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  • I guess another way of putting the point: There is a combination lock. The combination to open the lock for the ancients of that population is not the same combination to open the lock for the present day folks. Our autosomal DNA is an ever changing combination.

    It is true that we ultimately must descend from our ancestors in the sense that we don't pop up out of thin air. But, we would not be able to identify them as our ancestors within the sense of IBD (identical by descent) segments of autosomal DNA.

    The math shows this to be impossible.

    If for 5,000 years (i.e. 167 to 200 generations)

    exp(-(22+33*199)/2^199) = exp(-(6589)/8.0346902e+59) = exp(-8.2006896e-57) = 1
    exp(-(22+33*166)/2^166) = exp(-(5500)/9.3536105e+49) = exp(-5.8800823e-47) = 1

    then,

    2^199 *(1-1) = 8.0346902e+59 * 0 = 0 genetic ancestors
    2^166 *(1-1) = 0 genetic ancestors

    Comment


    • Originally posted by Barreldriver View Post
      I do think that FTDNA needs to explain their reasoning on some things, in particular time span. e.g. Western and Central Europe represents the diverse groups brought together over the past 5,000 years, as Germans, Celts, and Slavs have moved in with their cattle, and the Romans brought their mills and cities (Khan 2014).

      The claim doesn't seem to make sense when one calculates the # of genetic ancestors vs. the # of ancestors-on-paper. Our admixtures reflect very recent events in the context of history. It is true that we must have came forth from the ancients, however it is not true that they are our "genetic ancestors" - they've been erased as a result of recombination.

      I will be referring to the CoopLab of UC Davis.

      *A note: the following assumes a case of non-inbreeding.

      According to CoopLab, the probability that you inherit zero of your autosomal genome from a particular ancestor is approximately exp(-(22+33*(k-1))/2^(k-1)) (Coop 2013). K = the number of generations. I will make examples for recent (k = 7, k = 6, & k = 5) and ancient (k = 167 & k = 200).

      If k = 7, then the probability that you inherit zero of your autosomal genome from a particular ancestor is exp(-(22+33*6)/2^6) = exp(-(220)/64) = exp(-3.4375) = 0.03214494732. Next, 2^7 *(1- 0.03214494732) = 128 *(0.96785505268) = 123.885446743. Even though this is approaching an equal number of genetic vs. genealogical ancestors, there are still a bit less genetic ancestors than genealogical ones.

      If k = 6, then the probability that you inherit zero of you autosomal genome from a particular ancestor is exp(-(22+33*5)/2^5). exp(-(22+33*5)/2^5) = exp(-(187)/32) = exp(-5.84375) = 0.00289795489. Next, 2^6 *(1- 0.00289795489) = 64 *(0.99710204511) = 63.814530887. Even though the number of expected genetic ancestors is not precisely equal to the number of genealogical ancestors, the difference is quite small.

      If k =5, then the probability that you inherit zero of you autosomal genome from a particular ancestor is exp(-(22+33*4)/2^4). exp(-(22+33*4)/2^4) = exp(-(154)/16) = exp(-9.625) = 0.0000660565. Next, 2^5 *(1- 0.0000660565) = 32 *(0.9999339435) = 31.997886192. The number of expected genetic ancestors is still not precisely equal to the number of genealogical ancestors, but it’s close enough to wager that 5 generations back all the ancestors in that time frame contributed to your autosomal DNA.


      In the Occidental world the typical generation is held to be 25 to 30 years as a rule of thumb. So one can roughly estimate that 5,000 years is around 167 to 200 generations. 2^k = the number of “ancestors-on-paper”. 2^167 = 1.8707221e+50 and 2^200 = 1.606938e+60.

      Now, for the number of genetic ancestors:

      First:

      exp(-(22+33*199)/2^199) = exp(-(6589)/8.0346902e+59) = exp(-8.2006896e-57) = 1
      exp(-(22+33*166)/2^166) = exp(-(5500)/9.3536105e+49) = exp(-5.8800823e-47) = 1

      Second:

      2^199 *(1-1) = 8.0346902e+59 * 0 = 0
      2^166 *(1-1) = 0

      In short, one has zero “genetic ancestors” 5,000 years back.


      References

      Coop, Graham. et al. 2013. How many genetic ancestors do I have? URL: http://gcbias.org/2013/11/11/how-doe...ack-over-time/

      Khan, Razib. 2014. Population Clusters in myOrigins. URL: https://www.familytreedna.com/learn/...tion-clusters/
      So we all materialized 5000 ybp. LOL

      Comment


      • Originally posted by 1798 View Post
        So we all materialized 5000 ybp. LOL
        No, that's taking the argument to the absurd extreme (reductio ad absurdum). My point (which has to be withdrawn anyways) was that within the past 5,000 years our autosomal DNA has become so scrambled over the generations that we wouldn't be identifiable as atDNA descendants from someone that we claimed was in our pedigree that far back. Which would in a sense be a kind of an erase-event.

        I will state that I ultimately have to withdraw the conclusion with regard to the time frame of 5,000 years back based upon the insight of others at another forum (though the math still fits within a time frame of 10 generations to approximate where genetic ancestors become less than ancestors-on-paper):

        Post #45

        http://www.anthrogenica.com/showthre...ll=1#post44777

        To quote:

        "The problem is not that math is wrong. It never ever is. It's applying an existing mathematical model that does not explain the physical world. The physical observation is first and the math and/or mathematical model is created to explain the event. In the two cases above the "bad" model was an approximation that becomes inaccurate at extreme criteria (size and speed in those two cases).

        In Telfermagne example it's very clear the extreme size of the number of ancestors causes the extremely low probability. So yes for a given ancestor the probability having measurable IBD is extremely low .. essentially zero, but the sum of all those probabilities from all the ancestors is still one.

        Our DNA is finite (many billions of pairs) so at the finest level each loci could in theory be mapped back to a single ancestor with a 50% chance of passed to the next generation (not counting the X/Y Chromosomes).

        This is expressed in math as a limit: As the number of ancestors approaches infinity the probability approaches one." - thetick
        Last edited by Barreldriver; 7 July 2014, 08:14 PM.

        Comment


        • Extending my error to imply "ex nihilo" is not necessary. The intent of my earlier conclusion was to say that atDNA has become so scrambled that practical identification of descent from pedigree ancestors 5,000 years ago wouldn't be possible, and in that sense the ancestry would be "erased".

          (*Pulverized hammer parable: There was a hammer. It was pulverized, what remains is not identifiable as a hammer, then do not call it a hammer. It is instead a pile of remnants. - if an alleged "descendant" is not identifiable as a "descendant" then why call them one?).

          Comment


          • Originally posted by Barreldriver View Post
            No, that's taking the argument to the absurd extreme (reductio ad absurdum). My point (which has to be withdrawn anyways) was that within the past 5,000 years our autosomal DNA has become so scrambled over the generations that we wouldn't be identifiable as atDNA descendants from someone that we claimed was in our pedigree that far back. Which would in a sense be a kind of an erase-event.

            I will state that I ultimately have to withdraw the conclusion with regard to the time frame of 5,000 years back based upon the insight of others at another forum (though the math still fits within a time frame of 10 generations to approximate where genetic ancestors become less than ancestors-on-paper):

            Post #45

            http://www.anthrogenica.com/showthre...ll=1#post44777

            To quote:

            "The problem is not that math is wrong. It never ever is. It's applying an existing mathematical model that does not explain the physical world. The physical observation is first and the math and/or mathematical model is created to explain the event. In the two cases above the "bad" model was an approximation that becomes inaccurate at extreme criteria (size and speed in those two cases).

            In Telfermagne example it's very clear the extreme size of the number of ancestors causes the extremely low probability. So yes for a given ancestor the probability having measurable IBD is extremely low .. essentially zero, but the sum of all those probabilities from all the ancestors is still one.

            Our DNA is finite (many billions of pairs) so at the finest level each loci could in theory be mapped back to a single ancestor with a 50% chance of passed to the next generation (not counting the X/Y Chromosomes).

            This is expressed in math as a limit: As the number of ancestors approaches infinity the probability approaches one." - thetick
            How does it work for the small unknown tribes in the Amazon?
            Don't they get their autosomal dna from the same group of ancestors for eons and they don't have many birth defects or do they? It is amazing how it all works. First cousins can have perfectly normal children.

            Comment


            • If I recall right the CoopLab had stated that the approximation assumes "non-inbreeding", which would make sense for the common genealogies within 10 generations or so. Of course there's going to be exceptional cases like extremely isolated tribes or families, but they aren't the norm hence approximation instead of determination (Something that I had meant to convey in the original renditions, but I think it got buried/overshadowed. If it wasn't included, then there was an error on my part.).

              Originally posted by 1798 View Post
              How does it work for the small unknown tribes in the Amazon?
              Don't they get their autosomal dna from the same group of ancestors for eons and they don't have many birth defects or do they? It is amazing how it all works. First cousins can have perfectly normal children.

              Comment


              • Myorigins

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