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    <p>Dear all,</p>
    <p>I always try to refrain myself from getting into these
      discussions, but I cannot resist more the temptation. Here are
      some more ideas that I hope bring more light than confusion:</p>
    <p>- There must be some functional relationship between the FSC and
      the SNR, but the exact analytical form of this relationship is
      unknown (I suspect that it must be at least monotonic, the worse
      the SNR, the worse FSC; but even this is difficult to prove). The
      relationship we normally use FSC=SNR/(1+SNR) was derived in a
      context that does not apply to CryoEM (1D stationary signals in
      real space; our molecules are not stationary), and consequently
      any reasoning of any threshold based on this relationship is
      incorrect (see our review).</p>
    <p>- Still, as long as we all use the same threshold, the reported
      resolutions are comparable to each other. In that regard, I am
      happy that we have set 0.143 (although any other number would have
      served the purpose) as the standard.</p>
    <p>- I totally agree with Steve that the full FSC is much more
      informative than its crossing with the threshold. Specially,
      because we should be much more worried about its behavior when it
      has high values than when it has low values. Before crossing the
      threshold it should be as high as possible, and that is the "true
      measure" of goodness of the map. When it crosses the threshold of
      0.143, it has too low SNR, and by definition, that is a very
      unstable part of the FSC, resulting in relatively unstable reports
      of resolution. We made some tests about the variability of the FSC
      (refining random splits of the dataset), trying to put the error
      bars that Steve was asking for, and it turned out to be pretty
      reproducible (rather low variance except in the region when it
      crosses the threshold) as long as the dataset was large enough
      (which is the current state).</p>
    <p>- @Marin, I always suffer with your reference to sloppy
      statistics. If we take your paper of 2005 where the 1/2 bit
      criterion was proposed (<a
href="https://www.sciencedirect.com/science/article/pii/S1047847705001292">https://www.sciencedirect.com/science/article/pii/S1047847705001292</a>),
      Eqs. 4 to 15 have completely ignored the fact that you are dealing
      with Fourier components, that are complex numbers, and
      consequently you have to deal with random variables that have two
      components, which moreover the real and imaginary part are not
      independent and, in their turn, they are not independent of the
      nearby Fourier coefficients so that for computing radial averages
      you would need to account for the correlation among coefficients (<a
href="https://www.aimspress.com/fileOther/PDF/biophysics/20150102.pdf">https://www.aimspress.com/fileOther/PDF/biophysics/20150102.pdf</a>).
      For properly dealing the statistics, at least one needs to carry
      out a two-dimensional reasoning, including the complex conjugate
      multiplication which is all missing in your derivation, rather
      than treating everything as one-dimensional, real valued random
      variables. Additionally, embedded in your whole reasoning is the
      idea that the expected value of a ratio is the ratio of the
      expected values, that is a 0-th order Taylor approximation of the
      mean of the distribution of a ratio between two random variables.
      Finally, I always find an extreme difficulty to understand the 1
      bit or 1/2 bit criteria, that is, what is the relationship between
      the channel's capacity formula of Shannon (<a
        href="https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem">https://en.wikipedia.org/wiki/Shannon%E2%80%93Hartley_theorem</a>)
      and our FSC (we do not have any channel through which we are
      "transmitting" our volume, although it is true we have a model
      y=x+n that is the same as in signal transmission, it is not true
      that the average information of a signal is log2(1+SNR); for me,
      the only relationship is that the SNR appears in both formulas,
      FSC and channel capacity, but that does not automatically make
      them comparable and interchangeble). This is not a criticism on
      your work. I think the FSC is a very useful tool to measure some
      properties of the reconstruction process and the quality of the
      dataset (not everything is measured by the FSC) and it also has
      its drawbacks (for instance, systematic errors are rewarded by the
      FSC as they are reproducible in both halves). Moreover, I think
      you are an extremely intelligent person, who I consider a good
      friend, with a very good intuition about image processing and who
      has brought very interesting ideas and methodologies into the
      field. Only that we cannot become crazy about the FSC threshold
      and the reported resolution, as the most interesting part of the
      FSC is not when it is low, but when it is high.</p>
    <p>I hope I can keep refraining myself in the future :-)</p>
    <p>Cheers, Carlos Oscar<br>
    </p>
    <p>On 2/21/20 6:19 PM, Ludtke, Steven J. wrote:<br>
    </p>
    <blockquote type="cite"
      cite="mid:84516CBB-AE8E-49B4-A123-7C6724E93CE2@bcm.edu">
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      I've been steadfastly refusing to get myself dragged in this time,
      but with this very sensible statement (which I am largely in
      agreement with), I thought I'd throw in one thought, just to stir
      the pot a little more.
      <div class=""><br class="">
      </div>
      <div class="">This is not a new idea, but I think it is the most
        sensible strategy I've heard proposed, and addresses Marin's
        concerns in a more conventional way. What we are talking about
        here is the statistical noise present in the FSC curves
        themselves. Viewed from the framework of traditional error
        analysis and propagation of uncertainties, which pretty much
        every scientist should be familiar with since high-school, (and
        thus would not be confusing to the non statisticians)  the
        'correct' solution to this issue is not to adjust the threshold,
        but to present FSC curves with error bars. </div>
      <div class=""><br class="">
      </div>
      <div class="">One can then use a fixed threshold at a level based
        on expectation values, and simply produce a resolution value
        which also has an associated uncertainty. This is much better
        than using a variable threshold and still producing a single
        number with no uncertainty estimate!  Not only does this
        approach account for the statistical noise in the FSC curve, but
        it also should stop people from reporting resolutions as 2.3397
        Å, as it would be silly to say 2.3397 +- 0.2.  </div>
      <div class=""><br class="">
      </div>
      <div class="">The cross terms are not ignored, but are used in the
        production of the error bars. This is a very simple approach,
        which is certainly closer to being correct than the fixed
        threshold without error-bars approach, and it solves many of the
        issues we have with resolution reporting people do.  Of course
        we still have people who will insist that 3.2+-0.2 is better
        than 3.3+-0.2, but there isn't much you can do about them...
        (other than beat them over the head with a statistics textbook).</div>
      <div class="">
        <div class=""><br class="">
        </div>
        <div class="">The caveat, of course, is that like all
          propagation of uncertainty that it is a linear approximation,
          and the correlation axis isn't linear, so the typical Normal
          distributions with linear propagation used to justify
          propagation of uncertainty aren't _strictly_ true. However,
          the approximation is fine as long as the error bars are
          reasonably small compared to the -1 to 1 range of the
          correlation axis. Each individual error bar is computed around
          its expectation value, so the overall nonlinearity of the
          correlation isn't a concern.</div>
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                      <font class="" face="Courier"><span
                          style="font-size: 14px;" class="">--------------------------------------------------------------------------------------<br
                            class="">
                          Steven Ludtke, Ph.D. <<a
                            href="mailto:sludtke@bcm.edu" class=""
                            moz-do-not-send="true">sludtke@bcm.edu</a>> 
                                              Baylor College
                          of Medicine <br class="">
                          Charles C. Bell Jr., Professor of
                          Structural Biology<br class="">
                          Dept. of Biochemistry and Molecular Biology   
                                            (<a
                            href="http://www.bcm.edu/biochem" class=""
                            moz-do-not-send="true">www.bcm.edu/biochem</a>)<br
                            class="">
                          Academic Director, CryoEM Core               
                                                  (<a
                            href="http://cryoem.bcm.edu" class=""
                            moz-do-not-send="true">cryoem.bcm.edu</a>)<br
                            class="">
                          Co-Director CIBR Center                       
                                      (<a
                            href="http://www.bcm.edu/research/cibr"
                            class="" moz-do-not-send="true">www.bcm.edu/research/cibr</a>)<br
                            class="">
                          <br class="">
                        </span></font><br class="">
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            <div><br class="">
              <blockquote type="cite" class="">
                <div class="">On Feb 21, 2020, at 10:34 AM, Alexis Rohou
                  <<a href="mailto:a.rohou@gmail.com" class=""
                    moz-do-not-send="true">a.rohou@gmail.com</a>>
                  wrote:</div>
                <br class="Apple-interchange-newline">
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                      class="">***CAUTION:*** This email is not from a
                      BCM Source. Only click links or open attachments
                      you know are safe.</b></font><span
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                    Hi all,
                    <div class=""><br class="">
                    </div>
                    <div class="">For those bewildered by Marin's
                      insistence that everyone's been messing up their
                      stats since the bronze age, I'd like to offer what
                      my understanding of the situation. More details in
                      this thread from a few years ago on the exact same
                      topic: </div>
                    <div class=""><a
href="https://urldefense.proofpoint.com/v2/url?u=https-3A__mail.ncmir.ucsd.edu_pipermail_3dem_2015-2DAugust_003939.html&d=DwMFaQ&c=ZQs-KZ8oxEw0p81sqgiaRA&r=Dk5VoQQ-wINYVssLMZihyC5Dj_sWYKxCyKz9E4Lp3gc&m=UWn2RUCMENrXjn3JLSwlIU6Zmp_JYnRrXesjtsM1u2E&s=CZ3YcAV1LVKXsLT0KjCIRby6j3XPA6GqZcOVP3nMyK0&e="
                        class="" moz-do-not-send="true">https://mail.ncmir.ucsd.edu/pipermail/3dem/2015-August/003939.html</a><br
                        class="">
                    </div>
                    <div class=""><a
href="https://urldefense.proofpoint.com/v2/url?u=https-3A__mail.ncmir.ucsd.edu_pipermail_3dem_2015-2DAugust_003944.html&d=DwMFaQ&c=ZQs-KZ8oxEw0p81sqgiaRA&r=Dk5VoQQ-wINYVssLMZihyC5Dj_sWYKxCyKz9E4Lp3gc&m=UWn2RUCMENrXjn3JLSwlIU6Zmp_JYnRrXesjtsM1u2E&s=oG6lGnei74jC5VVGsfFAdiTpIxrZhs_IH2mH0re5QRM&e="
                        class="" moz-do-not-send="true">https://mail.ncmir.ucsd.edu/pipermail/3dem/2015-August/003944.html</a></div>
                    <div class=""><br class="">
                    </div>
                    <div class="">Notwithstanding notational problems
                      (e.g. strict equations as opposed to approximation
                      symbols, or omission of symbols to denote
                      estimation), I believe Frank & Al-Ali and
                      "descendent" papers (e.g. appendix of Rosenthal
                      & Henderson 2003) are fine. The cross terms
                      that Marin is agitated about indeed do in fact
                      have an expectation value of 0.0 (in the ensemble;
                      if the experiment were performed an infinite
                      number of times with different realizations of
                      noise). I don't believe Pawel or Jose Maria or any
                      of the other authors really believe that the
                      cross-terms are orthogonal.</div>
                    <div class=""><br class="">
                    </div>
                    <div class="">When N (the number of independent
                      Fouier voxels in a shell) is large enough,
                      mean(Signal x Noise) ~ 0.0 is only an
                      approximation, but a pretty good one, even for a
                      single FSC experiment. This is why, in my book,
                      derivations that depend on Frank & Al-Ali are
                      OK, under the strict assumption that N is large.
                      Numerically, this becomes apparent when Marin's
                      half-bit criterion is plotted - asymptotically it
                      has the same behavior as a constant threshold.</div>
                    <div class=""><br class="">
                    </div>
                    <div class="">So, is Marin wrong to worry about
                      this? No, I don't think so. There are indeed cases
                      where the assumption of large N is broken. And
                      under those circumstances, any fixed threshold
                      (0.143, 0.5, whatever) is dangerous. This is
                      illustrated in figures of van Heel & Schatz
                      (2005). Small boxes, high-symmetry, small objects
                      in large boxes, and a number of other conditions
                      can make fixed thresholds dangerous.</div>
                    <div class=""><br class="">
                    </div>
                    <div class="">It would indeed be better to use a
                      non-fixed threshold. So why am I not using the
                      1/2-bit criterion in my own work? While
                      numerically it behaves well at most resolution
                      ranges, I was not convinced by Marin's derivation
                      in 2005. Philosophically though, I think he's
                      right - we should aim for FSC thresholds that are
                      more robust to the kinds of edge cases mentioned
                      above. It would be the right thing to do.</div>
                    <div class=""><br class="">
                    </div>
                    <div class="">Hope this helps,</div>
                    <div class="">Alexis </div>
                    <div class=""><br class="">
                    </div>
                    <div class=""><br class="">
                    </div>
                  </div>
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                    <div dir="ltr" class="gmail_attr">On Sun, Feb 16,
                      2020 at 9:00 AM Penczek, Pawel A <<a
                        href="mailto:Pawel.A.Penczek@uth.tmc.edu"
                        class="" moz-do-not-send="true">Pawel.A.Penczek@uth.tmc.edu</a>>
                      wrote:<br class="">
                    </div>
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                      rgb(204, 204, 204); padding-left: 1ex;">
                      <div dir="auto" class="">Marin,
                        <div class=""><br class="">
                        </div>
                        <div class="">The statistics in 2010 review is
                          fine. You may disagree with assumptions, but I
                          can assure you the “statistics” (as you call
                          it) is fine. Careful reading of the paper
                          would reveal to you this much. <br class="">
                          <br class="">
                          <div dir="ltr" class="">Regards,
                            <div class="">Pawel</div>
                          </div>
                          <div dir="ltr" class=""><br class="">
                            <blockquote type="cite" class="">On Feb 16,
                              2020, at 10:38 AM, Marin van Heel <<a
                                href="mailto:marin.vanheel@googlemail.com"
                                target="_blank" class=""
                                moz-do-not-send="true">marin.vanheel@googlemail.com</a>>
                              wrote:<br class="">
                              <br class="">
                            </blockquote>
                          </div>
                          <blockquote type="cite" class="">
                            <div dir="ltr" class="">
                              <div class="">
                                <p class=""><font class="" size="3"
                                    face="arial, sans-serif" color="Red"><strong
                                      class="">**** EXTERNAL EMAIL ****</strong></font></p>
                              </div>
                              <div dir="ltr" class="">
                                <div class="">Dear Pawel and All others
                                  ....
                                  <div style="margin: 0cm 0cm 8pt;
                                    line-height: 17.1200008392334px;
                                    font-size: 12pt; font-family:
                                    "Times New Roman", serif;"
                                    class="">
                                    <br class="webkit-block-placeholder">
                                  </div>
                                  <p class="MsoNormal" style="margin:
                                    0cm 0cm 8pt; line-height:
                                    17.1200008392334px; font-size: 12pt;
                                    font-family: "Times New
                                    Roman", serif;">
                                    This 2010 review is - unfortunately
                                    - largely based on the flawed
                                    statistics I mentioned before,
                                    namely on the a priori assumption
                                    that the inner product of a signal
                                    vector and a noise vector are ZERO
                                    (an orthogonality assumption).  The
                                    (Frank & Al-Ali 1975) paper we
                                    have refuted on a number of
                                    occasions (for example in 2005, and
                                    most recently in our BioRxiv paper)
                                    but you still take that as the
                                    correct relation between SNR and FRC
                                    (and you never cite the
                                    criticism...). <span class=""></span></p>
                                  <span class=""></span></div>
                                <div class="">Sorry</div>
                                <div class="">Marin<br class="">
                                </div>
                              </div>
                              <br class="">
                              <div class="gmail_quote">
                                <div dir="ltr" class="gmail_attr">On
                                  Thu, Feb 13, 2020 at 10:42 AM Penczek,
                                  Pawel A <<a
                                    href="mailto:Pawel.A.Penczek@uth.tmc.edu"
                                    target="_blank" class=""
                                    moz-do-not-send="true">Pawel.A.Penczek@uth.tmc.edu</a>>
                                  wrote:<br class="">
                                </div>
                                <blockquote class="gmail_quote"
                                  style="margin: 0px 0px 0px 0.8ex;
                                  border-left-width: 1px;
                                  border-left-style: solid;
                                  border-left-color: rgb(204, 204, 204);
                                  padding-left: 1ex;">
                                  <div dir="auto" class="">
                                    <div dir="ltr" class="">Dear Teige,
                                      <div class=""><br class="">
                                      </div>
                                      <div class="">I am wondering
                                        whether you are familiar with</div>
                                      <div class="">
                                        <h2 style="box-sizing:
                                          border-box; font-family:
                                          "Lucida Grande",
                                          "Helvetica Neue",
                                          Helvetica, Arial; font-size:
                                          20px; margin: 8px auto;
                                          line-height: 24px;" class="">
                                          <br class="">
                                        </h2>
                                        <h2 style="box-sizing:
                                          border-box; font-family:
                                          "Lucida Grande",
                                          "Helvetica Neue",
                                          Helvetica, Arial; font-size:
                                          20px; margin: 8px auto;
                                          line-height: 24px;" class="">
                                          Resolution measures in
                                          molecular electron microscopy.</h2>
                                        <div
id="gmail-m_-325618919662780292gmail-m_4473107327319981263gmail-m_-4642610284911275282cit_summary"
                                          style="box-sizing: border-box;
                                          font-size: 14px; color:
                                          rgb(39, 39, 39); font-family:
                                          Arial;" class="">
                                          <div style="box-sizing:
                                            border-box;" class="">Penczek
                                            PA. Methods Enzymol. 2010.</div>
                                        </div>
                                        <div
id="gmail-m_-325618919662780292gmail-m_4473107327319981263gmail-m_-4642610284911275282cit_full"
                                          style="box-sizing: border-box;
                                          font-size: 14px; color:
                                          rgb(39, 39, 39); font-family:
                                          Arial;" class="">
                                          <h3 style="box-sizing:
                                            border-box; font-family:
                                            "Lucida Grande",
                                            "Helvetica Neue",
                                            Helvetica, Arial;
                                            margin-bottom: 0px;
                                            font-size: 14px;" class="">
                                            Citation</h3>
                                          <p style="box-sizing:
                                            border-box; margin: 2px
                                            0px;" class="">Methods
                                            Enzymol. 2010;482:73-100.
                                            doi:
                                            10.1016/S0076-6879(10)82003-8.</p>
                                        </div>
                                        <div class=""><br class="">
                                        </div>
                                        <div class="">You will find
                                          there answers to all questions
                                          you asked and much more. </div>
                                        <br class="">
                                        <div dir="ltr" class="">Regards,
                                          <div class="">Pawel Penczek</div>
                                        </div>
                                      </div>
                                    </div>
                                    <br class="">
                                    <br class="">
                                    <div dir="ltr" class="">Regards,
                                      <div class="">Pawel</div>
                                    </div>
                                  </div>
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      <pre class="moz-quote-pre" wrap="">_______________________________________________
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