nips nips2003 nips2003-51 knowledge-graph by maker-knowledge-mining

51 nips-2003-Design of Experiments via Information Theory

Source: pdf

Author: Liam Paninski

Abstract: We discuss an idea for collecting data in a relatively efﬁcient manner. Our point of view is Bayesian and information-theoretic: on any given trial, we want to adaptively choose the input in such a way that the mutual information between the (unknown) state of the system and the (stochastic) output is maximal, given any prior information (including data collected on any previous trials). We prove a theorem that quantiﬁes the effectiveness of this strategy and give a few illustrative examples comparing the performance of this adaptive technique to that of the more usual nonadaptive experimental design. For example, we are able to explicitly calculate the asymptotic relative efﬁciency of the “staircase method” widely employed in psychophysics research, and to demonstrate the dependence of this efﬁciency on the form of the “psychometric function” underlying the output responses. 1

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Summary: the most important sentenses genereted by tfidf model

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1 We prove a theorem that quantiﬁes the effectiveness of this strategy and give a few illustrative examples comparing the performance of this adaptive technique to that of the more usual nonadaptive experimental design. [sent-5, score-0.58]

2 For example, we are able to explicitly calculate the asymptotic relative efﬁciency of the “staircase method” widely employed in psychophysics research, and to demonstrate the dependence of this efﬁciency on the form of the “psychometric function” underlying the output responses. [sent-6, score-0.309]

3 1 Introduction One simple model of experimental design (we have neurophysiological experiments in mind, but our results are general with respect to the identity of the system under study) is as follows. [sent-7, score-0.18]

4 This knowledge is summarized in the form of a prior distribution, p0 (θ), on some space Θ of models θ. [sent-9, score-0.056]

5 A model is a set of probabilistic input-output relationships: regular conditional distributions p(y|x, θ) on Y , the set of possible output responses, given each x in X. [sent-10, score-0.04]

6 Thus the joint probability of stimulus and response is: p(x, y) = p(x, y, θ)dθ = p0 (θ)p(x)p(y|θ, x)dθ. [sent-11, score-0.121]

7 The “design” of an experiment is given by the choice of input probability p(x). [sent-12, score-0.045]

8 We want to design our experiment — choose p(x) — optimally in some sense. [sent-13, score-0.211]

9 One natural idea would be to choose p(x) in such a way that we learn as much as possible about the underlying model, on average. [sent-14, score-0.046]

10 Information theory thus suggests we choose p(x) to optimize the ∗ A longer version of this paper, including proofs, has been submitted and is available at http://www. [sent-15, score-0.046]

11 following objective function: I({x, y}; θ) = p(x, y, θ) log X×Y ×Θ p(x, y, θ) p(x, y)p(θ) (1) where I(. [sent-19, score-0.077]

12 In other words, we want to maximize the information provided about θ by the pair {x, y}, given our current knowledge of the model as summarized in the posterior distribution given N samples of data: pN (θ) = p(θ|{xi , yi }1≤i≤N ). [sent-22, score-0.12]

13 Somewhat surprisingly, we have not seen any applications of the information-theoretic objective function (1) to the design of neurophysiological experiments (although see the abstract by [7], who seem to have independently implemented the same idea in a simulation study). [sent-27, score-0.214]

14 The primary goal of this paper is to elucidate the asymptotic behavior of the a posteriori density pN (θ) when we choose x according to the recipe outlined above; in particular, we want to compare the adaptive strategy to the more usual case, in which the stimuli are drawn i. [sent-28, score-0.812]

15 Our main result (section 2) states that, under acceptably weak conditions on the models p(y|θ, x), the informationmaximization strategy leads to consistent and efﬁcient estimates of the true underlying model, in a natural sense. [sent-32, score-0.397]

16 Thus, instead of ﬁnding optimal distributions p(x), we need only ﬁnd optimal inputs x, in the sense of maximizing the conditional information between θ and y, given a single input x: I(y; θ|x) ≡ pN (θ)p(y|θ, x) log Y Θ p(y|x, θ) . [sent-36, score-0.04]

17 pN (θ)p(y|x, θ) Θ Our main result is a “Bernstein-von Mises” - type theorem [12]. [sent-37, score-0.15]

18 It turns out that the hard part is proving consistency (c. [sent-40, score-0.116]

19 section 4); we give the basic consistency lemma (interesting in its own right) ﬁrst, from which the main theorem follows fairly easily. [sent-42, score-0.333]

20 The loglikelihood log p(y|x, θ) is Lipschitz in θ, uniformly in x, with respect to some dominating measure on Y . [sent-47, score-0.2]

21 The prior measure p0 assigns positive measure to any neighborhood of θ0 . [sent-49, score-0.305]

22 The maximal divergence supx DKL (θ0 ; θ|x) is positive for all θ = θ0 . [sent-51, score-0.242]

23 Then the posteriors are consistent: pN (U ) → 1 in probability for any neighborhood U of θ0 . [sent-52, score-0.179]

24 Assume the conditions of Lemma 1, stengthened as follows: 1. [sent-54, score-0.065]

25 Θ has a smooth, ﬁnite-dimensional manifold structure in a neighborhood of θ 0 . [sent-55, score-0.107]

26 The loglikelihood log p(y|x, θ) is uniformly C 2 in θ. [sent-57, score-0.129]

27 In particular, the Fisher information matrices Iθ (x) = Y p(y|x, θ) ˙ p(y|x, θ) t p(y|x, θ) ˙ p(y|θ, x), p(y|x, θ) where the differential p is taken with respect to θ, are well-deﬁned and continuous ˙ in θ, uniformly in (x, θ) in some neighborhood of θ0 . [sent-58, score-0.229]

28 The prior measure p0 is absolutely continuous in some neighborhood of θ0 , with a continuous positive density at θ0 . [sent-60, score-0.43]

29 maxC∈co(Iθ0 (x)) det(C) > 0, where co(Iθ0 (x)) denotes the convex closure of the set of Fisher information matrices Iθ0 (x). [sent-62, score-0.04]

30 || denotes variation distance, N (µN , σN ) denotes the normal den2 sity with mean µN and variance σN , and µN is asymptotically normal with mean θ0 and 2 variance σN . [sent-64, score-0.474]

31 Thus, under these conditions, the information maximization strategy works, and works better than the i. [sent-66, score-0.178]

32 x strategy (where the asymptotic variance σ 2 is inversely related to an average, not a maximum, over x, and is therefore generically larger). [sent-69, score-0.431]

33 It should also be clear that we have not stated the results as generally as possible; we have chosen instead to use assumptions that are simple to understand and verify, and to leave the technical generalizations to the interested reader. [sent-73, score-0.046]

34 Our assumptions should be weak enough for most neurophysiological and psychophysical situations, for example, by assuming that parameters take values in bounded (though possibly large) sets and that tuning curves are not inﬁnitely steep. [sent-74, score-0.153]

35 The proofs of these three results are basically elaborations on Wald’s consistency method and Le Cam’s approach to the Bernstein-von Mises theorem [12]. [sent-75, score-0.267]

36 1 Applications Psychometric model As noted in the introduction, psychophysicists have employed versions of the informationmaximization procedure for some years [14, 9, 13, 6]. [sent-77, score-0.158]

37 Our results above allow us to precisely quantify the effectiveness of this stategy. [sent-79, score-0.056]

38 Let f be “sigmoidal”: a uniformly smooth, monotonically increasing function on the line, such that f (0) = 1/2, limt→−∞ f (t) = 0 and limt→∞ f (t) = 1 (this function represents the detection probability when the subject is presented with a stimulus of strength t). [sent-82, score-0.198]

39 Let fa,θ = f ((t − θ)/a); θ here serves as a location (“threshold”) parameter, while a sets the scale (we assume a is known, for now, although of course this can be relaxed [6]). [sent-83, score-0.069]

40 Finally, let p(x) and p 0 (θ) be some ﬁxed sampling and prior distributions, respectively, both with smooth densities with respect to Lebesgue measure on some interval Θ. [sent-84, score-0.362]

41 Now, for any ﬁxed scale a, we want to compare the performance of the informationmaximization strategy to that of the i. [sent-85, score-0.418]

42 We have by theorem 2 that the most efﬁcient estimator of θ is asymptotically unbiased with asymptotic variance 2 σinf o ≈ (N sup Iθ0 (x))−1 , x while the usual calculations show that the asymptotic variance of any efﬁcient estimator based on i. [sent-89, score-0.992]

43 sampling strategy is as asymptotically efﬁcient as informationmaximization strategy; for all other f , information maximization is strictly more efﬁcient. [sent-98, score-0.646]

44 [p(x)], and so f ∗ is the unique solution of the differential equation 1/2 df ∗ = c f ∗ (t)(1 − f ∗ (t)) , dt √ where the auxiliary constant c = I θ uniquely ﬁxes the scale a. [sent-101, score-0.127]

45 After some calculus, we obtain sin(ct) + 1 f ∗ (t) = 2 on the interval [−π/2c, π/2c] (and deﬁned uniquely, by monotonicity, as 0 or 1 outside this interval). [sent-102, score-0.069]

46 Since the support of the derivative of this function is compact, this result is quite dependent of the sampling density p(x); if p(x) places any of its mass outside of the interval 2 2 [−π/2c, π/2c], then σiid is always strictly greater than σinf o . [sent-103, score-0.264]

47 This recapitulates a basic theme from the psychophysical literature comparing adaptive and nonadaptive techniques: when the scale of the nonlinearity f is either unknown or smaller than the scale of the i. [sent-104, score-0.452]

48 2 Linear-nonlinear cascade model We now consider a model that has received increasing attention from the neurophysiology community (see, e. [sent-110, score-0.05]

49 Here the linear ﬁlter θ is some unit vector in X , the dual space of X (thus, the model space Θ is isomorphic to X), while the nonlinearity f is some nonconstant, nonnegative function on [−1, 1]. [sent-114, score-0.171]

50 We assume that f is uniformly smooth, to satisfy the conditions of theorem 2; we also assume f is known, although, again, this can be relaxed. [sent-115, score-0.248]

51 The response space Y — the space of possible spike counts, given the stimulus x — can be taken to be the nonnegative integers. [sent-116, score-0.16]

52 For simplicity, let the conditional probabilities p(y|x, θ) be parametrized uniquely by the mean ﬁring rate f (< θ, x >); the most convenient model, as usual, is to assume that p(y|x, θ) is Poisson with mean f (< θ, x >). [sent-117, score-0.098]

53 Finally, we assume that the sampling density p(x) is uniform on the unit sphere (this choice is natural for several reasons, mainly involving symmetry; see, e. [sent-118, score-0.305]

54 , [2, 8]), and that the prior p 0 (θ) is positive and continuous (and is therefore bounded away from zero, by the compactness of Θ). [sent-120, score-0.222]

55 The Fisher information for this model is easily calculated as Iθ (x) = (f (< θ, x >))2 f (< θ, x >) Px,θ , where f is the usual derivative of the real function f and Px,θ is the projection operator corresponding to x, restricted to the (d − 1)-dimensional tangent space to the unit sphere at θ. [sent-121, score-0.296]

56 1 apply here as well: the ratio σiid /σinf o grows roughly linearly in the inverse of the scale of the nonlinearity. [sent-124, score-0.069]

57 The more interesting asymptotics here, though, are in d. [sent-125, score-0.069]

58 This is because the unit sphere has a measure concentration property [11]: as d → ∞, the measure p(t) becomes exponentially concentrated around 0. [sent-126, score-0.353]

59 In fact, it is easy to show directly that, in this limit, p(t) converges in distribution to the normal measure with mean zero and variance d−2 . [sent-127, score-0.189]

60 The most surprising implication of this result is seen for nonlinearities f such that f (0) = 0, f (0) > 0; we have in mind, for example, symmetric nonlinearities like those often used to model complex cells in visual cortex. [sent-128, score-0.144]

61 For these nonlinearities, 2 σinf o = O(d−2 ) : 2 σiid that is, the information maximization strategy becomes inﬁnitely more efﬁcient than the usual i. [sent-129, score-0.304]

62 4 A Negative Example Our next example is more negative and perhaps more surprising: it shows how the information-maximation strategy can fail, in a certain sense, if the conditions of the consistency lemma are not met. [sent-133, score-0.458]

63 Let Θ be multidimensional, with coordinates which are “independent” in a certain sense, and assume the expected information obtained from one coordinate of the parameter remains bounded strictly away from the expected information obtained from one of the other coordinates. [sent-134, score-0.147]

64 5|, < 0 and 0 < θ1 < 1 are the parameters we want to learn. [sent-139, score-0.061]

65 Let the initial prior be absolutely continuous with respect to Lebesgue measure; this implies that all posteriors will have the same property. [sent-140, score-0.236]

66 Thus the information-maximization strategy will sample from x < 0 forever, leading to a linear information growth rate (and easily-proven consistency) for the left parameter and non-convergence on the right. [sent-142, score-0.13]

67 approach for choosing x (using any Lebesgue-dominating measure on the parameter space), which leads to the standard root-N rate for both parameters (i. [sent-146, score-0.071]

68 Note that this kind of inconsistency problem does not occur in the case of sufﬁciently smooth p(y|x, θ), by our main theorem. [sent-149, score-0.118]

69 Thus one way of avoiding this problem would be to ﬁx a ﬁnite sampling scale for each coordinate (i. [sent-150, score-0.205]

70 However, it is possible to ﬁnd other examples which show that the lack of consistency is not necessarily tied to the discontinuous nature of the conditional densities. [sent-154, score-0.156]

71 5 Directions In this paper, we have presented a rigorous theoretical framework for adaptively designing experiments using an information-theoretic objective function. [sent-155, score-0.174]

72 For example, our theorem 2 might suggest the use of a mixture-of-Gaussians representation as an efﬁcient approximation for the posteriors pN (θ) [5]. [sent-157, score-0.178]

73 Perhaps the most obvious such question concerns the use of non-information theoretic objective functions. [sent-159, score-0.077]

74 It turns out that many of our results apply with only modest changes if the experiment is instead designed to minimize something like the Bayes mean-square error (perhaps deﬁned only locally if Θ has a nontrivial manifold structure), for example: in this case, the results in sections 3. [sent-160, score-0.045]

75 2 remain completely unchanged, while the statement of our main theorem requires only slight changes in the asymptotic variance formula (see http://www. [sent-162, score-0.492]

76 Thus it seems our results here can add very little to any discussion of what objective function is “best” in general. [sent-166, score-0.077]

77 1 “Batch mode” and stimulus dependencies Perhaps our strongest assumption here is that the experimenter will be able to freely choose the stimuli on each trial. [sent-169, score-0.328]

78 Another common situation involves stimuli which vary temporally, for which the system is commonly modelled as responding not just to a given stimulus x(t), but also to all of its time-translates x(t − τ ). [sent-171, score-0.213]

79 Finally, if there is some cost C(x0 , x1 ) associated with changing the state of the observational apparatus from the current state x0 to x1 , the experimenter may wish to optimize an objective function which incorporates this cost, for example I(y; θ|x1 ) C(x0 , x1 ). [sent-172, score-0.146]

80 Here we restrict ourselves to the ﬁrst setting, and give a simple conjecture, based on the asymptotic results presented above and inspired by results like those of [1, 4, 10]. [sent-174, score-0.231]

81 First, we state more precisely the optimization problem inherent in designing a “batch” experiment: we wish to choose some sequence, {xi }1≤i≤k , to maximize I({xi , yi }1≤i≤k ; θ); the main difference here is that {xi }1≤i≤k must be chosen nonadaptively, i. [sent-175, score-0.129]

82 (Without such a smoothness condition — for example, if some input x could deﬁnitively decide between some given θ0 and θ1 — then no such asymptotic independence statement can hold, since no more than one sample from such an x would be necessary. [sent-179, score-0.272]

83 ) Thus, we can hope that we should be able to asymptotically approximate this optimal experiment by sampling in an i. [sent-180, score-0.295]

84 Moreover, we can make a guess as to the identity of this putative p(x): Conjecture (“Batch” mode). [sent-184, score-0.043]

85 Under suitable conditions, the empirical distribution corre- sponding to any optimal sequence {xi }1≤i≤k , pk (x) ≡ ˆ 1 k k δ(xi ), i=1 converges weakly as k → ∞ to S, the convex set of maximizers in p(x) of Eθ log(det(Ex Iθ (x))). [sent-185, score-0.052]

86 (2) Expression (2) above is an average over p(θ) of terms proportional to the negative entropy of the asymptotic Gaussian posterior distribution corresponding to each θ, and thus should be maximized by any optimal approximant distribution p(x). [sent-186, score-0.347]

87 ) In fact, it is not difﬁcult, using the results of Clarke and Barron [3] to prove the above conjecture under the conditions like those of Theorem 2, assuming that X is ﬁnite (in which case weak convergence is equivalent to pointwise convergence); we leave generalizations for future work. [sent-188, score-0.186]

88 Bayesian Statistics 4, chapter On priors that maximize expected information, pages 35–60. [sent-199, score-0.046]

89 Jeffreys’ prior is asymptotically least favorable under entropy risk. [sent-213, score-0.271]

90 Using mutual information to pre-process input data for a virtual sensor. [sent-221, score-0.07]

91 Shannon optimal priors on iid statistical experiments converge weakly to jeffreys’ prior. [sent-246, score-0.411]

92 Concentration of measure and isoperimetric inequalities in product spaces. [sent-254, score-0.071]

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