{"title": "Ambiguous Model Learning Made Unambiguous with 1/f Priors", "book": "Advances in Neural Information Processing Systems", "page_first": 1205, "page_last": 1212, "abstract": "", "full_text": "Ambiguous model learning made unambiguous\n\nwith 1/f priors\n\nG. S. Atwal\n\nDepartment of Physics\nPrinceton University\nPrinceton, NJ 08544\n\nWilliam Bialek\n\nDepartment of Physics\nPrinceton University\nPrinceton, NJ 08544\n\ngatwal@princeton.edu\n\nwbialek@princeton.edu\n\nAbstract\n\nWhat happens to the optimal interpretation of noisy data when there\nexists more than one equally plausible interpretation of the data? In a\nBayesian model-learning framework the answer depends on the prior ex-\npectations of the dynamics of the model parameter that is to be inferred\nfrom the data. Local time constraints on the priors are insuf\ufb01cient to\npick one interpretation over another. On the other hand, nonlocal time\nconstraints, induced by a 1/f noise spectrum of the priors, is shown to\npermit learning of a speci\ufb01c model parameter even when there are in-\n\ufb01nitely many equally plausible interpretations of the data. This transition\nis inferred by a remarkable mapping of the model estimation problem\nto a dissipative physical system, allowing the use of powerful statisti-\ncal mechanical methods to uncover the transition from indeterminate to\ndeterminate model learning.\n\n1\n\nIntroduction\n\nThe estimation of a model underlying the production of noisy data becomes highly non-\ntrivial when there exists more than one equally plausible model that could be responsible\nfor the output data. The viewing of ambiguous \ufb01gures, such as the Necker cube [1], is\na classical problem of this type in the \ufb01eld of visual psychology. Pitch perception when\nhearing a number of different harmonics is another example of ambiguous perception [2].\n\nPrevious studies [3] have reduced the problem of optimal interpretation of an ambiguous\nstimulus to the problem of estimating a single variable which may vary in time \u03b1(t), given\na time sequence of noisy data. Enforcing a prior belief that the local dynamics \u03b1(t) should\nnot vary too rapidly embodies the observer\u2019s knowledge that rapid variations in \u03b1(t) are\nunlikely in the natural world or in a given experiment. Such a prior prevents over\ufb01tting the\nmodel estimate to the data as it arrives. The statistically optimal interpretation of the data\nwas then found to consist of \u03b1(t) hopping randomly from one possible interpretation to\nanother. The rate of random switching between interpretations was found to be controlled\nnot by the noise level (e.g. in the neural hardware), as previously thought, but rather by\nthe observer\u2019s prior hypotheses. This hopping persists inde\ufb01nitely despite the fact that\nthe probability distribution of the incoming data remains the same.\nIn such cases it is\nimpossible to learn a speci\ufb01c model parameter.\n\n\fIn this paper we introduce another prior over the dynamics of \u03b1(t). We assume that \ufb02uc-\ntuations in \u03b1(t) have a 1/f spectrum, as observed ubiquitously in nature. Such a prior is\nshown to induce nonlocal time constraints on the trajectories of \u03b1(t) and, unlike the local\nconstraints, can result in speci\ufb01c model learning in the case of ambiguous models. The fact\nthat 1/f priors can induce unambiguous model learning is the central result of this work.\nThe analyses of the long-time dynamics with nonlocal priors is permitted by a surprising\nand remarkable mapping to a dissipative quantum system. This mapping not only guides\nour intuition of the optimal trajectories of \u03b1(t) but also permits the usage of powerful\nstatistical mechanical techniques.\nIn particular, the renormalization group (RG) can be\nemployed to uncover the conditions in which there is a transition from non-speci\ufb01c model\nlearning to speci\ufb01c model learning.\n\n2 Formalism\n\nSuppose that we are given a series of N measurements {xt} at discrete times t. Then Bayes\nrule gives us the conditional probability of {\u03b1t} giving rise to those data\n\nP [{\u03b1t}|{xt}] =\n\nP [{xt}|{\u03b1t}] P [{\u03b1t}]\n\nP [{xt}]\n\n,\n\n(1)\n\nwhere the probability of making the observations {xi} is given by summing up all the\npossible models that may give rise to them,\n\nP ({xt}) =Z d\u03b1 P [{xt}|{\u03b1t}]P [{\u03b1t}].\n\nWe further assume conditional independence of signals,\n\nN\n\nP [{xt}|{\u03b1t}] = P [x1x2...xN |{\u03b1t}] =\n\nP [xt|\u03b1t].\n\n(2)\n\n(3)\n\nYt=1\n\nA natural step is then to consider how close our estimate of the model \u03b1(t) lies to the true\nunderlying model \u03b1(t), which we take to be stationary \u03b1(t) = \u03b1. We can think of these\nprobability distributions as Boltzmann distributions in which some effective potential acts\nto hold \u03b1 close to \u00af\u03b1; thus we envision an energy landscape in the \u03b1 space with a minimum\nat \u00af\u03b1.\nA more interesting, and generalized, question arises when we consider the global properties\nof the extended energy landscape. In particular there may be M > 1 equally plausible\ninterpretations consistent with the input data1 in which case there exist degenerate minima\nat \u03b1m (m = 1, 2....M),\n\nP [xt|\u03b11] = P [xt|\u03b12] = ... = P [xt|\u03b1M ].\n\nTherefore we may write Eq. (3) as\n\nP [{xt}|{\u03b1t}] =\n\nN\n\nYt=1 M\nYm=1\n\nP [xt|\u03b1m]1/M! exp\" 1\n\nM\n\n(4)\n\n(5)\n\nln\n\nP [xt|\u03b1t]\n\nP [xt|\u03b1m]# .\n\nM\n\nN\n\nXm=1\n\nXt=1\n\nOn average, the term in square brackets is related to the Kullback-Leibler divergences be-\ntween distributions conditional on \u03b1(t) and distributions conditional on the true \u00af\u03b1. If the\n\n1Of course it may be the case that some interpretations may be more plausible than others, result-\ning in a non uniform probability distribution over possible models. In this paper we illustrate the case\nwhere all interpretations are equally likely, P [\u03b1m] = 1/M.\n\n\ftime variation of \u03b1 is slow, we effectively collect many samples of x before \u03b1 changes, and\nit makes sense to replace the sum over samples by its average:\n\nlim\nN\u2192\u221e\n\nM\n\nN\n\nXm=1\n\nXt=1\n\nln\n\nP [xt|\u03b1t]\nP [xt|\u03b1m]\n\n\u2248\n\n1\n\u03c40\n\nM\n\nXm=1Z dtZ dxP [x(t)|\u03b1m] ln\nXm=1Z dtDKL[\u03b1m||\u03b1(t)].\n\nM\n\n1\n\u03c40\n\n\u2261 \u2212\n\nP [x(t)|\u03b1(t)]\n\nP [x|\u03b1m]\n\n,\n\n(6)\n\nwhere \u03c40 is the average time between observations, and we take the continuum limit.\n\n2.1 Priors\n\nWe need to have some prior hypotheses about how \u03b1(t) can vary in time, serving as our\nprior probability distribution P [\u03b1(t)]. We introduce two different types of priors character-\nized by whether they constrain the local or nonlocal time dynamics,\n\nP [\u03b1(t)] = Plocal[\u03b1(t)]Pnonlocal[\u03b1(t)].\n\n(7)\n\nTo summarize our prior expectation that the local dynamics of \u03b1(t) vary slowly, we assume\nthat the time derivative of \u03b1(t) is chosen independently at each instant of time from a\nGaussian distribution,\n\nPlocal[\u03b1(t)] \u221d exp\"\u2212\n\n1\n\n4D Z dt(cid:18) \u2202\u03b1\n\n\u2202t(cid:19)2# .\n\n(8)\n\n(9)\n\nNote that this distribution corresponds to random walk with effective diffusion constant D.\nMotivated by the ubiquitous occurrence of 1/f \ufb02uctuations in nature we chose to encapsu-\nlate the nonlocal dynamics by a Gaussian distribution with a 1/f power spectrum of noise,\nconveniently expressed in Fourier coordinates \u03c9 as\n1\n\n|\u03b1(\u03c9)|2\n\nPnonlocal[\u03b1(t)] \u221d exp(cid:20)\u2212\n\n2Z d\u03c9\n\n2\u03c0\n\nS(\u03c9) (cid:21) ,\n\nwhere the spectral noise function takes the form\n\nS(\u03c9) =\n\n1\n\n\u03b7|\u03c9|\n\n.\n\n(10)\n\nNote that the spectrum must be even in \u03c9 since for any stationary process S(\u03c9) = S(\u2212\u03c9).\nThe parameter \u03b7 determines the strength of a priori belief in nonlocal dynamics, or as\nwe will see later, it can be equivalently viewed as a frictional constant determining the\ndissipation of the time trajectories of \u03b1(t). In the time-domain Eq. (9) becomes\n\nPnonlocal[\u03b1(t)] \u221d exp\"\u2212\n\n\u03b7\n\n4\u03c0 Z dtdt\u2032(cid:18) \u03b1(t) \u2212 \u03b1(t\u2032)\n\nt \u2212 t\u2032\n\n(cid:19)2# .\n\n(11)\n\nCombining Eq. (8) and Eq. (11) we then obtain the total prior expectation of the probability\ndistribution over the time-dependence of the model parameter \u03b1(t)\n\nP [\u03b1(t)] \u221d exp\"\u2212\n\n1\n\n\u2202t(cid:19)2\n4D Z dt(cid:18) \u2202\u03b1\n\n\u2212\n\n\u03b7\n\n4\u03c0 Z dtdt\u2032(cid:18) \u03b1(t) \u2212 \u03b1(t\u2032)\n\nt \u2212 t\u2032\n\n(cid:19)2# .\n\n(12)\n\nTaken together, the local and non-local terms describe \ufb02uctuations in \u03b1 which are 1/f up\nto a cutoff frequency, \u03c9c \u223c D\u03b7. Returning to the Bayesian conditional probability Eq. (1)\nwe then obtain a path-integral expression\n\nP [\u03b1(t)|{xi}] \u221d exp(\u2212S[\u03b1(t)]),\n\n(13)\n\n\fwhere the action S[\u03b1(t)] is given by\n\nS[\u03b1(t)] = Z dt\" 1\n\u2202t(cid:19)2\n4D (cid:18) \u2202\u03b1\nXm=1\n\n\u03c40M\n\nVe\ufb00 [\u03b1(t)] =\n\nM\n\n1\n\nDKL[\u03b1m||\u03b1(t)].\n\n+ \u03b7Z dt\u2032\n\n4\u03c0 (cid:18) \u03b1(t) \u2212 \u03b1(t\u2032)\n\nt \u2212 t\u2032\n\n(cid:19)2\n\n+ Ve\ufb00 [\u03b1(t)]# , (14)\n\n(15)\n\nThis is equivalent to the imaginary time path-integral for a quantum mechanical particle [4]\nof mass 1/2D , with coordinates given by \u03b1(t), moving in an effective potential Ve\ufb00 [\u03b1(t)]\nand subject to (linear) frictional forces with a damping constant \u03b7. This mapping provides\nan extremely useful guide to our intuition for the probable trajectories of \u03b1(t). Just as in the\nanalyses of particle dynamics in dissipative quantum mechanics [4] we anticipate that the\ntime-course of \u03b1(t) may exhibit qualitatively different types of behavior depending on the\nstrength of the non-local terms. In addition, the equivalence to a physical system permits\nexploitation of powerful techniques developed in the study of quantum mechanical systems\nwith in\ufb01nite degrees of freedom.\n\nIn the following we consider the cases of m = 1 and m = 2 and use the RG transformations\nto consider localization-delocalization transitions.\n\n2.2 M=1 : One true interpretation of data\n\nNow if \u03b1(t) differs from \u03b1 by a small \u2206\u03b1(t) we can Taylor expand the Kullback-Leibler\ndivergence to give a quadratic distance measure\n\nDKL(\u03b1||\u03b1) =\n\n1\n2\n\nF [\u03b1(t)]\u2206\u03b1(t)2 + O(\u2206\u03b13),\n\nwhere the metric is the Fisher information\n\nF [\u03b1(t)] =Z dx\n\n1\n\n\u2202\u03b1(t) (cid:19)2\nP [x|\u03b1(t)](cid:18) \u2202P [x|\u03b1(t)]\n\n.\n\n(16)\n\n(17)\n\nThus, close to the true parameter \u03b1 the potential energy term in Eq. (14) is simply a har-\nmonic oscillator with stiffness given by the Fisher information. Guided by the mapping to a\ndissipative quantum mechanical system we expect that if the initial distribution of \u03b1 already\nhappens to be closely centered around the correct value then the most likely trajectory will\nbe simply to move closer to the minima of the potential energy at \u03b11.\nThe important point to note is that had we chosen just the local constraints on our priors\nEq. (8) then the trajectory of \u03b1(t) would persistently \ufb02uctuate around \u03b11, representing a\ntrade-off between avoiding over\ufb01tting the data and inertia of our estimate. In the quan-\ntum mechanical picture this corresponds to the zero point \ufb02uctuations around the minima.\nAdding the dissipative term reduces the \ufb02uctuations around \u03b11 by an amount monotoni-\ncally dependent on \u03b7, thus improving on the optimal estimate.\nA RG treatment of the single-well problem, within the harmonic approximation, renor-\nmalizes the Fisher information such that the curvature of the potential well increases for\nall values of the \u03b7, and thus the \ufb01xed point of the dynamics is simply the convergence\nof \u03b1(t) to reduced \ufb02uctuations around the true parameter \u03b11. We explicitly carry out the\nRG calculation in the more interesting case where we have two global minima in the next\nsection.\n\n\f2.3 M=2 : Two equally possible interpretations of the data\n\nIn the case of two equally viable interpretations of the data, the potential energy term\nbecomes that of a double-well potential with degenerate minima at \u03b11 and \u03b12 and energy\nbarrier h\n\nh =\n\n1\n2\u03c40\n\n(DKL(\u03b11||(\u03b11 + \u03b12)/2) + DKL(\u03b12||\u03b11 + \u03b12)/2))\n\n(18)\n\n20\n\n15\n\n)\n\u03b1\n(\nV\n\n10\n\n5\n\n0\n-10\n\na1\n\n-5\n\nh\n\n0\n\na2\n\n5\n\n10\n\n \u03b1\n\nFigure 1: Potential energy landscape for \u03b1 where there exist two equally valid interpreta-\ntions. Eq. (19)\n\nWithout any dissipative dynamics, the optimal estimate of \u03b1(t) will switch between the\ntwo minima, representing instanton trajectories of a quantum particle tunnelling through\nthe energy barrier backwards and forwards [3]. In contrast, it is well known that, at least in\nsome regimes, the problem with dissipation has a phase transition to a truly localized state.\nPrevious work has demonstrated such a dynamical phase transition in the strong-coupling\nlimit (i.e. large barrier height limit) using semi-classical approximations for the dynamics\n[4,5,6], and in this section we will show that a perturbative RG treatment yields similar\nresults in the opposite weak-coupling limit.\n\nFor the sake of simplicity we employ the following simple quartic potential (see Fig.1),\nalthough the results will be independent of its exact form,\n\nV (\u03b1) =\n\n.\n\n(19)\n\nh\n\u03b14\n\n2\n\n1 (cid:0)\u03b12 \u2212 \u03b12\n1(cid:1)\n\nThe \u03b1 coordinates have been shifted such that \u03b11 = \u2212\u03b12, and the height h of the energy\nbarrier located at \u03b1 = 0 sets the overall energy scale. It is useful to write the effective\naction of Eq. (14) in dimensionless parameters\n\na =\n\n\u03b1\n\u03b11\n\n,\n\nb = \u03b7\u03b12\n1,\n\nc =\n\nh\n\u039b\n\n,\n\nwhere \u039b = D/\u03b12\n\n1 is the energy/frequency scale 2\n\nS =\n\n1\n\n2Z d\u03c9\n\n2\u03c0 (cid:18) 1\n\n2\u039b\n\n\u03c92 + b|\u03c9|(cid:19) |a(\u03c9)|2 + c\u039bZ dt V \u2032(a),\n\nV \u2032(a) = (a2 \u2212 1)2.\n\n(20)\n\n(21)\n\n(22)\n\n2The constant of proportionality between energy and frequency is set to 1, akin to the common\n\nphysics computation setting of \u00afh = 1.\n\n\fBy power counting in the \ufb01rst integral the dissipative term, at low frequencies, dominates\nover the kinetic energy term. In the language of RG, the kinetic energy term is an irrelevant\noperator and can thus be ignored if we now focus our attention to frequencies below some\ncut-off \u03bb. To determine the RG \ufb02ow of the dimensionless coupling parameters the high-\nfrequency components are integrated out from \u03c9 = \u03bb \u2212 d\u03bb to \u03c9 = \u03bb to give a new effective\naction \u02dcS over the low frequency modes \u03c9 < \u03bb. To accomplish this the function \u03b1(\u03c9) is\nsplit\n\na(\u03c9) = a<(\u03c9)\u03b8(|\u03c9| < \u03bb \u2212 d\u03bb) + a>(\u03c9)\u03b8(\u03bb \u2212 d\u03bb < |\u03c9| < \u03bb),\n\nand the new action is obtained by integrating over a>(\u03c9),\n\nZ = Z Da exp[\u2212S(a)],\n\n= Z Z Da exp[\u2212S(a< + a>)],\n= Z Da< exp[\u2212 \u02dcS(a<)].\n\nb\n\n2Z \u03bb\u2212d\u03bb\n\n0\n\nd\u03c9\n2\u03c0\n\n|\u03c9||a<(\u03c9)|2 + ln(cid:28)exp(cid:20)c\u039bZ dtV \u2032(a< + \u03b1>)(cid:21)(cid:29)a>\n\n,\n\nTherefore,\n\n\u02dcS(a<) =\n\nwhere the averaging is de\ufb01ned by\n\nhAia> \u221dZ Da> exp(\u2212\n\nb\n\n2Z \u039b\n\n\u039b\u2212d\u039b\n\nd\u03c9\n2\u03c0\n\n|\u03c9||a>(\u03c9)|2) A.\n\n(23)\n\n(24)\n\n(25)\n\n(26)\n\nIn the weak-coupling limit, we may expand the exponential term in Eq. (25) before per-\nforming the averaging,\n\n(cid:28)exp[c\u039bZ dtV \u2032(a< + a>)](cid:29)a>\n\n=(cid:28)1 + c\u039bZ dtV \u2032(a< + a>) + ...(cid:29)a>\n\n.\n\n(27)\n\nTerminating the expansion to \ufb01rst order in the potential represents a one-loop calculation\nin \ufb01eld theories.\n\nMaking use of\n\n\u03bb\u2212d\u03bb\nwe \ufb01nd that the potential term renormalizes as\n\n(cid:10)a2\n>(t)(cid:11)a>\n\n=Z \u03bb\n\nd\u03c9\n\u03c0\n\n1\n\nb|\u03c9|\n\n\u2248\n\n1\n\u03c0b\n\nd\u03bb\n\u03bb\n\n,\n\n(c\u039b(a2 \u2212 1)2)\u03bb \u21d2 (c\u039b(a2 \u2212 1)2)\u03bb\u2212d\u03bb \u2248 (c\u039b)\u03bb(cid:20)(a2\n\n< \u2212 1)2 + (3a2\n\n< \u2212 1)\n\n(28)\n\n2\n\u03c0b\n\nd\u03bb\n\n\u03bb (cid:21) , (29)\n\nwhere we have ignored terms including higher powers of d\u03bb/\u03bb. To recast the new lower-\nfrequency action into the same form as the original action the dimensionless coupling pa-\nrameters must be renormalised. In particular, we observe that the dimensionless barrier\nheight c can either grow or shrink depending on the value of the dimensionless dissipation\nb. Note that the coordinates must also be rescaled (also known as wavefunction renor-\nmalization) for the potential in Eq. (29) to maintain the same quartic form as in Eq. (22),\nthereby inducing a rescaling of b. We concentrate here on the renormalized potential cou-\npling term and \ufb01nd that, up to a constant,\n\nc\u03bb\u2212d\u03bb = c\u03bb(cid:20)1 +\n\nd\u03bb\n\n\u03bb (cid:18)1 \u2212\n\n6\n\n\u03c0b(cid:19)(cid:21) ,\n\n(30)\n\n\fgiving then the following differential RG \ufb02ow equation\n\ndc\n\nd ln \u03bb\n\n=(cid:18) b\u2217\n\nb\n\n\u2212 1(cid:19) c.\n\n(31)\n\nAs the (dimensionless) barrier height c renormalizes towards lower frequencies we observe\ntwo types of behavior depending on whether the parameter b is greater or smaller than the\ncritical value b\u2217 = 6/\u03c0 (the actual numerical value may well be slightly altered by going\nto higher orders in the perturbative expansion, but the important point to note that it is non-\nzero and thus gives rise to distinct dynamical phases). For b > b\u2217 the barrier height grows\nwithout bounds and thus effectively traps \u03b1(t) in one of the two minima, representing a\nlocalized phase. This localization can be brought about by increasing the magnitude of \u03b7,\nthe numerical prefactor of our dissipative nonlocal priors, and/or increasing \u03b11 the distance\nbetween the two possible interpretations of the data. On the other hand, for b < b\u2217 the\npotential becomes ineffective in localizing \u03b1, and thus \u03b1 freely tunnels between the two\nwells, representing indeterminancy of the correct true model parameter.\n\nIt is interesting to note that a \ufb02ow equation, similar to Eq. (31), has been reported for the\nopposite limit (strong-coupling) using the instanton method[5,6]. Arguably what we have\nreally shown is that even if one starts with weak coupling, so that it should be \u201deasy\u201d to\njump from one interpretation to another, for b > b\u2217 we will \ufb02ow to strong-coupling, at\nwhich point known results about localization take over.\n\nlocal\n\n8\n\nc\n\n0\n\n*\nnonlocal b\n\nb\n\nFigure 2: Schematic RG \ufb02ow of the potential energy coupling parameter for M \u2265 2. Note\nthat the \ufb02ow-lines are not expected to be strictly vertical due to wavefunction renormaliza-\ntion.\n\nThe qualitative picture does not change when there are more than two possible model in-\nterpretations, M > 2. In fact, the case of M = \u221e has been studied [7] where the potential\nenergy landscape is taken to be sinusoidal, and it has been demonstrated that there again\nexists a critical value b\u2217 which separates a localized phase from a nonlocalized phase. The\n\ufb02ow of the potential energy coupling constant c is shown in Fig.2 which is expected to be\nqualitatively correct across the whole range 2 \u2264 M \u2264 \u221e.\n\n3 Discussion\n\nIn summary, the optimal model estimate in the response of ambiguous signals always re-\nsults in random perceptual switching when the priors only constrain the local dynamics.\nWe have shown that when we allow the possibility of 1/f noise in our priors then a speci\ufb01c\nmodel is learnt amongst the many possible models.\n\nThe connection between estimation theory and statistical mechanics is well known. One\nof the key results in statistical mechanics is that local interactions in one dimension can\n\n\fnever lead to a phase transition. Thus if we are interested in, for example, learning a\nsingle parameter by making repeated observations, then there can be no phase transition\nto certainty about the value of this parameter as long as our prior hypotheses about its\ndynamics are equivalent to local models in statistical mechanics. Markov models, Gaussian\nprocesses with rational spectra, and other common priors all fall in this local class.\n\nThe common occurrence of 1/f \ufb02uctuations in nature motivates the analyses of estimation\ntheory with such priors. Crucially, 1/f spectra do not correspond to local models. In fact\nthey correspond exactly to the addition of friction to the path integral describing a quantum\nmechanical particle, a problem of general interest in condensed matter physics and more\nrecently in quantum computing. Here we note one important consequence of these priors,\nnamely that we can process data in a model which admits the possibility of time variation\nfor the underlying parameter, but nonetheless \ufb01nd that our best estimate of this parameter\nis localized for all time to one of many equally plausible alternatives. It seems that 1/f\npriors may provide a way to understand the emergence of certainty more generally as a\nphase transition.\n\nReferences\n\n[1] G. H. Fisher, Perception & Psychophysics 4, 189 (1968)\n\n[2] E. de Boer, Handbk. Sens. Physiol. 3, 479 (1976)\n\n[3] W. Bialek and M. DeWeese, M. Phys. Rev. Lett. 74, 3079 (1995)\n\n[4] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981)\n\n[5] A. J. Bray and M. A. Moore, Phys. Rev. Lett 49, 1545 (1982)\n\n[6] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zwerger, Rev. Mod.\nPhys. 59, 1 (1987)\n\n[7] M. P. A. Fisher and W. Zwerger, Phys. Rev. Lett 32, 6190 (1985)\n\n\f", "award": [], "sourceid": 2459, "authors": [{"given_name": "Gurinder", "family_name": "Atwal", "institution": null}, {"given_name": "William", "family_name": "Bialek", "institution": null}]}