- #1

ChrisVer

Gold Member

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## Main Question or Discussion Point

Hi everyone. I am reading through these very interesting (in terms of topics) notes:

https://arxiv.org/abs/1807.05996

And so far I am at Section 5. The author gives me the impression they don't seem to fear to call what is Bayesian and what is Frequentist, making the distinction in applications quite clear. Section 5 however gave me quite a few things I had never thought about, because I felt they were very intuitively correct.

For example I always thought that a uniform prior [itex]p(\mu)[/itex] can reflect complete ignorance (I think I've read it also as the highest entropy choice or so), as for any value [itex][\mu,\mu+d\mu[/itex] has the same probability as any other. Although, now I start doubting of how sure one can be by making such a claim, as this is parametrization-dependent. I.e. if we say we are ignorant on [itex]\mu^2[/itex] by choosing a uniform prior in [itex]\mu^2[/itex], this is no longer true for [itex]\mu[/itex] (Sec 5.3, 2nd paragraph's last sentence). I think this is the case because of the Jacobian that shows up once we make a transformation of variables. I find that quite unintuitive: "I have no idea what your temperature is, it can be anything between 33-46C. But that is because I chose to take your temperature as my parameter. I have prior preferences of what your temperature squared might be instead!".

Does anyone have a good explanation for that?

I think this is also somewhat related with Sec5.5, although that is specific on the "non-subjective" priors (such as Jeffrey's prior), which is also weird (I have to admit I didn't look through the references). "Weird" because we constructed the non-subjective priors to obtain the least-information (maximize our ignorance) out of a specific measurement.

https://arxiv.org/abs/1807.05996

And so far I am at Section 5. The author gives me the impression they don't seem to fear to call what is Bayesian and what is Frequentist, making the distinction in applications quite clear. Section 5 however gave me quite a few things I had never thought about, because I felt they were very intuitively correct.

For example I always thought that a uniform prior [itex]p(\mu)[/itex] can reflect complete ignorance (I think I've read it also as the highest entropy choice or so), as for any value [itex][\mu,\mu+d\mu[/itex] has the same probability as any other. Although, now I start doubting of how sure one can be by making such a claim, as this is parametrization-dependent. I.e. if we say we are ignorant on [itex]\mu^2[/itex] by choosing a uniform prior in [itex]\mu^2[/itex], this is no longer true for [itex]\mu[/itex] (Sec 5.3, 2nd paragraph's last sentence). I think this is the case because of the Jacobian that shows up once we make a transformation of variables. I find that quite unintuitive: "I have no idea what your temperature is, it can be anything between 33-46C. But that is because I chose to take your temperature as my parameter. I have prior preferences of what your temperature squared might be instead!".

Does anyone have a good explanation for that?

I think this is also somewhat related with Sec5.5, although that is specific on the "non-subjective" priors (such as Jeffrey's prior), which is also weird (I have to admit I didn't look through the references). "Weird" because we constructed the non-subjective priors to obtain the least-information (maximize our ignorance) out of a specific measurement.