Prospects for narrowing ECS bounds

I was just going to briefly mention a new paper by Stevens, Sherwood, Bony and Webb in which they present [p]rospects for narrowing bounds on Earth’s equilibrium climate sensitivity. The basic argument is similar to what has – at times – been discussed here; consider what would be required for the Equilibrium Climate Sensitivity (ECS) to be, for example, small. If those conditions are not satisified, then one can eliminate that region of parameter space. Similarly for large ECS values.

For example, an ECS < 1.5K would require:

  • cooling of climate associated with anthropogenic aerosols would have to have been modest, and/or a historical “pattern effect” would have to be less important than indicated by models
  • tropical sea-surface temperatures during the last glacial maximum (LGM, 21 kya) would need to have been at the warm end of the expected range, and/or a “pattern effect” for the LGM would have to be more important than current models predict
  • climate feedbacks would have to have been much larger in past hot climates than they are at present, or else climate forcing at those times has been significantly underestimated; and
  • cloud feedbacks from warming would have to be negative.

Given that all of these are quite unlikely suggests that an ECS smaller than 1.5K is unlikely. A similar argument can be made for an ECS greater than 4.5K. This is illustrated in the figure below.

Credit: Stevens et al. (2016)

Credit: Stevens et al. (2016)

Credit: Stevens et al. (2016)

Credit: Stevens et al. (2016)

The idea then is that you can combine the different lines of evidence, and use Bayesian inference, to determine a range for the ECS. I couldn’t quite tell if their analysis was intended to simply be illustrative, or not, but as the figure on the right shows, it suggests a 95% range of 1.6K – 4.1K and a median of 2.6K. This may not sound all that different from the IPCC’s 1.5K – 4.5K, but that is presented as a 66% range; the IPCC only regards less than 1K, and greater than 6K, as extremely/very unlikely. What’s presented in Stevens et al. (2016) also seems to rule out quite a bit of Nic Lewis’s range, based on energy balance estimates.

That’s all I was going to say. I think the basic idea is very sensible; when combining different estimates you should at least consider whether, or not, you’ve end up with a range that includes regions that are probably precluded due to physical arguments. You might argue that if some estimates include that portion of the range, then it should be physically plausible. That’s not really correct, since all estimates could be influenced by factors for which they can’t easily compensate (for example, energy balance methods don’t consider the full doubling of CO2 and can’t correct for possible influences from the pattern effect). Therefore using physical storylines to narrow the ECS bounds seems quite sensible.

Update: As Thorsten Mauritsen points out in the first comment, the second figure is probably just illustrative and shouldn’t be seen as a definitive analysis that has narrowed the range by using the physical storylines.

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64 Responses to Prospects for narrowing ECS bounds

  1. Thorsten Mauritsen says:

    As much as I like their Figure 1, even if don’t agree with every detail of it, I think it was a mistake to show Figure 2 which as I understand the text is only for illustration. The problem is that one needs to convert the “story-line” into a probability and the numbers that are used are mostly made up, at least for now. By making the plot, though, the less careful reader is easily lead to believe that the study has actually constrained ECS.

  2. Thorsten,
    Thanks, I wasn’t entirely sure about that. I will add an update to make that clearer.

  3. danialcblog says:

    What’s presented in Stevens et al. (2016) also seems to rule out quite a bit of Nic Lewis’s range, based on energy balance estimates. Nic please reply. I have money on this.

  4. Nic Lewis says:

    I sent Bjorn Stevens a 2000+ word comment on this new paper earlier this month. Pendng a full response from Bjorn, I don’t wish to say too much here. But I would make two points.

    The first point is that, as Thorsten Mauritsen points out, the numbers used are mostly made up: the primary point of the paper is to propose an approach to more narrowly constraining ECS, not to actually do so.

    The second point is that the standard Bayesian method that Bjorn uses, whilst fine for representing a single individual’s subjective personalistic beliefs, is fundamentally wrong if one wants the results to reflect observed reality (correctly reflecting observational and other uncertainties).
    As Andrew Gelman says, correspondence to observed reality is the ultimate source of scientific consensus. I have developed an objective Bayesian method for combining evidence about ECS that does meet this scientific requirement. The slides (with notes) from a seminar I gave at Reading University last month that covers my new method are available here: https://niclewis.files.wordpress.com/2016/10/bayesian-parameter-estimation-with-weak-data-ecs_reading-fulll.pdf

  5. BBD says:

    What’s presented in Stevens et al. (2016) also seems to rule out quite a bit of Nic Lewis’s range, based on energy balance estimates.

    Well, actual palaeoclimate behaviour does do that, which is a pretty solid indicator that NL and other lukewarm estimates are incorrect. We always come back to a fast-feedbacks sensitivity fairly close to 3C per doubling, presumably because that’s about what it is.

  6. Nic,
    I’ll have a look at your slides later (am out at the moment) but my impression is that you’re somewhat selectively representing Andrew Gel man’s endorsement. It’s probably fair to suggest that one’s analysis should produce a result that is consistent with what one might call an objective reality. Using the term “objevtive” in your method, however, doesn’t somehow guarantee that it will.

  7. BBD says:

    Not very objective if it doesn’t reconcile with palaeo behaviour.

    We’ll see if James Annan feels it necessary to update on this comment on vocabulary:

    Nic uses the emotionally appealing terminology of “objective probability” for this method. I don’t blame him for this (he didn’t invent it) but I do wonder whether many people have been seduced by the language without understanding what it actually does.

  8. Fully agree with James Annan. There is no such thing as an “objective” Bayesian method.

  9. Nic Lewis says:

    ATTP,
    I didn’t say Andrew Gelman endorsed my particualr method. I wrote that he says correspondence to observed reality is the ultimate source of scientific consensus. To put this in slightly more context, he wrote:

    “The world outside the observer’s mind plays a key role in usual concepts of objectivity. Finding out about the real world is seen by many as the major objective of science, and this suggests correspondence to reality as the ultimate source of scientific consensus.”

  10. Nic,

    I didn’t say Andrew Gelman endorsed my particualr method. I wrote that he says correspondence to observed reality is the ultimate source of scientific consensus.

    I didn’t really say he endoresed your method, but quoting him positively – as you did – would seem to imply that either he would endorse what you’re doing, or that what you’re doing is consistent with what he is saying. A more appropriate quote might be

    Despite what the Wikipedia entry says, there’s no objective prior or subjective prior, nor is there any reason to think the Jeffreys prior is a good idea in any particular example. A prior distribution, like a data distribution, is a model of the world. It encodes information and must be taken as such. Inferences can be sensitive to the prior distribution, just as they can be sensitive to the data model. That’s just life (and science): we’re always trying to learn what we can from our data.

    I still haven’t had a chance to look at your link, but will do so.

  11. Nic,
    I’ve probably expressed this view to you before, but it seems to me that you’re arguing for a process in which we try to use a method that you regard as “objective”. The problem – as I see it – is maybe twofold. My understanding is that one of the strengths of the Bayesian method is that it allows you to include prior information. Why would one not take advantage of that aspect of Bayesian inference? The other issue, is that your method has the potential to be used in a “turn the handle, get a result” kind of way. Developing an elegant anlaysis method is not that useful if the result is not physically plausible, or returns results that seem more likely than is generally regarded as the case. If your method returns that there is a non-negligible change of an ECS < 1K, then I would argue that you should think again.

  12. Fergus Brown says:

    Nic; I’m curious to know the (real) distinction between a subjective prior and an objective one based on weak data? In the latter case, is it not still necessary to make subjective judgments? I’m also unclear as to how your ‘new’ method constrains the prior, if that is what is needed.

  13. “As Andrew Gelman says, correspondence to observed reality is the ultimate source of scientific consensus. I have developed an objective Bayesian method for combining evidence about ECS that does meet this scientific requirement.”

    I agree that this juxtaposition is rather misleading (and that it could be construed as a potential endorsement). The Jeffrey’s prior is not objective in the sense of correspondence to observed reality, but in the sense of “not influenced by personal feelings or opinions in considering and representing facts.”. As the Jeffrey’s prior gives greatest weight to an ECS of zero, it is clearly not obviously in correspondence with observed reality! That doesn’t mean that it is incorrect, or invalid, but it certainly doesn’t mean that objective (in the Bayesian sense) is a synonym for “correct” (or “good” or “rational” or “scientific” etc.).

    The real problem with objective Bayes (in this particular case) is that we do have prior knowledge, and we need to include it into our analysis if we want to be objective in the sense of maximising correspondence to observed reality. However doing this in an objective (in the sense of “not influenced by…”) manner is rather difficult, but that is not a good reason for ignoring the prior knowledge.

  14. I think there was a rather lengthy discussion of this basic issue in the comments of this post.

  15. ATTP “Why would one not take advantage of that aspect of Bayesian inference?” indeed. The place where “uninformative” priors are really useful is in expressing the knowledge that there is some quantity that you know don’t know anything about, so the effect of that uncertainty is properly accounted for in the uncertainty in the conclusions of the analysis. An objective approach that ignores relevant prior knowledge certainly is no more rational or scientific than a “subjective” approach that takes this prior knowledge into account. I suspect that the latter is likely to be more in correspondence to observed reality, provided the prior is an accurate representation of our prior knowledge (and that the prior knowledge is accurate). What we really need is an objective method that incorporates the laws of physics, observations of paleoclimate and modern observations, but that is very difficult. Alternatively there is the approach taken by Stevens et al. …

  16. Fergus Brown says:

    Am I being dense? Can’t we just run a comparison series using different subjective priors then compare the results to observations? Or is this putting the cart before the horse?

  17. Fergus,
    I guess the problem is that we don’t have an observation of CO2 doubling. We have estimates based on paleo climate, we have model estimates, we have estimates using volcanoes, and we have estimates that use the instrumental record. I don’t really know what observation could resolve the various discrepancies.

  18. Nic Lewis says:

    ATTP,

    “My understanding is that one of the strengths of the Bayesian method is that it allows you to include prior information. Why would one not take advantage of that aspect of Bayesian inference?”

    My method does so. May I suggest that you consider studying my presentation before commenting further on my proposals?

    The other issue, is that your method has the potential to be used in a “turn the handle, get a result” kind of way. Developing an elegant anlaysis method is not that useful if the result is not physically plausible, or returns results that seem more likely than is generally regarded as the case. If your method returns that there is a non-negligible change of an ECS < 1K, then I would argue that you should think again."

    What uncertainty ranges my method produces depends on the characteristics of the ECS estimates that are being combined, and hence indirectly on those of the observational data underlying those estimates. If any of those ECS estimates is based on data that, with uncertainty being adequately allowed for, imply a negligible probability of ECS being < 1 K, then provided the set of estimates is not mutually incompatible my method will give a combined-evidence ECS estimate that likewise gives a negligible probability of ECS being < 1 K.

  19. “My method does so.”

    does the new method incorporate our prior knowledge from paleoclimate studies, or that an ECS of zero is almost certainly incorrect? Genuine question, it would resolve a lot of my concerns regarding your previous work (which as I have said before I already considered to be a useful contribution).

  20. May I suggest that you consider studying my presentation before commenting further on my proposals?

    I was going to make a suggestion to you, but I’ll give this discussion one more chance to be constructive.

    My method does so.

    In what way does it do so? My understanding is that your prior lets the “data dominate”. However, there could be prior information that isn’t included in the data. So, unless I’m mistaken, your method doesn’t allow you to incorporate prior information that isn’t somehow in the data. For example, as Dikran points out, an ECS of 0 is essentially completely ruled out, and yet your prior does not preclude this.

    What uncertainty ranges my method produces depends on the characteristics of the ECS estimates that are being combined, and hence indirectly on those of the observational data underlying those estimates.

    But I think this is largely the point that Stevens et al. are getting at. You can end up combining estimates that produce a result that is somehow statistically consistent with all these estimates, but that includes region that are precluded from physical arguments. That’s essentially my point about “turning the handle”. An elegant method that produces a result that isn’t physically plausible is a method that maybe one should think about again.

  21. I don’t know if this is still the method that Nic is proposing, but as BBD has already highlighted, James Annan has commented before on Nic’s “objective” approach, and concludes with

    Nic’s method insists that no trace remains of the Kamakura era, and I don’t see any point in a probabilistic method that generates such obvious nonsense.

    I realise that it was for a different application, but it is still the point. A method that returns something that is nonsense is pretty useless, however elegant it might be.

  22. As I’ve said before, as a statistician (of sorts), I am much more persuadable by physics than by statistics.

  23. BBD says:

    A method that returns something that is nonsense is pretty useless, however elegant it might be.

    That depends on what you want: a result consistent with physics or with a (shall well call it) prior commitment to a particular political viewpoint. If what you are really doing is servicing the latter, then cranking the handle over and over again is perhaps the best one can hope for.

  24. BBD says:

    typo: “(shall we call it)”

  25. Eli Rabett says:

    Nic borrowed Tol’s dog

  26. izen says:

    @-BBD
    “That depends on what you want: a result consistent with physics or with a (shall we call it) prior commitment to a particular political viewpoint”

    I think the implication of political motivation is unwarranted.
    (although there is a risk of co-option)

    The slides/talk/presentation show a very clever person who has figured out that most everyone else in the field is doing it wrong. Only he has found the correct way to calculate the true value and he is willing to explain his method.

    The inclusion of prior information or updating a Bayesian model can change the result. By comparison the NL method seems to produce almost the same result with increased probability with any input information. Clearly(?) a method that produces such a consistent result must be more accurate than a method that indicates it is uncertain and context dependent.
    Mathematical models, even Bayesian ones, can be so much more beautiful than ugly facts.

  27. BBD says:

    I think the implication of political motivation is unwarranted.

    Do you now. Please see eg. this ‘report’ commissioned by the GWPF (which is a political lobby group engaged in constant misrepresentations of the science for political ends). NL accused the IPCC of scientific misconduct. He accused the IPCC of hiding evidence that sensitivity is low. Hiding it. It doesn’t get much more political than that until we reach ‘it’s a hoax’ territory.

  28. Eli Rabett says:

    @ Izen: By comparison the NL method seems to produce almost the same result with increased probability with any input information.

    Clearly a good indication that some part of the prior is dominating the output, just as in Allen’s uniform prior out to 20C the high end dominated. The way to test this is to feed it artificial data sets whose ECS is known and see what Nic’s prior does with it.

  29. Steven Mosher says:

    “That depends on what you want: a result consistent with physics or with a (shall well call it) prior commitment to a particular political viewpoint. ”

    First to take things to the gutter talk.

    u never disappoint BBD

  30. BBD says:

    Defending the indefensible again Steven. You never learn.

  31. izen says:

    @-BBD
    ” NL accused the IPCC of scientific misconduct. He accused the IPCC of hiding evidence that sensitivity is low. Hiding it. It doesn’t get much more political than that until we reach ‘it’s a hoax’ territory.”

    It is a strong case.
    But I would suggest that Hanlons razor should allow the consideration that such accusatory behaviour can derive from a conviction you posses the mathematical method for determining ‘Truth’.
    A self-belief in your infallibility, and conversely the foolish myopia of everyone else, avoids accusations of intentional fraud; and assumptions of academic superiority may be more common than Machiavellianism.

    On the other hand, perhaps he is right.
    Eli suggests a test.

  32. It would be rather dishonest to pretend that people associated with the GW Policy Foundation or Forum act in good faith. If people have a scientific interest, they can simply distance themselves from the GWPF.

  33. angech says:

    Broadly speaking,
    “there are two views on Bayesian probability that interpret the probability concept in different ways. According to the objectivist view, the rules of Bayesian statistics can be justified by requirements of rationality and consistency and interpreted as an extension of logic.[1][6] According to the subjectivist view, probability quantifies a “personal belief”.
    In probability theory and statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes’ rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event.
    The sequential use of Bayes’ formula: when more data become available, calculate the posterior distribution using Bayes’ formula; subsequently, the posterior distribution becomes the next prior.

    ATTP says rightly “The idea then is that you can combine the different lines of evidence, and use Bayesian inference, to determine a range for the ECS.””

    dikranmarsupial says: November 30, 2016 at 4:06 pm
    “The real problem with objective Bayes (in this particular case) is that we do have prior knowledge, and we need to include it into our analysis if we want to be objective in the sense of maximizing correspondence to observed reality. However doing this in an objective (in the sense of “not influenced by…”) manner is rather difficult, but that is not a good reason for ignoring the prior knowledge.”
    No problem if following the method above

    Thorsten Mauritsen says: November 30, 2016 at 9:18 am
    “the less careful reader is easily lead to believe that the study has actually constrained ECS”.
    The whole purpose is to develop a method to actually constrain the ECS

    Victor Venema says: November 30, 2016 at 1:09 pm
    “Fully agree with James Annan. There is no such thing as an “objective” Bayesian method.”
    It is possible, though your view would count against it in an objective Bayesian method.

    dikranmarsupial says: November 30, 2016 at 4:17 pm
    ATTP “Why would one not take advantage of that aspect of Bayesian inference?
    The place where “uninformative” priors are really useful is in expressing the knowledge that there is some quantity that you know don’t know anything about, so the effect of that uncertainty is properly accounted for in the uncertainty in the conclusions of the analysis”
    -.
    -A prior event is either informative or uninformative.
    If uninformative it is irrelevant.
    It has no place in any assessment and it is not an uncertainty.
    Prior events on the other hand are full of uncertainty and are very important to Bayesian assessment. They have to be included and as new information comes to light clarifying the uncertainty a new, more rigorous and tighter ECS range can be established.
    Despite ATTP and yourself refusing to consider impossible a priors the Bayesian approach handles this with no fuss.They do not matter.
    Take a negative ECS or a 20C positive ECS. Apply the concept. No past evidence of either? scrub them and add in a positive ECS 0.1 and a high ECS 12 C . Still not possible move up into the real ranges. Will it be 3C?
    Who knows but the method if used by Nic or Stevens, Sherwood, Bony and Webb does not care about the starting point. Every bit of informative evidence improves the prior towards the right range]. As Dikran says “an objective method that incorporates the laws of physics, observations of paleoclimate and modern observations is needed”
    This is scientific practical and another bit of the puzzle. embrace it. Point out to Nic which bits you feel he has missed out and incorporate them as well. If you do not like his ratings that very salient observation often made to me by Mosher springs to mind. Do the work yourself.

  34. -1=e^iπ says:

    @ Dikran – “As the Jeffrey’s prior gives greatest weight to an ECS of zero, it is clearly not obviously in correspondence with observed reality! ”

    Well, if we were to argue that physically an ECS of 0 or less is unphysical, then could one approach be to take the Jeffrey’s prior for the logarithm of ECS instead of ECS itself?

  35. BBD says:

    @izen

    What Victor said. You cannot make common cause with Bishop Hill, Anthony Watts and the GWPF and pretend to an apolitical, objective, scientific stance.

  36. -1 I suspect that as the Jeffrey’s prior is designed to make the analysis independent of transformations of the variable of interest, you would still get the same result (the form of the Jeffrey’s prior would be different, but it would encode the same information), but I’d have to work through the mathematics to be sure.

  37. angech wrote “-A prior event is either informative or uninformative.
    If uninformative it is irrelevant.”

    O.K. so not only does angech think he understand the carbon cycle better than Archer or Broeker, he now thinks he knows more about Bayesian statistics than Jeffreys, Jayenes, Bernado and many others. As Einstein said, “There are only two things that are infinite, the universe and angech’s hubris, … but I’m not sure about angech;s hubris”,

    “Despite ATTP and yourself refusing to consider impossible a priors the Bayesian approach handles this with no fuss.”

    angech, I am a Bayesian, I have published papers on Bayesian inference with Jeffrey’s priors. Of course the Bayesian framework allows you to apply priors that are at variance with physics “with no fuss”, but GIGO still applies, garbage in, garbage out. If you use a prior that ignores what we know about the physics, then the conclusions are going to be unreliable. This isn’t rocket science.

    “Do the work yourself.”

    I’d rather see Nic doing it, he is a bright chap and has made a very useful contribution already, and he is in a better position than I am to move the work forward efficiently, and a bit of advice about what his “opponents” would find convincing is one of the ways academic work progresses (hint it is one of the things peer review does).

  38. Angech, your idea of sequential Bayesian analysis is very naive (in fact it is often called “naive Bayes”). To see why it is naive, consider the tragic case of Sally Clark where that sort of reasoning ended up in a serious miscarriage of justice. If combining all the source of information properly in a Bayesian framework were that easy, it would have been done long ago.

  39. angech quores: “there are two views on Bayesian probability that interpret the probability concept in different ways. According to the objectivist view, the rules of Bayesian statistics can be justified by requirements of rationality and consistency and interpreted as an extension of logic.”

    The rules are objective. Your priors on how the system works and the probabilities of the parameters are not.

    Anyway, Nic Lewis’ overconfidence about his prior is just one of the many problems of using simplified statistical models to estimate the climate sensitivity (what some like to call “observational methods”, but comprehensive climate models also fit to the historical observations). We now know of several reasons who these methods give too low values of the climate sensitivity. If you take them all into account, first studies suggest that these “observational methods” even produce above average estimates.

    http://variable-variability.blogspot.com/2016/07/climate-sensitivity-energy-balance-models.html

    This at the very least suggests that the tiny uncertainty range of Nic Lewis is much, much too low. He should read the complete teachings of the Uncertainty Monster.

  40. VV wrote “The rules are objective. Your priors on how the system works and the probabilities of the parameters are not.”

    I think Edwin Jaynes (and others like him) would disagree. There is nothing subjective about a prior that encodes some law of physics, for example an inability to travel faster than the speed of light. Jayne’s book makes the point that we can define Bayesian probabilities in a way in which automata with the same prior information would agree on the conclusions, in which case it is difficult to see how it can be described as “subjective”. It helps to use terms like “state of knowledge” rather than e.g. “belief”, which suggest subjectivity.

  41. Just to be clear, the prior that Nic Lewis uses is “objective”, but that doesn’t mean it is correct. A belief that objective priors are always preferable on the other hand would be entirely subjective (AFAICS).

  42. angech says:

    Angech, your idea of sequential Bayesian analysis is very naive.
    Strange, The quotes come straight from Wiki on Bayesian analysis.
    Your attempt to make light of my reasoning is not logical.
    The fact that a sub branch is called “naive Bayes” does not imply that it is in fact naive.
    The sad case you allude to is a consequence of misapplied statistics otherwise known as an error, nothing to do with the use of simple or “naive Bayes”, merely the misuse of an erroneous calculation by someone who should have known better.

    see
    ” The prosecution case relied on significantly flawed statistical evidence presented by pediatrician Professor Sir Roy Meadow, who testified that the chance of two children from an affluent family suffering sudden infant death syndrome was 1 in 73 million. He had arrived at this figure erroneously by squaring 1 in 8500, as being the likelihood of a cot death in similar circumstances. The Royal Statistical Society later issued a statement arguing that there was “no statistical basis” for Meadow’s claim, and expressing its concern at the “misuse of statistics in the courts”.
    The example you provided which shows that Bayesian theory was not at fault, as you are doubtless aware.

    ” If combining all the source of information properly in a Bayesian framework were that easy, it would have been done long ago.”

    The fact is that that Bayesian frameworks work by incorporating new information as it comes along, that changes the prior that one had assumed not by having all the information. Bayesian theory works on using limited information to improve ones knowledge.
    We do not have all the sources of information and if we did we would no longer need Bayesian techniques to deduce a probability range, We would have a definitive answer.
    The fact that this discussion will lead you to a better understanding of Bayesian theory is actually an example of the theory in practice. One learns and adapts. Happy to be of service.

  43. “Your attempt to make light of my reasoning is not logical.”

    I wasn’t making light of it, I was pointing out that it is naive, as in “showing a lack of experience/knowledge”, which it is, and which I pointed out.

    “The fact that a sub branch is called “naive Bayes” does not imply that it is in fact naive.”

    actually it does, it is called that because it makes the assumption of independence, which is a naive assumption because it is rarely actually true.

    “The sad case you allude to is a consequence of misapplied statistics otherwise known as an error, nothing to do with the use of simple or “naive Bayes”, merely the misuse of an erroneous calculation by someone who should have known better.”

    One of the errors in question being to treat the observations as being independent an updating the posterior accordingly, i.e. naive Bayes.

    “He had arrived at this figure erroneously by squaring 1 in 8500, as being the likelihood of a cot death in similar circumstances. “

    which is just what your sequential Bayes would do, if you had actually tried to work through the arithmetic involved you would know that.

    “The fact that this discussion will lead you to a better understanding of Bayesian theory is actually an example of the theory in practice. One learns and adapts. Happy to be of service.”

    O.K., this is now just an obvious attempt to wind me up. How very childish. As I said, I research this stuff for a living, and have published journal papers on it. The person that ought to be trying to learn is you, rather than making a fool of yourself by your hubris.

  44. Just to spell it out, if you have two events A and B and two hypotheses H0 and H1, then the full Bayesian probability for H0 is:

    P(H0|A,B) = P(A,B|H0)P(H0)/P(A,B) (1)

    where P(A,B) = P(A,B|H0)P(H0) + P(A,B|H1)P(H1).

    Now P(A,B|H0) = P(A|B,H0) * P(B|H0). Notice that this is not just multiplying P(A|H0) and P(B|H0).

    Now in the sequential Bayes approach (i.e. naive Bayes), we start with the first event (B):

    P(H0|B) = P(B|H0)P(H0)/P(B). (2)

    No problems so far. Then we use our old posterior as the new prior and feed in the second event (A):

    P(H0|A,B) = P(A|H0)P(H0|B)/P(A)

    substituting (2) we get

    P(H0|A,B) = P(A|H0)P(B|H0)P(H0)/P(A)P(B) (3)

    Comparing (1) and (3) we can see that naive Bayes uses P(A,B|H0) = P(A|H0)P(B|H0) instead of P(A|B,H0)P(B|H0), which is only true if P(A|B,H0) = P(A|H0), i.e. the events are independent.

    This is where the squaring in the Sally Clark case comes in. The “expert witness” treated the two deaths (A and B) as being independent events and so assumed that the probability of two deaths was just the square of the probability of one cot death.

    Note if you are bothered by the denominator, then consider the Bayes Factor, K, which gives the ratio of the probabilities of the two hypotheses, in which case the denominators cancel.

  45. dikranmarsupial says: “There is nothing subjective about a prior that encodes some law of physics, for example an inability to travel faster than the speed of light.

    But that is not your complete prior. Otherwise you would not need statistics in the first place.

  46. VV the point I am making that that priors can be constructed in an objective manner, e.g. using physical laws, using MaxEnt, using transformation groups etc. You can have objective knowledge to put into the prior, there are objective ways of specifying that you don’t know something. There are both objectivist and subjectivist schools of Bayesian statistics and both have their place.

  47. The point I am making is that even in such an ideal case this would be science by induction. You always make simplifications, if only for physics that you did not know to be important at the time. Especially in case of the climate system such subjective factors will always be important and any claim that a method is objective is no more than rhetorics.

  48. -1=e^iπ says:

    @ Dikran
    “-1 I suspect that as the Jeffrey’s prior is designed to make the analysis independent of transformations of the variable of interest, you would still get the same result (the form of the Jeffrey’s prior would be different, but it would encode the same information), but I’d have to work through the mathematics to be sure.”

    Are you sure it’s not just invertible transformations? You obviously have a better statistical background, I am just trying to understand. According to wiki “When using the Jeffreys prior, inferences about θ → {\displaystyle {\vec {\theta }}} {\vec \theta } depend not just on the probability of the observed data as a function of θ → {\displaystyle {\vec {\theta }}} {\vec \theta }, but also on the universe of all possible experimental outcomes, as determined by the experimental design, because the Fisher information is computed from an expectation over the chosen universe.” Your variable being real in the logarithm of climate sensitivity space corresponds to your variable being positive in climate sensitivity space. So aren’t you restricting the chosen universe if you take the logarithm?

  49. I don’t see how making inductive inferences is inherently subjective. All science makes simplifications, it isn’t only all statistical models that are wrong (but some are useful), it applies to physics as well. If things that you don’t know you don’t know cause something to be subjective rather than objective, then all science is subjective (because we can never rule out the possibility of something we didn’t know we didn’t know – ask Newton). Besides, whether a prior is objective or not doesn’t depend on whether it matches our actual prior knowledge, it is about its construction and meaning (see Jaynes)

    The use of objective in the sense used to describe schools of Bayesianism is not rhetoric, but a substantive technical distinction between the two approaches.

  50. -1 “Are you sure it’s not just invertible transformations?” no, I’m not ;o) As I said I’d need to look into it and do the maths (if I ever finish my marking… ;o)

  51. dikranmarsupial: “I don’t see how making inductive inferences is inherently subjective.”

    When I wrote “The point I am making is that even in such an ideal case this would be science by induction”, I did not mean that it would be inductive, but that it would not be science. For the reasons you then explain yourself.

    I have no problem with using the term “objective” in the scientific literature, there everyone (hopefully) knows the limitations of the term. I object to its use as a rhetorical bludgeon in the public debate to pretend that one very limited study is the pinnacle of science and all other lines of evidence can be ignored, including all the studies that show that this limited field of study produces biased results.

  52. Eli Rabett says:

    It looks like the peak at zero is a feature of Jeffrey’s prior and it basically says that we start from the POV that there is no connection between the prior and the data, e.g. nothing that changes will affect the ECS. The smooth decline from that point says that the linkage is more likely to be weak than strong. In that sense it is ignorant, not objective.

  53. We already had so many threads on the problems of energy balance models. Maybe we should go back to the good idea in the above post.

    Making and improving climate models and validating them is important work and needs to be done. But it may not be the best way to get more accurate estimates of the climate sensitivity. Science is at its most productive when you make hypotheses and try to disprove them and also when many scientists work on the same problem. The above study by Bjorn Stevens gives us questions/hypotheses to work on. In the Forecast podcast Bjorn Stevens talks about this contrast.

    This contrast of model improvement/validation and hypothesis testing in my last paragraph is unfairly black and white. For example, already now we focus much of the improvements/validation on clouds, while Bjorn Stevens also mentions the cloud feedback as important for limiting the climate sensitivity. It is no coincidence that work on improvements/validation is about clouds that is because the hypothesis driven work has shown that to be the main source of uncertainty.

    Still it would be good if we could nudge the work more into the direction of hypothesis driven research. Unfortunately, validating and comparing models are relatively easy and risk free papers, while finding a productive hypothesis and studying it is hard and will often fail (to give any answer, positive or negative).

  54. VV ” I did not mean that it would be inductive, but that it would not be science. For the reasons you then explain yourself.”

    I’m sorry, I don’t understand, I didn’t say anything wouldn’t be science.

  55. Sorry, writing error. Let’s try again.

    dikranmarsupial: “I don’t see how making inductive inferences is inherently subjective.”

    When I wrote “The point I am making is that even in such an ideal case this would be science by induction”, I did not mean that it would be *subjective*, but that it would not be science. For the reasons you then explain yourself.

    Or for the famous black swan reason. No matter how much white swans you observe, that is not enough evidence to exclude black swans. Induction is nice, but in the end you need to venture a conjecture to do science.

  56. angech says:

    dikranmarsupial says: December 1, 2016 at 3:55 pm
    “I don’t see how making inductive inferences is inherently subjective”.

    Quoting re induction and its usage,
    “Inductive reasoning (as opposed to deductive reasoning ) is reasoning in which the premises are viewed as supplying strong evidence for the truth of the conclusion. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.
    the premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensure it.
    Unlike deductive arguments, inductive reasoning allows for the possibility that the conclusion is false, even if all of the premises are true.”

    Subjectivity is built in to inductive inferences, It is akin to cherrypicking in that the inference chosen, even if true in itself, may not be true when a subjective inference is tied to it. It is therefore impossible to see the subjective element when one makes an inductive inference.
    You have to step back a little.

  57. VV none of the things I mentioned stop science from being science, even though they apply to science as well, that was the point.

    “Or for the famous black swan reason. No matter how much white swans you observe, that is not enough evidence to exclude black swans. “

    Indeed, but as I said that applies to science as well as statistics. A taxonomist with no evidence for a black swan doesn’t make a space for a black swan on the cladogram just because they haven’t been ruled out. Note statisticians have had methods for rare events for a long time, the “black swan” argument has been rather over-sold (or bought into too much) IMHO.

  58. JCH says:

    Prospects for a prolonged slowdown in global warming in the early 21st century

    The synthetic series in Fig. 5a also show examples of greatly accelerated warming lasting a decade or more, which are evidently spring-back effects as an internal variability cooling episode is followed by a strong internal variability warming episode. The strong warming episodes are further amplified by the underlying forced warming trend. One extreme example shows a warming of almost 1 °C in 15 years—a much greater 15-year warming rate than has occurred in the observations to date (red curves). These spring-back warmings illustrate another important potential consequence of strong internal multidecadal variability as simulated in CM3, and reinforce the need to better understand whether such internal variability actually occurs in the real world. …

    Just curious, a system that is sensitive to NV (assuming the above describes an unusually sensitive system,) would be to be ____ _______ to CO2? (more sensitive or less sensitive)

  59. BBD says:

    Doubtless a rhetorical question from the Collings-pickin’ ole boy there :-), but here’s Kyle Swanson answering it at RC:

    It first needs to be emphasized that natural variability and radiatively forced warming are not competing in some no-holds barred scientific smack down as explanations for the behavior of the global mean temperature over the past century. Both certainly played a role in the evolution of the temperature trajectory over the 20th century, and significant issues remain to be resolved about their relative importance. However, the salient point, one that is oftentimes not clear in arguments about variability in the climate system, is that all else being equal, climate variability and climate sensitivity are flip sides of the same coin. (see also the post Natural Variability and Climate Sensitivity)

    A climate that is highly sensitive to radiative forcing (i.e., responds very strongly to increasing greenhouse gas forcing) by definition will be unable to quickly dissipate global mean temperature anomalies arising from either purely natural dynamical processes or stochastic radiative forcing, and hence will have significant internal variability. The opposite also holds. It’s painfully easy to paint oneself logically into a corner by arguing that either (i) vigorous natural variability caused 20th century climate change, but the climate is insensitive to radiative forcing by greenhouse gases; or (ii) the climate is very sensitive to greenhouse gases, but we still are able to attribute details of inter-decadal wiggles in the global mean temperature to a specific forcing cause. Of course, both could be wrong if the climate is not behaving as a linear forced (stochastic + GHG) system.

  60. JCH says:

    BBD – well, yeah, I have about a dozen of them to pick from.

    As for 7 to15 years with a very high rate of warming because of synchronously coupled modes of variability and higher than expected climate sensitivity, I think we’re already about three years into it.

  61. BBD says:

    BBD – well, yeah, I have about a dozen of them to pick from.

    A gentleman never brags about the size of his… collection 🙂

    As for the hot decade hypothesis, well, perhaps yes. It would almost be a relief. I have become irrationally vexed by the words ‘pause’ and ‘hiatus’ even in entirely unrelated contexts.

  62. JCH says:

    I didn’t brag… I left out the my old Martin and my old Gibsons!

  63. JCH says:

    I think the surges in 20th-21st century warming are clearly related to the periodic surges in the PDO. The longest hyper warming is roughly 1933 to 1945… 11 years at .43 ℃ per decade.

    Using their parlance, spring-back warming events:

    1933 to 1945… ~11 years at .43 ℃ per decade
    1974 to 1981… ~6 years at .6 ℃ per decade
    1986 to 1991… ~4 years at .39 ℃ per decade
    1991 to 1999… ~7 years at .36 ℃ per decade
    2000 to 2006… ~5 years at .36 ℃ per decade
    2012 to present… ~3 years at .97 ℃ per decade

    In the 1980s and 1990s there were the volcanoes that caused cooling… they help hide the extent of the above, and cooling caused by periodic downturns in the PDO.

    Sea level rise appears to agree with what I am saying.

    The AMO? It has little to nothing to do with it. It is simply behaving like the GMST.

  64. Eli Rabett says:

    As long as we are trading Gelman quote here, he actually said something about the use of the Jeffrey’s prior in the context of ECS

    Despite what the Wikipedia entry says, there’s no objective prior or subjective prior, nor is there any reason to think the Jeffreys prior is a good idea in any particular example. A prior distribution, like a data distribution, is a model of the world. It encodes information and must be taken as such. Inferences can be sensitive to the prior distribution, just as they can be sensitive to the data model. That’s just life (and science): we’re always trying to learn what we can from our data.

    I’m no expert on climate sensitivity so I can’t really comment on the details except to say that I think all aspects of the model need to be evaluated on their own terms. And there is no reason to privilege “the likelihood,” which itself will be based on modeling assumptions.

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