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Thread Subject:
How to combine two probability density functions (PDFs)?

Subject: How to combine two probability density functions (PDFs)?

From: Ryan

Date: 24 Aug, 2010 19:29:23

Message: 1 of 6

I am writing to seek help with modeling the probability of an event (age), which is bracketed by two ages, where each age is defined by a Gaussian probability density function (PDF).

To start simply, consider the case where an event is known to have occurred between two brackets. For example:
-The upper age bracket is described by a Gaussian age with a mean value of 5000 (yr) and a 1-sigma value of 200 (yr).
-The lower age bracket is also described by a Gaussian age, but with a mean value of 2000 (yr) and a 1-sigma value of 400 (yr).

My 1st question: how can I use the two age brackets to define a new PDF reflecting the probability of the event, which is known to have occurred between the two features? Note: my intuition is that the shape of the PDF should be a plateau between the brackets with Gaussian tails.

My 2nd question: how does the solution change where the distance between the end-member brackets decreases and the sigma values increase (e.g., when the tails of each age bracket overlap, especially given that the event defined by the old bracket has to have occurred prior to the event defined by the younger bracket)

Thanks for any help that you can provide!

Sincerely,
Ryan

Subject: How to combine two probability density functions (PDFs)?

From: Roger Stafford

Date: 24 Aug, 2010 21:30:35

Message: 2 of 6

"Ryan " <rgold@usgs.gov> wrote in message <i516ij$opj$1@fred.mathworks.com>...
> I am writing to seek help with modeling the probability of an event (age), which is bracketed by two ages, where each age is defined by a Gaussian probability density function (PDF).
>
> To start simply, consider the case where an event is known to have occurred between two brackets. For example:
> -The upper age bracket is described by a Gaussian age with a mean value of 5000 (yr) and a 1-sigma value of 200 (yr).
> -The lower age bracket is also described by a Gaussian age, but with a mean value of 2000 (yr) and a 1-sigma value of 400 (yr).
>
> My 1st question: how can I use the two age brackets to define a new PDF reflecting the probability of the event, which is known to have occurred between the two features? Note: my intuition is that the shape of the PDF should be a plateau between the brackets with Gaussian tails.
>
> My 2nd question: how does the solution change where the distance between the end-member brackets decreases and the sigma values increase (e.g., when the tails of each age bracket overlap, especially given that the event defined by the old bracket has to have occurred prior to the event defined by the younger bracket)
>
> Thanks for any help that you can provide!
>
> Sincerely,
> Ryan
- - - - - - - - -
  I am struggling to understand what you are asking, Ryan. It doesn't seem like a well-defined question. You don't appear to have given the "event" (between the brackets) any a-priori probability distribution and therefore no question about its conditional probability given that it lies between the two Gaussians can be answered. Also you haven't stated whether these Gaussians are mutually independent or at least jointly Gaussian. I think you need to describe your problem a lot more carefully before anyone can give you meaningful help.

Roger Stafford

Subject: How to combine two probability density functions (PDFs)?

From: Ross W

Date: 25 Aug, 2010 04:41:04

Message: 3 of 6

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <i51dlr$5bf$1@fred.mathworks.com>...
> "Ryan " <rgold@usgs.gov> wrote in message <i516ij$opj$1@fred.mathworks.com>...
> > I am writing to seek help with modeling the probability of an event (age), which is bracketed by two ages, where each age is defined by a Gaussian probability density function (PDF).
> >
> > To start simply, consider the case where an event is known to have occurred between two brackets. For example:
> > -The upper age bracket is described by a Gaussian age with a mean value of 5000 (yr) and a 1-sigma value of 200 (yr).
> > -The lower age bracket is also described by a Gaussian age, but with a mean value of 2000 (yr) and a 1-sigma value of 400 (yr).
> >
> > My 1st question: how can I use the two age brackets to define a new PDF reflecting the probability of the event, which is known to have occurred between the two features? Note: my intuition is that the shape of the PDF should be a plateau between the brackets with Gaussian tails.
> >
> > My 2nd question: how does the solution change where the distance between the end-member brackets decreases and the sigma values increase (e.g., when the tails of each age bracket overlap, especially given that the event defined by the old bracket has to have occurred prior to the event defined by the younger bracket)
> >
> > Thanks for any help that you can provide!
> >
> > Sincerely,
> > Ryan
> - - - - - - - - -
> I am struggling to understand what you are asking, Ryan. It doesn't seem like a well-defined question. You don't appear to have given the "event" (between the brackets) any a-priori probability distribution and therefore no question about its conditional probability given that it lies between the two Gaussians can be answered. Also you haven't stated whether these Gaussians are mutually independent or at least jointly Gaussian. I think you need to describe your problem a lot more carefully before anyone can give you meaningful help.
>
> Roger Stafford

Hi Ryan,

Is it true that the age you seek is UNIFORMLY distributed between the upper age and the lower age? Or is it distributed in some other way between those two ages?

Ross

Subject: How to combine two probability density functions (PDFs)?

From: Ryan

Date: 27 Aug, 2010 21:02:23

Message: 4 of 6

"Ross W" <rosswoodskiwi@hotmail.com> wrote in message <i526t0$ruu$1@fred.mathworks.com>...
> "Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <i51dlr$5bf$1@fred.mathworks.com>...
> > "Ryan " <rgold@usgs.gov> wrote in message <i516ij$opj$1@fred.mathworks.com>...
> > > I am writing to seek help with modeling the probability of an event (age), which is bracketed by two ages, where each age is defined by a Gaussian probability density function (PDF).
> > >
> > > To start simply, consider the case where an event is known to have occurred between two brackets. For example:
> > > -The upper age bracket is described by a Gaussian age with a mean value of 5000 (yr) and a 1-sigma value of 200 (yr).
> > > -The lower age bracket is also described by a Gaussian age, but with a mean value of 2000 (yr) and a 1-sigma value of 400 (yr).
> > >
> > > My 1st question: how can I use the two age brackets to define a new PDF reflecting the probability of the event, which is known to have occurred between the two features? Note: my intuition is that the shape of the PDF should be a plateau between the brackets with Gaussian tails.
> > >
> > > My 2nd question: how does the solution change where the distance between the end-member brackets decreases and the sigma values increase (e.g., when the tails of each age bracket overlap, especially given that the event defined by the old bracket has to have occurred prior to the event defined by the younger bracket)
> > >
> > > Thanks for any help that you can provide!
> > >
> > > Sincerely,
> > > Ryan
> > - - - - - - - - -
> > I am struggling to understand what you are asking, Ryan. It doesn't seem like a well-defined question. You don't appear to have given the "event" (between the brackets) any a-priori probability distribution and therefore no question about its conditional probability given that it lies between the two Gaussians can be answered. Also you haven't stated whether these Gaussians are mutually independent or at least jointly Gaussian. I think you need to describe your problem a lot more carefully before anyone can give you meaningful help.
> >
> > Roger Stafford
>
> Hi Ryan,
>
> Is it true that the age you seek is UNIFORMLY distributed between the upper age and the lower age? Or is it distributed in some other way between those two ages?
>
> Ross

Hi Ross,

Thanks for your response.

My expectation is that the age I seek is UNIFORMLY distributed.

Here's a description of a scenario: There is a stratigraphic horizon which I want to date (e.g., represent an earthquake in the geologic record). Below that horizon, I have dated deeper layer of sediment that provides a maximum constraint on the age of the earthquake. Similarly, above the earthquake horizon, I have dated a shallower layer of sediment that provides a minimum constraint on the age of the earthquake. The question: how do I describe the age of the earthquake based on these two boundaries?

Here is a modified scenario, from the one I previously described: Maximum age constraint = 10000 (yr) with a 1-sigma value of 400 (yr) [normally distributed] / Minimum age constraint = 5000 (yr) and a 1-sigma value of 200 (yr) [normally distributed].

My intuition is that I lack sufficient knowledge about sedimentation rates, erosion rates, etc., to state with confidence where the earthquake event that occurs between the boundaries is closer to the maximum or minimum bracket. Thus, I expect to see a uniformly distributed zone in the region between the constraints (e.g., flat PDF from 6000-8500 yr, in the example above), with tails that taper toward the boundaries.

Many thanks for additional input!

Sincerely,
Ryan

Subject: How to combine two probability density functions (PDFs)?

From: Roger Stafford

Date: 28 Aug, 2010 00:32:23

Message: 5 of 6

"Ryan " <rgold@usgs.gov> wrote in message <i5994v$koj$1@fred.mathworks.com>...
> My expectation is that the age I seek is UNIFORMLY distributed.
>
> Here's a description of a scenario: There is a stratigraphic horizon which I want to date (e.g., represent an earthquake in the geologic record). Below that horizon, I have dated deeper layer of sediment that provides a maximum constraint on the age of the earthquake. Similarly, above the earthquake horizon, I have dated a shallower layer of sediment that provides a minimum constraint on the age of the earthquake. The question: how do I describe the age of the earthquake based on these two boundaries?
>
> Here is a modified scenario, from the one I previously described: Maximum age constraint = 10000 (yr) with a 1-sigma value of 400 (yr) [normally distributed] / Minimum age constraint = 5000 (yr) and a 1-sigma value of 200 (yr) [normally distributed].
>
> My intuition is that I lack sufficient knowledge about sedimentation rates, erosion rates, etc., to state with confidence where the earthquake event that occurs between the boundaries is closer to the maximum or minimum bracket. Thus, I expect to see a uniformly distributed zone in the region between the constraints (e.g., flat PDF from 6000-8500 yr, in the example above), with tails that taper toward the boundaries.
>
> Many thanks for additional input!
>
> Sincerely,
> Ryan
- - - - - - - -
  We can make the assumption that before we are given that the three events are in sequence, their a priori distributions are all mutually independent. Given that the earthquake occurs at age t, then the probability that this lies between the ages of the other two events is:

 p(t) = normcdf(t,5000,200)*(1-normcdf(t,10000,400));

This is just the product of the probabilities, given that the earthquake is at time t, that the "10000" layer age was greater than t and the "5000" layer age was less than t.

  I claim that p(t) is proportional to your desired pdf quantity, but to make this a valid probability density it should be normalized by being divided by its integral w.r. to t from -inf to +inf, since you have assumed a uniform a priori distribution for the earthquake age. Offhand I cannot think of any fast way to do this integration, so you might have to resort to using one of matlab's numerical quadrature routines to get the normalizing factor.

  A plot of p(t) supports your claim that there is a "plateau" in between the earlier and later ages. Given all the assumptions above, the same formula would still be valid if these distributions are adjusted so that the tails overlap by appreciable amounts.

  Even if sedimentation rates are uncertain, I am surprised that no account is being taken of the thickness of the sediment layers in this problem. I would think the age of the earthquake would still be correlated to a significant degree with the respective thicknesses above and below to the dated layers. However, I have to admit my knowledge of geology is rather minimal. (My brother who is in fact a geologist can testify to that fact.)

Roger Stafford

Subject: How to combine two probability density functions (PDFs)?

From: Ryan

Date: 30 Aug, 2010 19:55:05

Message: 6 of 6

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <i59len$b87$1@fred.mathworks.com>...
> "Ryan " <rgold@usgs.gov> wrote in message <i5994v$koj$1@fred.mathworks.com>...
> > My expectation is that the age I seek is UNIFORMLY distributed.
> >
> > Here's a description of a scenario: There is a stratigraphic horizon which I want to date (e.g., represent an earthquake in the geologic record). Below that horizon, I have dated deeper layer of sediment that provides a maximum constraint on the age of the earthquake. Similarly, above the earthquake horizon, I have dated a shallower layer of sediment that provides a minimum constraint on the age of the earthquake. The question: how do I describe the age of the earthquake based on these two boundaries?
> >
> > Here is a modified scenario, from the one I previously described: Maximum age constraint = 10000 (yr) with a 1-sigma value of 400 (yr) [normally distributed] / Minimum age constraint = 5000 (yr) and a 1-sigma value of 200 (yr) [normally distributed].
> >
> > My intuition is that I lack sufficient knowledge about sedimentation rates, erosion rates, etc., to state with confidence where the earthquake event that occurs between the boundaries is closer to the maximum or minimum bracket. Thus, I expect to see a uniformly distributed zone in the region between the constraints (e.g., flat PDF from 6000-8500 yr, in the example above), with tails that taper toward the boundaries.
> >
> > Many thanks for additional input!
> >
> > Sincerely,
> > Ryan
> - - - - - - - -
> We can make the assumption that before we are given that the three events are in sequence, their a priori distributions are all mutually independent. Given that the earthquake occurs at age t, then the probability that this lies between the ages of the other two events is:
>
> p(t) = normcdf(t,5000,200)*(1-normcdf(t,10000,400));
>
> This is just the product of the probabilities, given that the earthquake is at time t, that the "10000" layer age was greater than t and the "5000" layer age was less than t.
>
> I claim that p(t) is proportional to your desired pdf quantity, but to make this a valid probability density it should be normalized by being divided by its integral w.r. to t from -inf to +inf, since you have assumed a uniform a priori distribution for the earthquake age. Offhand I cannot think of any fast way to do this integration, so you might have to resort to using one of matlab's numerical quadrature routines to get the normalizing factor.
>
> A plot of p(t) supports your claim that there is a "plateau" in between the earlier and later ages. Given all the assumptions above, the same formula would still be valid if these distributions are adjusted so that the tails overlap by appreciable amounts.
>
> Even if sedimentation rates are uncertain, I am surprised that no account is being taken of the thickness of the sediment layers in this problem. I would think the age of the earthquake would still be correlated to a significant degree with the respective thicknesses above and below to the dated layers. However, I have to admit my knowledge of geology is rather minimal. (My brother who is in fact a geologist can testify to that fact.)
>
> Roger Stafford

Hi Roger,

Many thanks for the constructive suggestion. I think the solution works well (i.e., it produces a PDF that is consistent with my expectations).

Regarding your inquiry into sedimentation rates, you are correct that we are able to sometimes able to use information such as sample position, sediment type, and sediment package thickness to gain insight into the age of an earthquake, relative to the bracketing ages. Unfortunately, I lack that information for this particular project.

Thanks for your help.

Sincerely,
Ryan

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