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This is not a very typical MO question, but I hope you bear with me. It concerns a recent disagreement in the biology literature about how many different odors humans can discriminate. The authors of a paper in Science from March 2014 claimed that, based on their experiments, "humans can discriminate more than 1 trillion olfactory stimuli," a number which puts our other senses to shame. A recent critique of the Science paper, posted on arXiv, takes issue with the way the Science authors interpreted their data, and claims that even a number as small as merely 10 discriminable stimuli is consistent with the data. From what I can tell, the disagreement boils down to a very clear difference in assumptions, which I will try to explain in a simplified way that probably misses a lot of the more minute details, but hopefully gives a good idea of the mathematical question which is at play.

The Science authors give a subject three vials to smell. Each vial contains a random, equal-parts mixture of 30 compounds out of a repertoire of 128 compounds. Two of the vials have identical mixtures, and the subject is asked to pick the odd vial out. Think of each mixture as a binary vector $\mathbf{x}\in X=\{0,1\}^{128}$ of Hamming weight 30. The authors estimate a critical Hamming distance $D$ at which 50% of mixture pairs at this distance are discriminable. Not many tests need to be run to obtain a very good estimate of $D$.

Think of each discriminable "odor" as a set $S\subset X$, and assume that the mixture space is completely partitioned into $N$ odors $S_1,\ldots,S_N$. Mixture $\mathbf{x}$ and $\mathbf{y}$ can be told apart by a subject if and only if they are not in the same odor $S$. The main task is to infer a likely value for $N$. If odors are all roughly Hamming balls of equal radius, we can directly obtain the radius from the critical distance $D$, from the radius we get the volume of the ball, and we obtain $N\approx|X|/|S|$. This analysis appears to give an estimate of $N>10^{12}$.

The critique points out that there is no reason to assume that the odors are roughly spherical in the absence of detailed mechanistic knowledge about the olfactory system, which apparently we do not have. One interesting section of the critique shows that a similar analysis applied to a hypothetical experiment using color stimuli yields nonsensical results. In the color vision system, we know that if $X$ is a binary vector space describing mixtures of optical spectra, then only the projection of $\mathbf{x}$ into a real three dimensional space can be sensed, $(s,m,l)(\mathbf{x}) = \sum_i x_i (s_i,m_i,l_i)$. Therefore, the colors are far from spheres in $X$; they are more like the preimages of balls under this projection. Clearly, for this kind of highly anisotropic shape, the same critical distance $D$ corresponds to a much larger volume. I don't know enough about the biology to tell whether it is reasonable to expect that something similar could happen in the olfactory system, but the critique points out that when allowing for nonspherical shapes, the same value of $D$ is consistent with many-order-of-magnitudes-larger odors. A particular construction even yields a number $N=10$ as being consistent with the data.

Now, ignoring the biology, measuring $D$ is clearly a bad way to constrain $N$, since one value of $D$ is apparently consistent with both $N>10^{12}$ and $N=10$. The question I have for the MO crowd is what extra quantity, preferably obtainable from the same kind of odd-vial-out tests (maybe even extractable from the existing data), can actually constrain $N$ to a reasonable range.

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    $\begingroup$ I don't think one can shed much light on this by mathematical means -- rather I think this is a genuine biology question to which, if at all, you can find an answer by investigating the olfactory system. But also bear in mind that there might be not a well-defined answer, as things may differ a lot between individuals, and for the same individual between different times of measurement. $\endgroup$
    – Stefan Kohl
    Nov 8 '14 at 11:15
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    $\begingroup$ en.wikipedia.org/wiki/Digital_scent_technology "In 1999, DigiScents developed a computer peripheral device called iSmell, which was designed to emit a smell when a user visited a web site or opened an email. The device contained a cartridge with 128 "primary odors", which could be mixed to replicate natural and man-made odors. DigiScents had indexed thousands of common odors, which could be coded, digitized, and embedded into web pages or email." Well, I'm not sure what spam (or a virus) would be would this system... $\endgroup$ Nov 8 '14 at 13:19
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    $\begingroup$ @StefanKohl, do you say that because you think there is no quantity that would constrain N in the model I set up in the question? Do you have a reason to think that? Or is it more that you think this model is inadequate? The model comes straight out of the Science paper, so it is definitely one that biologists take seriously. $\endgroup$ Nov 8 '14 at 13:54
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    $\begingroup$ @StefanKohl I thought that I did. To summarize: X is partitioned into sets S_1,...,S_N. The critical distance above which less than 0.5 of pairs of a fixed distance in X lie in different partitions does not constrain N well. What measurement does? $\endgroup$ Nov 8 '14 at 15:44
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    $\begingroup$ I'm dismayed at the votes to close and Stefan's critique. Yoav carefully explained the issues in such a way that a mathematician with no knowledge of this biological system could make a meaningful contribution. If we close this question, we might as well put a banner on the front page saying "applied math questions not welcome here". $\endgroup$ Nov 9 '14 at 16:34
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Suppose $\Omega = \{0,1\}^{128}$ is partitioned into $N$ different sets $S_j$, with $|S_j|/|X| = p_j$. Suppose you take $n$ random pairs of points of $\Omega$ and see how many of these pairs can't be distinguished (presumably because they are in the same partition).
The probability that two given points are in the same partition is $S_2 = \sum_{j=1}^N p_j^2$, so the expected number of indistinguishable pairs is $n S_2$; the fraction of pairs that are indistinguishable is an unbiased estimator of $S_2$. By Cauchy-Schwarz, $S_2 \ge 1/N$ (which would be the value if all $p_j$ are equal), so a good estimate on $S_2$ gives you a lower bound on $N$.

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    $\begingroup$ While this is correct, the number of measurements $n$ needed to estimate $S_2$ well is extremely large if indeed $N>10^{12}$. The good thing about $D$ was that it was cheap to estimate well. $\endgroup$ Nov 9 '14 at 14:20

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