Solution to the paradox of climate sensitivity

Abstract

Most countries endorse a limit of either 2°C or 1.5°C global warming above pre-industrial levels. However, for several reasons, there is still a significant uncertainty in the climate sensitivity parameter, which relates greenhouse gas concentration (or other forcings) to steady-state temperature. One key source of uncertainty is the disagreement about the appropriate prior for Bayesian estimation. A common choice is the uniform distribution, often thought to contain no information. However, when used to estimate sensitivity it leads to paradoxical results, which have been interpreted as revealing an inherent indeterminacy in the prior of choice. If this were the case, part of the uncertainty would be irreducible. Here I develop an objective Bayesian approach to this problem. I show that both Jaynes’ invariant groups criterion and a new criterion based on information theory lead to the conclusion that there is a uniquely defined non-informative prior of climate sensitivity, which is distinct from the uniform and solves the paradox. This prior distribution is the log-uniform. Furthermore, this result is supported empirically by the observation that other comparable non-equilibrium parameters display a scale-invariant, log-uniform-like frequency distribution. Rather than advocating a direct use of this prior, I recommend to refine it with a limited use of expert elicitation or other methods. A sound prior is a key ingredient in the process to reach a consensus low-uncertainty estimate of climate sensitivity to inform climate policy.

Appendix: Mathematical proofs

Theorem 1

Invariance under change of units implies that S follows a log-uniform distribution (Eq. 11).

Proof

Invariance under change of units means that, for every positive constant C, the distribution of S and of CS are identical. Therefore, according to Eq. 7:

$$ f(S) = f\left( {CS} \right)\left| {\frac{{d\left( {CS} \right)}}{{dS}}} \right|, $$

being f is the same function on both sides of the equation. It is equally true that

$$ f(C) = f\left( {CS} \right)\left| {\frac{{d\left( {CS} \right)}}{{dC}}} \right|, $$

following that

$$ f\left( {CS} \right) = f(C){\left| S \right|^{{ - 1}}}, $$

i.e.

$$ f\left( {CS} \right) = \left[ {Cf(C)} \right]{\left| {CS} \right|^{{ - 1}}}, $$

Since Cf(C) is a constant, this equation generalizes to

$$ f(S) = \left[ {Cf(C)} \right]{\left| S \right|^{{ - 1}}}. $$

Defining θ = Cf(C), this is equivalent to Eq. 11. Since, according to Eq. 11, f(C) ∝C ⁻¹, θ is independent of the choice of the constant C, as expected.

Remark 1

In Theorem 1 it is understood that the change of units is the only transformation taking place. For example, it does not apply if there is a simultaneous change of coordinates, a case treated by Baker and Christakos (2007) which would have no evident interpretation when referring to climate sensitivity.

Theorem 2

Given the relation ∆T = S∆F (Eq. 1), the log-uniform (Eq. 11) is the only distribution of S for which ΔF gives no information about ΔT (Eq. 14).

Proof

Let us use the symbol f for the PDF of S, and g for the PDF of ΔT. According to Eq. 7,

$$ g\left( {\Delta T} \right) = f(S)\left| {\frac{{dS}}{{d\Delta T}}} \right|. $$

Since, from Eq. 1, $\frac{{dS}}{{d\Delta T}} = {\left( {\Delta F} \right)^{{ - 1}}}$, for any given ΔF:

$$ g\left( {\Delta T\left| {\Delta F} \right.} \right) = f\left( {S\left[ {\Delta T,\Delta F} \right]} \right){\left| {\Delta F} \right|^{{ - 1}}}, $$

(16)

where

$$ S\left[ {\Delta T,\Delta F} \right] = {{{\Delta T}} \left/ {{\Delta F}} \right.}. $$

(17)

Using again Eq. 1, Eq. 16 can also be expressed as

$$ g\left( {\Delta T\left| {\Delta F} \right.} \right) = \left[ {f\left( {S\left[ {\Delta T,\Delta F} \right]} \right)\left| {S\left[ {\Delta T,\Delta F} \right]} \right|} \right]{\left| {\Delta T} \right|^{{ - 1}}}. $$

(19)

Given Eq. 17, it follows from Eq. 14 that the part of Eq. 19 within brackets has to be a constant, i.e. $f(S)\left| S \right| = C$ has to be a constant, which leads us to Eq. 11.