this error is not negligible. And not all observations are directly measured; some of them are estimated through proxies. Proxies have statistical models behind them, so it is easy enough to estimate their uncertainty. Sometimes proxies from different sources are averaged (or have some other function applied to them), and this introduces new error that must be quantified. New measurements from the satellite era are not immune to error. There is different quality in observational fidelity between earlier versus mid-technology, versus current satellites; all packages suffer from instrument drift which must be corrected statistically; the coverage areas average and alias large swaths in a manner that is not always best for characterizing the variability of the real atmosphere; measurements from satellites are inverse problems, which must be solved by numerical approximation, itself subject to quantifiable error, and the algorithms used to solve the inverse problem have changed through time, e.g. Enting (2002); Tarantola (2005).

So, some possible numbers, erring on the side of generosity, are

$$\Pr\{ \} = \Pr\{ \}\times\Pr\{ \}\times\Pr\{ \}\times\Pr\{ \}$$

$$= 0.99\times0.9\times0.9\times0.9 \approx 0.70$$ (9)

Bumping up the probability of ``Locations" etc. to 0.95 gives us an approximately 85% chance the OBS is true.

Explanations

Just as with observations, there are several layers of uncertainty about our explanations. The most important are: the fundamental equations of motion, or motion physics; the physical processes describing heat and radiative transfer, cloud dynamics, and the like, or for shorthand, heat physics; trust that the algorithms used to solve the physical equations are accurate plus the computer code is error free or insignificant; that chaos is unimportant on the time scale of future predictions; and that there is no experimenter effect, which I will explain below. Again, each of these categories is not atomic and may be broken down into subcategories. If EXP is true, then each of its components must be true, or

$$\Pr\{ \} = \Pr\{ \}\times\Pr\{ \}$$

$$\times \Pr\{ \}\times\Pr\{ \}$$

$$\times \Pr\{ \}$$ (10)

This toy equation suffers just as eq. (8) does from the same two faults: the individual components are assumed to be indepenent when they are not, and each component is assumed to be equally probative on EXP. Again, the more serious of the two is the second assumption and it must be kept in mind when we eventually assign numbers to the individual probabilities.

We are sure of the fundamental equations of motion, e.g. Trenberth (1993). This certainty, which arises from a deduction based upon first principles, is unfortunate in a sense. Because we know with certainty how a theoretical non-turbulent fluid flows, this knowledge tends to increase the subjective certainty that scientists have in the other components of their models, such as the heat physics and so on, and especially predictions on out-of-sample data. This gives a false sense of security. There is the possibility that flow that is turbulent is not well represented, conveying a slight decrease in the overall certainty of the mechanics. Modelling the ocean-atmosphere exchange (Curry and Webster, 1999), and land-atmosphere exchange is also in its relative infancy, e.g. (Mencuccini et al., 2004).

Radiative transfer equations are also well know (Chandrasekhar, 1960), but suffer from the same uncertainties as the equations of motion do. The sun, typically through statistical arguments, is usually said to be relatively unimportant, though not all agree with this. In any case, it is true that the changes in solar irradiation are the result from processes that are not fully known.

Many other physical processes, such as precipition and the release of latent heat due to condensation are also know to a high degree, however, in climate models, these are ``sub scale" processes and are not directly mathematically represented and instead appear in the form of parameterizations and thus are sources of tremendous uncertainty. Small tweaks in these parameterizations can result in small changes in output. It is also not clear that all the parameterizations that are used are complete or ideal. Improvements in them appear constantly, which obiviously implies that the previous ones were sub-optimal, meaning our certainty in what we have being ``the best" should be tempered.

Even if we know how non-turblent fluid flows theoretically, we do not know how to best solve the equations of motion, nor do we know that the methods of approximation so far developed are ideal, nor can we say for certain whether the approximation algorithms encoded into computer are ideal or even that they are error free. Climate model software does not typically face the same level of auditing that, say, your bank's compound interest calculator does.

Again, though the equations of motion are known, and even supposing the algorithms etc. used to represent them are sufficient, the chaotic nature of the climate is usually ignored or underappreciated (Orrell, 2005). Of course, the effects of chaos depend closely on the time frame of the predictions made. One or two ``steps" into the future, whether these steps be yearly or cententially, do not usually diverge enough to be important. But several steps into the future might. Then, small changes in an observation or parameterization can lead to an enormous changes in predictions.

There is also the ever-lurking danger of the experimenter effect, pointed out elsewhere by, for example, Michael Crichton (2005). One crude way to ilustrate this is: a scientist hypothesizes that increasing CO$_2$ plus a positive feedback leads to increasing temperature; this supposition is turned into a computer model which is run at ``pre-industrial" versus two times ``pre-industrial" levels of CO$_2$ ; it is discovered that the temperature in the two-times model shows a warmer atmosphere. The scientist then publishes a paper which states something to the effect of, "Our new model shows that increasing CO2 and feedback warms the air." Well, the model couldn't do anything but show that, since that is what it was programmed to show. But, somehow, the fact the model shows just what it was programmed to show is used as evidence that the assumptions underlying the model were correct, when, in fact, the only true test is whether this model can skillfully predict data that has not been used in any way to build the model.

Thus, there is the chance that over ``tweaking" has taken place. It is a observation in philosophy and theorem in statistics that, for any set of data whatsoever, a model may always be found which fits this data to any degree of accuracy that is desired, even perfectly. This is true both for probability and physical models. The complexity of the data does not enter into it: that is, the data may be simple like in public health data such as Bi and Parton's study or like the climate, hugely multi-dimensional and time dependent. So it would not be, and should not be, surprising the climate models can be found to fit different sets and ranges of historical observational data. Of course, the true model of climate, or any reasonably approximate one, will also fit the same historical data. The point is there is less probative information that EXP is true because the models fit the historically observed data well than is commonly thought.

Not forgetting the tendency that people who work in a particular area tend to exaggerate the probability the people who work in other areas do a better job at approximating the truth than they do, leads us to our guesses:

$$\Pr\{ \} = \Pr\{ \}\times\Pr\{ \}$$

$$\times \Pr\{ \}\times\Pr\{ \}$$

$$\times \Pr\{ \}$$

$$= 0.99 \times 0.9 \times 0.9 \times 0.9 \times 0.9 = 0.65$$ (11)

If all the 0.9s become 0.95s, then the overall probability is about 0.8.

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Again, it is a commonplace that no matter how well a model fits past data it is of little use unless it can predict data which has not been used in any way to fit that model. In other words, climate models have to be able to skillfuly predict future data. Climate models do not have a stellar record here so far, e.g. Douglas et al. (2007); Palmer et al. (2005); Legates and Jr. (1999); John and Soden (2008); Stainforth et al. (2005). This gives

$$\Pr\{ \} = \Pr\{ \}\times\Pr\{ \}$$

$$= 0.75 \times 0.65 < 0.5$$ (12)

If we instead used all 0.95s everywhere like above, $\Pr\{$AGW$\} \approx 0.7$ .

This current level of certainty in climate change predictions is fine if we are interested in nothing more than an intellectual argument between the statements ``It will warm" versus ``It will not." It might even be sufficiently high for us to ask or encourage people to voluntarily change their behavior. But has it reached the level to require legislation--and thereby force--changes in behavior? This depends on the uncertainty of the effects of AGW (like A) and, it should never be forgotten, on the certainty that these kinds of official mandates do exactly what they were designed to do. Given all of human history, we have to figure that our economic predictions are not going to turn out exactly as intended, but I say nothing more about them here.

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Back to B&P. We still need to estimate $\Pr\{$A$\vert$   AGW$\}$ . They claim, for their specific statement AGW, that some, but not all, climate models predict an increase in number of heat waves on the order of a few years from now. Some climate models predict more warming at nighttime and a more evening out of temperatures (reducing the diurnal swing of temperatures) than an increase in severe weather such as heat waves. So we might choose to say that for this A at this time scale etc., that $\Pr\{$AGW$\} = 0.5$ , but I will not insist on this.

We now have to consider the conclusions of their study. For example, did the authors look through the data to find diseases that increased in frequency during heat waves? If so, it is highly improbable that if we look at future heat wave data, we would see the same high levels of the diseases, most would have ``regressed" to their mean level. And other diseases that they did not study will be found to have increased in frequency. What period of data was used? Presumably, the epidemiology of these diseases have changed through time, certainly ``ambulance driving" has. The time series component to these data were not, so far as I can tell, accounted for. How many diseases did they find that did not increase in frequency during heat waves? These should have been noted. How many diseases did they find that decreased in frequency during heat waves? These should have been touted as benefits of warming. How many diseases increased in frequency during ``cold waves"? These should have been touted as benefits of warming.

Are there rigorously clear and certain connections between humans living in heat waves and the diseases noted? If not, then the uncertainty associated with each should be considered. Again, the diseases increasing in frequency under cold waves were ignored. What benefits for other maladies are there for increased warming? It is foolish to say there are none, for, at the least, fewer people would die from extreme cold.

It is not at all certain that, given that heat waves will increase in frequency, people will suffer in them as they suffer now. It is highly probable that technological advances will, for example, increase the availability and efficiency of air conditioning. Medical science, too, will almost certainly increase in efficacy and, with high probability, lessen the number of people susceptible to the diseases under question, therefore, even if heat waves increase, the rate at which people suffer might decrease.

Let us very generously say that $\Pr\{$A$\vert$   AGW$\} = 0.7$ . Thus, the unconditional probability that A is true is

$$\Pr\{ \} = \Pr\{ \vert \}\Pr\{ \}$$

$$= 0.7 \times 0.45 \approx 0.30.$$ (13)

And that number still seems high to me.

We not only have Bi and Parton's study, but we have hundrends of others as noted in Section 1, each making claims of effects assuming that global warming is true. Let these be denoted

$$_1 =$$ More kidney and liver disease, ambulance trips, etc.

$$_2 =$$ Fewer African circumcisions

$$\cdots$$

$$_n =$$ More nasty bugs

What we want is probability that all these are true, which is

$$\Pr\{$$A$$_1 \&$$   A$$_2 \& \cdots \&$$   A$$_n \vert$$   AGW$$\}\Pr\{$$AGW$$\}$$

It isn't entirely clear that we can do this, since we cannot assume strict independence between each A$_i$ , but to first approximation this equals

$$\Pr\{$$A$$_1\vert$$AGW$$\}\Pr\{$$A$$_2 \vert$$AGW$$\} \cdots \Pr\{$$A$$_n \vert$$   AGW$$\}\Pr\{$$AGW$$\}$$

We don't need to plug in any numbers to see that as $n$ gets even moderately big, the chance that all ill effects being true is near 0.

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Attempting to quantify, to the level of precision given, the uncertainties in effects caused by global warming, particularly through the use of mathematical equations that imply a level of certainty which is not felt, can lead to charges that I have done nothing more than build an AGW version of the infamous Drake equation (Drake and Sobel, 1992). I would not dispute that argument. I will claim that the estimates I arrived at are at least within an order of magnitude of the actual uncertainties. For example, the probability that AGW is true might not be 0.8, but it is certainly higher than 0.08.

The equations given, then, are not meant to be authoritative or complete. Their purpose is to concentrate attention of what exactly is being asked. It is too easy to conflate questions of what will happen if AGW is true with questions of is AGW true. And it is just as simple to confuse questions of the veracity and accuracy of observations and with the accuracy of the models or their components. People who work on a particular component are often aware of its boundaries and restrictions, and so are more willing to reduce the probability that this component is an adequate descrition of the physical world, but they are usually likely to assume that the areas on which they do not have daily familiarity are more certain than they are. Ideally, experts in each of the areas I have listed should supply a measure of uncertainty for that area alone. I would welcome a debate and discussion on this topic.

I also would not make the claim that I have accurately listed all the avenues where uncertainty arises (for example, I did not even touch on the uncertainty inherent in classical statisitcal models). But the ones I did list are relevant, though not necessarily of equal importance. We do have uncertainty in the observations we make and we do have uncertainty in the models of these observations. At the very least, we know empirically that we cannot predict the future perfectly. Further, the claims made about global warming's effects are also uncertain. Taken together, then, it is indisputable that we are less certain that both global warming and its claimed effects are true than in either AGW or its effects alone.

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Robustness and uncertainties of climate change predictions

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Footnotes

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Thanks to Dan Hughes for clarifying the role of computational implementation of models.