That might have been a sensible reply if we were dealing with an abstract random variable but we aren’t; we intrinsically know that increasing sensitivity is decreasingly likely and the mode confirms the most probable result as being low. Using the median can give exactly the same result whether the most probable value is close to 1.5 or close to 4.5. So what he is implicitly asserting is that the skew in the distribution is unimportant.

Yet just looking at the distributions (rather than attempting to describe them by a single number) tells you the mode is more important. Use instead a median value in print and most folk seem to assume that it is the centre point of a normal distribution.

Such bold and unsupported assertions may have some validity in the financial world or otherwise where the underlying science is unknown or data is scarce but not everywhere and certainly not with climate sensitivity!

]]>The book I am just about finished reading is titled “Doing Bayesian Data Analysis” by John K Kruschke of Indiana University. He writes very upbeat and starts each chapter with a poem. The one on specifying a beta prior starts with:

I built up my courage to ask her to dance

By drinking too much before taking a chance.

I fell on my butt when she said see ya later;

Less priors might make my posterior beta.

Thanks again

]]>Without the normalization one has something proportional to a PDF for the probability function of A|B and I would suppose that one could obtain a 95% probability range from that curve, but then one needs to integrate it in order to get the area under the curve. Perhaps in Nic Lewis’ absence here you can direct me to where these calculations were made.

]]>Corrigendum.

My second paragraph should start with:-

Say you work out your conditional probability function of “A given B” as the product of a likelihood function for B given A times a PRIOR pdf for A.

Sorry for any confusion.

No, I was not saying that the normalisation constant is unity. I was saying that it needs to be calculated so that the integral of the pdf of interest is equal to unity, an essential requirement for the result to be deemed a proper pdf.

Say you work out your conditional probability function of “A given B” as the product of a likelihood function for B given A times a posterior pdf for A. There is no guarantee that this function integrates to unity. When you integrate this conditional probability function across the space of A you will generally find that you obtain a number K which is NOT equal to unity. But the integral of any pdf must equal unity. So, to convert the conditional probability function of A given B into a proper pdf, you then need to divide the function by the constant K. The result then retains the original relative properties of the A variable within the conditional probability function, but now integrates to unity, as any self-respecting pdf should.

The constant of proportionality which you asked for is given therefore by (1/K). This is “the inverse of the integral of the product of the likelihood function of B given A times the prior probability of A, integrated over the space of the A variable”, which was what I stated before. It just sounds complicated when you write it out textually. It is a straightforward routine step that generally is not over-described.

]]>Hi Kenneth,

You wrote:

“I cannot find how, you, in your paper, or Frame/Allen or Annan/Hargreaves, in their papers linked here, determined the normalization constant required to give the posterior PDF in terms of probabilities. Is this something so simple that none of the authors described it and I am missing something here? I see you and Annan using the proportional sign in Bayes equations.”

Say the variable of interest is A, and the conditioning variable is B. The integral of the posterior pdf of A given B must be equal to unity.

If the prior pdf of A is a “proper” pdf but the likelihood function is not (it very often is defined only to yield correct proportionality relative to other realisations), then the result needs to be normalised by the constant factor given by inverse of the integral of the product of the likelihood function of B given A times the prior probability of A, integrated over the space of the A variable. This ensures that the posterior pdf of A given B is proper in the sense that it integrates to unity.

Mmm. This sounds more complicated than it was intended to be. Hope it helps.

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