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Assembly Bias Fitting Paper
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monotonic statement is still unclear to me #31

Closed vdbosch69 closed 8 years ago

vdbosch69 commented 8 years ago

In caption of Table 2, and somewhere in the text, you write

In cases for which the posterior is monotonic within the physical parameter space, we show one-sided percentiles.

I do NOT understand what you mean by the posterior being monotonic within the physical parameter space.....

andrew-zentner commented 8 years ago

Purely increasing or decreasing. It doesn't have a turnover, so it doesn't really make much sense to have two-sided constraints.

vdbosch69 commented 8 years ago

with what???? I do know what monotomic means… You are talking about a 1D distribution….so monotonic in what sense?????

Prof. Frank C. van den Bosch e-mail: frank.vandenbosch@yale.edu Astronomy Department, Yale University phone: +1-203-432-0196 P.O. Box 208101 fax: +1-203-432-5048

New Haven, CT 06520-8101 http://www.astro.yale http://www.astro.yale/.edu/vdbosch

On Jun 17, 2016, at 2:11 PM, Andrew Zentner notifications@github.com wrote:

Purely increasing or decreasing.

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andrew-zentner commented 8 years ago

OK, let's back up.

  1. I'm talking about the 1D posterior so that is a function of a single variable, agreed?
  2. The variable is the value of the parameter of interest, say Acen, agreed?
  3. Most posteriors are not monotonic within the physical parameter space, they have a peak and we define a parameter confidence region based on the shape of the posterior function. Agreed?
  4. If a posterior is monotonic, then there is no obvious sense in a 2-sided confidence region. The posterior will peak at one or the other end of the viable region of parameter space. Agreed?
  5. If a posterior is a monotonic function, then it is more informative to quote the 1-sided confidence region. Agreed? Let me know where this breaks down. I honestly don't know a better way to say it.
vdbosch69 commented 8 years ago

OK. I now understand what you are trying to say, and I agree with that.

I do not think, though, that your way of saying it is clear. I will think of alternative ways of formulating this. If unsuccesful, I will leave as is.

An working on Section 4 now…

F

Prof. Frank C. van den Bosch e-mail: frank.vandenbosch@yale.edu Astronomy Department, Yale University phone: +1-203-432-0196 P.O. Box 208101 fax: +1-203-432-5048

New Haven, CT 06520-8101 http://www.astro.yale http://www.astro.yale/.edu/vdbosch

On Jun 17, 2016, at 2:25 PM, Andrew Zentner notifications@github.com wrote:

OK, let's back up.

  1. I'm talking about the 1D posterior so that is a function of a single variable, agreed?
  2. The variable is the value of the parameter of interest, say Acen, agreed?
  3. Most posteriors are not monotonic within the physical parameter space, they have a peak and we define a parameter confidence region based on the shape of the posterior function. Agreed?
  4. If a posterior is monotonic, then there is no obvious sense in a 2-sided confidence region. The posterior will peak at one or the other end of the viable region of parameter space. Agreed?
  5. If a posterior is a monotonic function, then it is more informative to quote the 1-sided confidence region. Agreed? Let me know where this breaks down. I honestly don't know a better way to say it.

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andrew-zentner commented 8 years ago

yes, please do.

Andrew R. Zentner Associate Professor Director of Graduate Studies Department of Physics and Astronomy & Pittsburgh Particle physics, Astrophysics and Cosmology Center (PITT PACC) University of Pittsburgh 320 Allen Hall Pittsburgh, PA 15260

zentner@pitt.edumailto:zentner@pitt.edu http://www.phyast.pitt.edu/~zentner/Home.html

On Jun 17, 2016, at 2:34 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.com> wrote:

OK. I now understand what you are trying to say, and I agree with that.

I do not think, though, that your way of saying it is clear. I will think of alternative ways of formulating this. If unsuccesful, I will leave as is.

An working on Section 4 now…

F

Prof. Frank C. van den Bosch e-mail: frank.vandenbosch@yale.edumailto:frank.vandenbosch@yale.edu Astronomy Department, Yale University phone: +1-203-432-0196 P.O. Box 208101 fax: +1-203-432-5048

New Haven, CT 06520-8101 http://www.astro.yale http://www.astro.yale/.edu/vdbosch

On Jun 17, 2016, at 2:25 PM, Andrew Zentner notifications@github.com<mailto:notifications@github.com> wrote:

OK, let's back up.

  1. I'm talking about the 1D posterior so that is a function of a single variable, agreed?
  2. The variable is the value of the parameter of interest, say Acen, agreed?
  3. Most posteriors are not monotonic within the physical parameter space, they have a peak and we define a parameter confidence region based on the shape of the posterior function. Agreed?
  4. If a posterior is monotonic, then there is no obvious sense in a 2-sided confidence region. The posterior will peak at one or the other end of the viable region of parameter space. Agreed?
  5. If a posterior is a monotonic function, then it is more informative to quote the 1-sided confidence region. Agreed? Let me know where this breaks down. I honestly don't know a better way to say it.

— You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_andrew-2Dzentner_abmcmc_issues_31-23issuecomment-2D226844902&d=AwMCaQ&c=-dg2m7zWuuDZ0MUcV7Sdqw&r=akBy6VKVSXuVF7qPhX_uKgCTzeIo3SdhrMQi2H7IAuI&m=NtSN-qJNz4uxHifqiaO9aYkCQRtF8NQbUWYv3mlm_Pk&s=SNMjSDp-reNmOOKFOleZQNksAqwWGVEMTAWyQiIhKSE&e=, or mute the thread https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_notifications_unsubscribe_AIWS8X4hstp8WPTP9c1FFujFW3ZYRAtFks5qMuZ9gaJpZM4I4mdo&d=AwMCaQ&c=-dg2m7zWuuDZ0MUcV7Sdqw&r=akBy6VKVSXuVF7qPhX_uKgCTzeIo3SdhrMQi2H7IAuI&m=NtSN-qJNz4uxHifqiaO9aYkCQRtF8NQbUWYv3mlm_Pk&s=jWKcLn2G2IsWNznGslJGtzZb6R074mcFX7ySARjYZdI&e=.

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vdbosch69 commented 8 years ago

How about this change:

In cases where the posterior distribution of either $A{\rm cen}$ or $A{\rm sat}$ is a monotonic within the physical parameter range, we quote an upper or lower limit on the parameter

to

In cases where the posterior distribution of either $A{\rm cen}$ or $A{\rm sat}$ peaks at the physical boundaries ($\pm 1)$, we only quote an upper or lower limit on the parameter

F

Prof. Frank C. van den Bosch e-mail: frank.vandenbosch@yale.edu Astronomy Department, Yale University phone: +1-203-432-0196 P.O. Box 208101 fax: +1-203-432-5048

New Haven, CT 06520-8101 http://www.astro.yale http://www.astro.yale/.edu/vdbosch

On Jun 17, 2016, at 2:36 PM, Andrew Zentner notifications@github.com wrote:

yes, please do.

Andrew R. Zentner Associate Professor Director of Graduate Studies Department of Physics and Astronomy & Pittsburgh Particle physics, Astrophysics and Cosmology Center (PITT PACC) University of Pittsburgh 320 Allen Hall Pittsburgh, PA 15260

zentner@pitt.edumailto:zentner@pitt.edu http://www.phyast.pitt.edu/~zentner/Home.html

On Jun 17, 2016, at 2:34 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.com> wrote:

OK. I now understand what you are trying to say, and I agree with that.

I do not think, though, that your way of saying it is clear. I will think of alternative ways of formulating this. If unsuccesful, I will leave as is.

An working on Section 4 now…

F

Prof. Frank C. van den Bosch e-mail: frank.vandenbosch@yale.edumailto:frank.vandenbosch@yale.edu Astronomy Department, Yale University phone: +1-203-432-0196 P.O. Box 208101 fax: +1-203-432-5048

New Haven, CT 06520-8101 http://www.astro.yale http://www.astro.yale/.edu/vdbosch

On Jun 17, 2016, at 2:25 PM, Andrew Zentner notifications@github.com<mailto:notifications@github.com> wrote:

OK, let's back up.

  1. I'm talking about the 1D posterior so that is a function of a single variable, agreed?
  2. The variable is the value of the parameter of interest, say Acen, agreed?
  3. Most posteriors are not monotonic within the physical parameter space, they have a peak and we define a parameter confidence region based on the shape of the posterior function. Agreed?
  4. If a posterior is monotonic, then there is no obvious sense in a 2-sided confidence region. The posterior will peak at one or the other end of the viable region of parameter space. Agreed?
  5. If a posterior is a monotonic function, then it is more informative to quote the 1-sided confidence region. Agreed? Let me know where this breaks down. I honestly don't know a better way to say it.

— You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_andrew-2Dzentner_abmcmc_issues_31-23issuecomment-2D226844902&d=AwMCaQ&c=-dg2m7zWuuDZ0MUcV7Sdqw&r=akBy6VKVSXuVF7qPhX_uKgCTzeIo3SdhrMQi2H7IAuI&m=NtSN-qJNz4uxHifqiaO9aYkCQRtF8NQbUWYv3mlm_Pk&s=SNMjSDp-reNmOOKFOleZQNksAqwWGVEMTAWyQiIhKSE&e=, or mute the thread https://urldefense.proofpoint.com/v2/url?u=https-3A__github.com_notifications_unsubscribe_AIWS8X4hstp8WPTP9c1FFujFW3ZYRAtFks5qMuZ9gaJpZM4I4mdo&d=AwMCaQ&c=-dg2m7zWuuDZ0MUcV7Sdqw&r=akBy6VKVSXuVF7qPhX_uKgCTzeIo3SdhrMQi2H7IAuI&m=NtSN-qJNz4uxHifqiaO9aYkCQRtF8NQbUWYv3mlm_Pk&s=jWKcLn2G2IsWNznGslJGtzZb6R074mcFX7ySARjYZdI&e=.

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vdbosch69 commented 8 years ago

Ignore my previous suggestion...rather I suggest the following:

In cases where the 16 or 84 percentiles coincide with a physical or prior bound, we only quote a lower or upper limit on the parameter, respectively.

andrew-zentner commented 8 years ago

That can never happen, by definition, so that rewrite doesn't work.

Andrew R. Zentner Associate Professor & Director of Graduate Studies Department of Physics & Astronomy The University of Pittsburgh

On Jun 17, 2016, at 4:49 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.com> wrote:

Ignore my previous suggestion...rather I suggest the following:

In cases where the 16 or 84 percentiles coincide with a physical or prior bound, we only quote a lower or upper limit on the parameter, respectively.

You are receiving this because you commented. Reply to this email directly, view it on GitHubhttps://na01.safelinks.protection.outlook.com/?url=https%3a%2f%2fgithub.com%2fandrew-zentner%2fabmcmc%2fissues%2f31%23issuecomment-226878012&data=01%7c01%7czentner%40pitt.edu%7c3a061b3b43f841c8abed08d396f0e3a0%7c9ef9f489e0a04eeb87cc3a526112fd0d%7c1&sdata=B6a5C9Dmi0F2xTorJmnRNrAmyraRXwEdBd%2forg5Mhq8%3d, or mute the threadhttps://na01.safelinks.protection.outlook.com/?url=https%3a%2f%2fgithub.com%2fnotifications%2funsubscribe%2fAIhmmw9eqElzB9BxyNgnGv8KKSk39-sqks5qMwhcgaJpZM4I4mdo&data=01%7c01%7czentner%40pitt.edu%7c3a061b3b43f841c8abed08d396f0e3a0%7c9ef9f489e0a04eeb87cc3a526112fd0d%7c1&sdata=vb9k%2fcmzerlpK1IKQmlmuWPRPU6PHPaPXh%2fq%2fbY5Jqw%3d.

vdbosch69 commented 8 years ago

That can never happen, by definition, so that rewrite doesn't work.

Yes it can. Consider a posterior distribution, say for A_{cen}, which peaks at 0.95 or so.
Suppose the value of the posterior probability at this peak is Pmax.

For each value of A_{cen} there is a corresponding value of the posterior probability, P.

In order to determine the 16 and 84 percentiles, what do you do? Well, you find the posterior probability P{68} such that 68 percent of the chain elements have P>P{68}. In the example, you see that the values of A{cen} that correspond to this P{68} are A_{16} (the 16 percentile) and the prior constraint +1. I hope the example demonstrates that 84 percentiles can coincide with the upper limit…..

Frank

Andrew R. Zentner Associate Professor & Director of Graduate Studies Department of Physics & Astronomy The University of Pittsburgh

On Jun 17, 2016, at 4:49 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.com> wrote:

Ignore my previous suggestion...rather I suggest the following:

In cases where the 16 or 84 percentiles coincide with a physical or prior bound, we only quote a lower or upper limit on the parameter, respectively.

You are receiving this because you commented. Reply to this email directly, view it on GitHubhttps://na01.safelinks.protection.outlook.com/?url=https%3a%2f%2fgithub.com%2fandrew-zentner%2fabmcmc%2fissues%2f31%23issuecomment-226878012&data=01%7c01%7czentner%40pitt.edu%7c3a061b3b43f841c8abed08d396f0e3a0%7c9ef9f489e0a04eeb87cc3a526112fd0d%7c1&sdata=B6a5C9Dmi0F2xTorJmnRNrAmyraRXwEdBd%2forg5Mhq8%3d, or mute the threadhttps://na01.safelinks.protection.outlook.com/?url=https%3a%2f%2fgithub.com%2fnotifications%2funsubscribe%2fAIhmmw9eqElzB9BxyNgnGv8KKSk39-sqks5qMwhcgaJpZM4I4mdo&data=01%7c01%7czentner%40pitt.edu%7c3a061b3b43f841c8abed08d396f0e3a0%7c9ef9f489e0a04eeb87cc3a526112fd0d%7c1&sdata=vb9k%2fcmzerlpK1IKQmlmuWPRPU6PHPaPXh%2fq%2fbY5Jqw%3d. — You are receiving this because you authored the thread. Reply to this email directly, view it on GitHub, or mute the thread.

andrew-zentner commented 8 years ago

What you describe is not a percentile. It is a probability threshold.

Andrew R. Zentner Associate Professor & Director of Graduate Studies Department of Physics & Astronomy The University of Pittsburgh

On Jun 18, 2016, at 4:02 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.com> wrote:

That can never happen, by definition, so that rewrite doesn't work.

Yes it can. Consider a posterior distribution, say for A_{cen}, which peaks at 0.95 or so. Suppose the value of the posterior probability at this peak is Pmax.

For each value of A_{cen} there is a corresponding value of the posterior probability, P.

In order to determine the 16 and 84 percentiles, what do you do? Well, you find the posterior probability P{68} such that 68 percent of the chain elements have P>P{68}. In the example, you see that the values of A{cen} that correspond to this P{68} are A_{16} (the 16 percentile) and the prior constraint +1. I hope the example demonstrates that 84 percentiles can coincide with the upper limit.....

Frank

Andrew R. Zentner Associate Professor & Director of Graduate Studies Department of Physics & Astronomy The University of Pittsburgh

On Jun 17, 2016, at 4:49 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.commailto:notifications@github.com> wrote:

Ignore my previous suggestion...rather I suggest the following:

In cases where the 16 or 84 percentiles coincide with a physical or prior bound, we only quote a lower or upper limit on the parameter, respectively.

You are receiving this because you commented.

Reply to this email directly, view it on GitHubhttps://na01.safelinks.protection.outlook.com/?url=https%3a%2f%2fgithub.com%2fandrew-zentner%2fabmcmc%2fissues%2f31%23issuecomment-226878012&data=01%7c01%7czentner%40pitt.edu%7c3a061b3b43f841c8abed08d396f0e3a0%7c9ef9f489e0a04eeb87cc3a526112fd0d%7c1&sdata=B6a5C9Dmi0F2xTorJmnRNrAmyraRXwEdBd%2forg5Mhq8%3d, or mute the threadhttps://na01.safelinks.protection.outlook.com/?url=https%3a%2f%2fgithub.com%2fnotifications%2funsubscribe%2fAIhmmw9eqElzB9BxyNgnGv8KKSk39-sqks5qMwhcgaJpZM4I4mdo&data=01%7c01%7czentner%40pitt.edu%7c3a061b3b43f841c8abed08d396f0e3a0%7c9ef9f489e0a04eeb87cc3a526112fd0d%7c1&sdata=vb9k%2fcmzerlpK1IKQmlmuWPRPU6PHPaPXh%2fq%2fbY5Jqw%3d.

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vdbosch69 commented 8 years ago

What you describe is not a percentile. It is a probability threshold.

Fair enough….I guess what I was describing is the 68 % Confidence interval. So then perhaps what you can say is

In cases where the upper or lower bound of the 68 percent confidence interval coincides with a physical or prior bound, we only quote a lower or upper limit on the parameter, respectively.

Sorry to go on about this, but I find that statement about the monotonicity incomprehensible…

F

Andrew R. Zentner Associate Professor & Director of Graduate Studies Department of Physics & Astronomy The University of Pittsburgh

On Jun 18, 2016, at 4:02 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.com> wrote:

That can never happen, by definition, so that rewrite doesn't work.

Yes it can. Consider a posterior distribution, say for A_{cen}, which peaks at 0.95 or so. Suppose the value of the posterior probability at this peak is Pmax.

For each value of A_{cen} there is a corresponding value of the posterior probability, P.

In order to determine the 16 and 84 percentiles, what do you do? Well, you find the posterior probability P{68} such that 68 percent of the chain elements have P>P{68}. In the example, you see that the values of A{cen} that correspond to this P{68} are A_{16} (the 16 percentile) and the prior constraint +1. I hope the example demonstrates that 84 percentiles can coincide with the upper limit.....

Frank

Andrew R. Zentner Associate Professor & Director of Graduate Studies Department of Physics & Astronomy The University of Pittsburgh

On Jun 17, 2016, at 4:49 PM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.commailto:notifications@github.com> wrote:

Ignore my previous suggestion...rather I suggest the following:

In cases where the 16 or 84 percentiles coincide with a physical or prior bound, we only quote a lower or upper limit on the parameter, respectively.

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andrew-zentner commented 8 years ago

The reason I don't like saying that is that there is no unambiguous meaning to the phrase "68% confidence interval." There are many possible choices for such an interval and different authors make different choices. The only requirement is that the interval contain 68% of the posterior. To me, the statement that the posterior PDF increases monotonically toward a physical boundary of the parameter space is completely unambiguous.

vdbosch69 commented 8 years ago

The reason I don't like saying that is that there is no unambiguous meaning to the phrase "68% confidence interval." There are many possible choices for such an interval and different authors make different choices. The only requirement is that the interval contain 68% of the posterior. To me, the statement that the posterior PDF increases monotonically toward a physical boundary of the parameter space is completely unambiguous.

Well, it is your call. The posterior is on A_{sat} for the -19 sample (Fig 5) is NOT monotonic, yet, you quote only a lower bound in Table 2. But if you want to stick with your description, that is fine with me.

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andrew-zentner commented 8 years ago

I'll think it over.

Andrew R. Zentner Associate Professor & Director of Graduate Studies Department of Physics & Astronomy The University of Pittsburgh

On Jun 20, 2016, at 9:02 AM, Frank C. van den Bosch notifications@github.com<mailto:notifications@github.com> wrote:

The reason I don't like saying that is that there is no unambiguous meaning to the phrase "68% confidence interval." There are many possible choices for such an interval and different authors make different choices. The only requirement is that the interval contain 68% of the posterior. To me, the statement that the posterior PDF increases monotonically toward a physical boundary of the parameter space is completely unambiguous.

Well, it is your call. The posterior is on A_{sat} for the -19 sample (Fig 5) is NOT monotonic, yet, you quote only a lower bound in Table 2. But if you want to stick with your description, that is fine with me.

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andrew-zentner commented 8 years ago

Tried one more phrasing. Check it out.

vdbosch69 commented 8 years ago

YES. Much better. I like it. Please also rewrite this in the caption of Table 2!

Thanks!!!!

Prof. Frank C. van den Bosch e-mail: frank.vandenbosch@yale.edu Astronomy Department, Yale University phone: +1-203-432-0196 P.O. Box 208101 fax: +1-203-432-5048

New Haven, CT 06520-8101 http://www.astro.yale http://www.astro.yale/.edu/vdbosch

On Jun 20, 2016, at 3:10 PM, Andrew Zentner notifications@github.com wrote:

Tried one more phrasing. Check it out.

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