Abstract

Recovery from contrast adaptation was studied in psychophysical experiments. We measured detection thresholds for a test pulse presented on a photopic background as a function of the time after the offset of a high-contrast flicker of the background. The decrease of thresholds with time is well described by a power-law function. Thresholds for tests presented at 640 ms after the offset of the background contrast are still significantly elevated above the threshold measured when the observers have completely adapted to a steady background. We compare the psychophysical data with contrast estimates of ideal-observer models. A match between the results for human and ideal observers can be obtained when the ideal observer is limited by noise. For a quantitative match, we assume that the ideal observer performs a Bayesian calculation on its noise-perturbed input, sampled every 10–20 ms. For the Bayesian calculation we assume a prior probability distribution function for the input contrast that has a lower cutoff at the standard deviation of the noise.

© 2003 Optical Society of America

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References

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    [CrossRef] [PubMed]
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2002 (3)

D. G. Albrecht, W. S. Geisler, R. A. Frazor, A. M. Crane, “Visual cortex of monkey and cats: temporal dynamics of the contrast response function,” J. Neurophysiol. 88, 888–913 (2002).
[PubMed]

L. Meier, M. Carandini, “Masking by fast gratings,” J. Vision 2, 293–301 (2002).
[CrossRef]

S. A. Baccus, M. Meister, “Fast and slow contrast adaptation in retinal circuitry,” Neuron 36, 909–919 (2002).
[CrossRef] [PubMed]

2001 (7)

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

R. J. Snowden, “Contrast gain mechanism or transient channel? Why the effects of a background pattern alter over time,” Vision Res. 41, 1879–1883 (2001).
[CrossRef] [PubMed]

A. L. Fairhall, G. D. Lewen, W. Bialek, R. R. de Ruyter van Steveninck, “Efficiency and ambiguity in an adaptive neural code,” Nature 412, 787–792 (2001).
[CrossRef] [PubMed]

S. P. Brown, R. H. Masland, “Spatial scale and cellular substrate of contrast adaptation by retinal ganglion cells,” Nature Neurosci. 4, 44–51 (2001).
[CrossRef] [PubMed]

D. Chander, E. J. Chichilnisky, “Adaptation to temporal contrast in primate and salamander retina,” J. Neurosci. 21, 9904–9916 (2001).
[PubMed]

K. J. Kim, F. Rieke, “Temporal contrast adaptation inthe input and output signals of salamander retinal ganglion cells,” J. Neurosci. 21, 287–299 (2001).
[PubMed]

F. Rieke, “Temporal contrast adaptation in salamander bipolar cells,” J. Neurosci. 21, 9445–9454 (2001).
[PubMed]

2000 (4)

H. P. Snippe, L. Poot, J. H. van Hateren, “A temporal model for early vision that explains detection thresholds for light pulses on flickering backgrounds,” Visual Neurosci. 17, 449–462 (2000).
[CrossRef]

N. Brady, D. J. Field, “Local contrast in natural images: normalisation and coding efficiency,” Perception 29, 1041–1055 (2000).
[CrossRef]

Y. Tadmor, D. J. Tolhurst, “Calculating the contrasts that retinal ganglion cells and LGN neurones encounter in natural scenes,” Vision Res. 40, 3145–3157 (2000).
[CrossRef] [PubMed]

S. S. Wolfson, N. Graham, “Exploring the dynamics of light adaptation: the effects of varying the flickering background’s duration in the probed-sinewave paradigm,” Vision Res. 40, 2277–2289 (2000).
[CrossRef]

1998 (1)

M. DeWeese, A. Zador, “Asymmetric dynamics in optimal variance adaptation,” Neural Comput. 10, 1179–1202 (1998).
[CrossRef]

1997 (4)

J. H. van Hateren, “Processing of natural time series of intensities by the visual system of the blowfly,” Vision Res. 37, 3407–3416 (1997).
[CrossRef]

L. Poot, H. P. Snippe, J. H. van Hateren, “Dynamics of adaptation at high luminances: adaptation is faster after luminance decrements than after luminance increments,” J. Opt. Soc. Am. A 14, 2499–2508 (1997).
[CrossRef]

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[CrossRef] [PubMed]

J. D. Victor, M. M. Conte, K. P. Purpura, “Dynamic shifts of the contrast-response function,” Visual Neurosci. 14, 577–587 (1997).
[CrossRef]

1996 (1)

M. E. Rudd, L. G. Brown, “Stochastic retinal mechanisms of light adaptation and gain control,” Spatial Vision 10, 125–148 (1996).
[CrossRef] [PubMed]

1994 (1)

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

1993 (1)

J. M. Foley, G. M. Boynton, “Forward pattern masking and adaptation: effects of duration, interstimulus interval, contrast, and spatial and temporal frequency,” Vision Res. 33, 959–980 (1993).
[CrossRef] [PubMed]

1992 (2)

N. Graham, D. C. Hood, “Modeling the dynamics of light adaptation: The merging of two traditions,” Vision Res. 32, 1373–1393 (1992).
[CrossRef] [PubMed]

M. M. Hayhoe, M. E. Levin, R. J. Koshel, “Subtractive processes in light adaptation,” Vision Res. 32, 323–333 (1992).
[CrossRef] [PubMed]

1991 (2)

M. W. Greenlee, M. A. Georgeson, S. Magnussen, J. P. Harris, “The time course of adaptation to spatial contrast,” Vision Res. 31, 223–236 (1991).
[CrossRef] [PubMed]

D. G. Albrecht, W. S. Geisler, “Motion selectivity and the contrast-response function of simple cells in the visual cortex,” Visual Neurosci. 7, 531–546 (1991).
[CrossRef]

1987 (1)

J. D. Victor, “The dynamics of the cat retinal X cell centre,” J. Physiol. (London) 386, 219–246 (1987).

1985 (1)

S. Magnussen, M. W. Greenlee, “Marathon adaptation to spatial contrast: saturation in sight,” Vision Res. 25, 1409–1411 (1985).
[CrossRef] [PubMed]

1982 (1)

D. Rose, I. Lowe, “Dynamics of adaptation to contrast,” Perception 11, 505–528 (1982).
[CrossRef] [PubMed]

1981 (1)

1974 (1)

J. Thorson, M. Biederman-Thorson, “Distributed relaxation processes in sensory adaptation,” Science 183, 161–172 (1974).
[CrossRef] [PubMed]

1961 (1)

1947 (1)

B. H. Crawford, “Visual adaptation in relation to brief conditioning stimuli,” Proc. R. Soc. London 134, 283–302 (1947).
[CrossRef]

Albrecht, D. G.

D. G. Albrecht, W. S. Geisler, R. A. Frazor, A. M. Crane, “Visual cortex of monkey and cats: temporal dynamics of the contrast response function,” J. Neurophysiol. 88, 888–913 (2002).
[PubMed]

D. G. Albrecht, W. S. Geisler, “Motion selectivity and the contrast-response function of simple cells in the visual cortex,” Visual Neurosci. 7, 531–546 (1991).
[CrossRef]

Baccus, S. A.

S. A. Baccus, M. Meister, “Fast and slow contrast adaptation in retinal circuitry,” Neuron 36, 909–919 (2002).
[CrossRef] [PubMed]

Berry, M. J.

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[CrossRef] [PubMed]

Bialek, W.

A. L. Fairhall, G. D. Lewen, W. Bialek, R. R. de Ruyter van Steveninck, “Efficiency and ambiguity in an adaptive neural code,” Nature 412, 787–792 (2001).
[CrossRef] [PubMed]

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[CrossRef] [PubMed]

F. Rieke, D. K. Warland, R. R. de Ruyter van Steveninck, W. Bialek, Spikes: Exploring the Neural Code (MIT Press, Cambridge, Mass., 1996).

Biederman-Thorson, M.

J. Thorson, M. Biederman-Thorson, “Distributed relaxation processes in sensory adaptation,” Science 183, 161–172 (1974).
[CrossRef] [PubMed]

Boynton, G. M.

J. M. Foley, G. M. Boynton, “Forward pattern masking and adaptation: effects of duration, interstimulus interval, contrast, and spatial and temporal frequency,” Vision Res. 33, 959–980 (1993).
[CrossRef] [PubMed]

Brady, N.

N. Brady, D. J. Field, “Local contrast in natural images: normalisation and coding efficiency,” Perception 29, 1041–1055 (2000).
[CrossRef]

Brown, L. G.

M. E. Rudd, L. G. Brown, “Stochastic retinal mechanisms of light adaptation and gain control,” Spatial Vision 10, 125–148 (1996).
[CrossRef] [PubMed]

Brown, S. P.

S. P. Brown, R. H. Masland, “Spatial scale and cellular substrate of contrast adaptation by retinal ganglion cells,” Nature Neurosci. 4, 44–51 (2001).
[CrossRef] [PubMed]

Burbeck, C. A.

Carandini, M.

L. Meier, M. Carandini, “Masking by fast gratings,” J. Vision 2, 293–301 (2002).
[CrossRef]

Chander, D.

D. Chander, E. J. Chichilnisky, “Adaptation to temporal contrast in primate and salamander retina,” J. Neurosci. 21, 9904–9916 (2001).
[PubMed]

Chichilnisky, E. J.

D. Chander, E. J. Chichilnisky, “Adaptation to temporal contrast in primate and salamander retina,” J. Neurosci. 21, 9904–9916 (2001).
[PubMed]

Conte, M. M.

J. D. Victor, M. M. Conte, K. P. Purpura, “Dynamic shifts of the contrast-response function,” Visual Neurosci. 14, 577–587 (1997).
[CrossRef]

Crane, A. M.

D. G. Albrecht, W. S. Geisler, R. A. Frazor, A. M. Crane, “Visual cortex of monkey and cats: temporal dynamics of the contrast response function,” J. Neurophysiol. 88, 888–913 (2002).
[PubMed]

Crawford, B. H.

B. H. Crawford, “Visual adaptation in relation to brief conditioning stimuli,” Proc. R. Soc. London 134, 283–302 (1947).
[CrossRef]

de Ruyter van Steveninck, R. R.

A. L. Fairhall, G. D. Lewen, W. Bialek, R. R. de Ruyter van Steveninck, “Efficiency and ambiguity in an adaptive neural code,” Nature 412, 787–792 (2001).
[CrossRef] [PubMed]

F. Rieke, D. K. Warland, R. R. de Ruyter van Steveninck, W. Bialek, Spikes: Exploring the Neural Code (MIT Press, Cambridge, Mass., 1996).

DeWeese, M.

M. DeWeese, A. Zador, “Asymmetric dynamics in optimal variance adaptation,” Neural Comput. 10, 1179–1202 (1998).
[CrossRef]

Fairhall, A. L.

A. L. Fairhall, G. D. Lewen, W. Bialek, R. R. de Ruyter van Steveninck, “Efficiency and ambiguity in an adaptive neural code,” Nature 412, 787–792 (2001).
[CrossRef] [PubMed]

Field, D. J.

N. Brady, D. J. Field, “Local contrast in natural images: normalisation and coding efficiency,” Perception 29, 1041–1055 (2000).
[CrossRef]

Foley, J. M.

J. M. Foley, G. M. Boynton, “Forward pattern masking and adaptation: effects of duration, interstimulus interval, contrast, and spatial and temporal frequency,” Vision Res. 33, 959–980 (1993).
[CrossRef] [PubMed]

Frazor, R. A.

D. G. Albrecht, W. S. Geisler, R. A. Frazor, A. M. Crane, “Visual cortex of monkey and cats: temporal dynamics of the contrast response function,” J. Neurophysiol. 88, 888–913 (2002).
[PubMed]

Geisler, W. S.

D. G. Albrecht, W. S. Geisler, R. A. Frazor, A. M. Crane, “Visual cortex of monkey and cats: temporal dynamics of the contrast response function,” J. Neurophysiol. 88, 888–913 (2002).
[PubMed]

D. G. Albrecht, W. S. Geisler, “Motion selectivity and the contrast-response function of simple cells in the visual cortex,” Visual Neurosci. 7, 531–546 (1991).
[CrossRef]

Georgeson, M. A.

M. W. Greenlee, M. A. Georgeson, S. Magnussen, J. P. Harris, “The time course of adaptation to spatial contrast,” Vision Res. 31, 223–236 (1991).
[CrossRef] [PubMed]

Graham, N.

S. S. Wolfson, N. Graham, “Exploring the dynamics of light adaptation: the effects of varying the flickering background’s duration in the probed-sinewave paradigm,” Vision Res. 40, 2277–2289 (2000).
[CrossRef]

N. Graham, D. C. Hood, “Modeling the dynamics of light adaptation: The merging of two traditions,” Vision Res. 32, 1373–1393 (1992).
[CrossRef] [PubMed]

Greenlee, M. W.

M. W. Greenlee, M. A. Georgeson, S. Magnussen, J. P. Harris, “The time course of adaptation to spatial contrast,” Vision Res. 31, 223–236 (1991).
[CrossRef] [PubMed]

S. Magnussen, M. W. Greenlee, “Marathon adaptation to spatial contrast: saturation in sight,” Vision Res. 25, 1409–1411 (1985).
[CrossRef] [PubMed]

Harris, J. P.

M. W. Greenlee, M. A. Georgeson, S. Magnussen, J. P. Harris, “The time course of adaptation to spatial contrast,” Vision Res. 31, 223–236 (1991).
[CrossRef] [PubMed]

Hayhoe, M. M.

M. M. Hayhoe, M. E. Levin, R. J. Koshel, “Subtractive processes in light adaptation,” Vision Res. 32, 323–333 (1992).
[CrossRef] [PubMed]

Hood, D. C.

N. Graham, D. C. Hood, “Modeling the dynamics of light adaptation: The merging of two traditions,” Vision Res. 32, 1373–1393 (1992).
[CrossRef] [PubMed]

Kelly, D. H.

Kim, K. J.

K. J. Kim, F. Rieke, “Temporal contrast adaptation inthe input and output signals of salamander retinal ganglion cells,” J. Neurosci. 21, 287–299 (2001).
[PubMed]

Koshel, R. J.

M. M. Hayhoe, M. E. Levin, R. J. Koshel, “Subtractive processes in light adaptation,” Vision Res. 32, 323–333 (1992).
[CrossRef] [PubMed]

Laming, D.

D. Laming, Sensory Analysis (Academic, London, 1986).

Levin, M. E.

M. M. Hayhoe, M. E. Levin, R. J. Koshel, “Subtractive processes in light adaptation,” Vision Res. 32, 323–333 (1992).
[CrossRef] [PubMed]

Lewen, G. D.

A. L. Fairhall, G. D. Lewen, W. Bialek, R. R. de Ruyter van Steveninck, “Efficiency and ambiguity in an adaptive neural code,” Nature 412, 787–792 (2001).
[CrossRef] [PubMed]

Lowe, I.

D. Rose, I. Lowe, “Dynamics of adaptation to contrast,” Perception 11, 505–528 (1982).
[CrossRef] [PubMed]

Magnussen, S.

M. W. Greenlee, M. A. Georgeson, S. Magnussen, J. P. Harris, “The time course of adaptation to spatial contrast,” Vision Res. 31, 223–236 (1991).
[CrossRef] [PubMed]

S. Magnussen, M. W. Greenlee, “Marathon adaptation to spatial contrast: saturation in sight,” Vision Res. 25, 1409–1411 (1985).
[CrossRef] [PubMed]

Masland, R. H.

S. P. Brown, R. H. Masland, “Spatial scale and cellular substrate of contrast adaptation by retinal ganglion cells,” Nature Neurosci. 4, 44–51 (2001).
[CrossRef] [PubMed]

Meier, L.

L. Meier, M. Carandini, “Masking by fast gratings,” J. Vision 2, 293–301 (2002).
[CrossRef]

Meister, M.

S. A. Baccus, M. Meister, “Fast and slow contrast adaptation in retinal circuitry,” Neuron 36, 909–919 (2002).
[CrossRef] [PubMed]

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[CrossRef] [PubMed]

Poot, L.

H. P. Snippe, L. Poot, J. H. van Hateren, “A temporal model for early vision that explains detection thresholds for light pulses on flickering backgrounds,” Visual Neurosci. 17, 449–462 (2000).
[CrossRef]

L. Poot, H. P. Snippe, J. H. van Hateren, “Dynamics of adaptation at high luminances: adaptation is faster after luminance decrements than after luminance increments,” J. Opt. Soc. Am. A 14, 2499–2508 (1997).
[CrossRef]

H. P. Snippe, L. Poot, J. H. van Hateren are preparing a manuscript to be called “Asymmetric dynamics of adaptation after onset and offset of flicker.”

Purpura, K. P.

J. D. Victor, M. M. Conte, K. P. Purpura, “Dynamic shifts of the contrast-response function,” Visual Neurosci. 14, 577–587 (1997).
[CrossRef]

Rieke, F.

F. Rieke, “Temporal contrast adaptation in salamander bipolar cells,” J. Neurosci. 21, 9445–9454 (2001).
[PubMed]

K. J. Kim, F. Rieke, “Temporal contrast adaptation inthe input and output signals of salamander retinal ganglion cells,” J. Neurosci. 21, 287–299 (2001).
[PubMed]

F. Rieke, D. K. Warland, R. R. de Ruyter van Steveninck, W. Bialek, Spikes: Exploring the Neural Code (MIT Press, Cambridge, Mass., 1996).

Rose, D.

D. Rose, I. Lowe, “Dynamics of adaptation to contrast,” Perception 11, 505–528 (1982).
[CrossRef] [PubMed]

Rudd, M. E.

M. E. Rudd, L. G. Brown, “Stochastic retinal mechanisms of light adaptation and gain control,” Spatial Vision 10, 125–148 (1996).
[CrossRef] [PubMed]

Ruderman, D. L.

D. L. Ruderman, “The statistics of natural images,” Network 5, 517–548 (1994).
[CrossRef]

Schwartz, O.

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

Simoncelli, E. P.

O. Schwartz, E. P. Simoncelli, “Natural signal statistics and sensory gain control,” Nat. Neurosci. 4, 819–825 (2001).
[CrossRef] [PubMed]

Smirnakis, S. M.

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[CrossRef] [PubMed]

Snippe, H. P.

H. P. Snippe, L. Poot, J. H. van Hateren, “A temporal model for early vision that explains detection thresholds for light pulses on flickering backgrounds,” Visual Neurosci. 17, 449–462 (2000).
[CrossRef]

L. Poot, H. P. Snippe, J. H. van Hateren, “Dynamics of adaptation at high luminances: adaptation is faster after luminance decrements than after luminance increments,” J. Opt. Soc. Am. A 14, 2499–2508 (1997).
[CrossRef]

H. P. Snippe, L. Poot, J. H. van Hateren are preparing a manuscript to be called “Asymmetric dynamics of adaptation after onset and offset of flicker.”

Snowden, R. J.

R. J. Snowden, “Contrast gain mechanism or transient channel? Why the effects of a background pattern alter over time,” Vision Res. 41, 1879–1883 (2001).
[CrossRef] [PubMed]

Tadmor, Y.

Y. Tadmor, D. J. Tolhurst, “Calculating the contrasts that retinal ganglion cells and LGN neurones encounter in natural scenes,” Vision Res. 40, 3145–3157 (2000).
[CrossRef] [PubMed]

Thorson, J.

J. Thorson, M. Biederman-Thorson, “Distributed relaxation processes in sensory adaptation,” Science 183, 161–172 (1974).
[CrossRef] [PubMed]

Tolhurst, D. J.

Y. Tadmor, D. J. Tolhurst, “Calculating the contrasts that retinal ganglion cells and LGN neurones encounter in natural scenes,” Vision Res. 40, 3145–3157 (2000).
[CrossRef] [PubMed]

van Hateren, J. H.

H. P. Snippe, L. Poot, J. H. van Hateren, “A temporal model for early vision that explains detection thresholds for light pulses on flickering backgrounds,” Visual Neurosci. 17, 449–462 (2000).
[CrossRef]

J. H. van Hateren, “Processing of natural time series of intensities by the visual system of the blowfly,” Vision Res. 37, 3407–3416 (1997).
[CrossRef]

L. Poot, H. P. Snippe, J. H. van Hateren, “Dynamics of adaptation at high luminances: adaptation is faster after luminance decrements than after luminance increments,” J. Opt. Soc. Am. A 14, 2499–2508 (1997).
[CrossRef]

H. P. Snippe, L. Poot, J. H. van Hateren are preparing a manuscript to be called “Asymmetric dynamics of adaptation after onset and offset of flicker.”

Victor, J. D.

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Figures (5)

Fig. 1
Fig. 1

Detection thresholds for the test pulse as a function of time t after the offset of the flicker of the background for two observers. Steady-state values of the detection thresholds during the background flicker are 5930±150 td for HS and 5080±180 td for RV. The dashed horizontal lines indicate the detection threshold for the test pulse measured when the observers are completely adapted to the steady background.

Fig. 2
Fig. 2

Threshold elevation, Eq. (2), as a function of time t after the offset of the flicker of the background for two observers. Note the log–log scaling of the axes. The straight lines are power-law fits, Eq. (3), to the psychophysical data. The dashed and the dotted curves are ideal-observer predictions, as described in Subsection 4.C.

Fig. 3
Fig. 3

Divisive gain control for contrast. The input I(t) is divided by a gain-control signal Cˆ(t), a dynamic estimate of the current contrast C(t) of the input. This divisive operation produces an output O(t)=I(t)/Cˆ(t).

Fig. 4
Fig. 4

Posterior pdf’s [relation (8)] of the stimulus contrast C for an ideal observer that uses a prior p.d.f. with a lower cutoff at C=L [Eq. (9), with m=0], after receiving N noise-free samples si=0 of a stimulus with contrast C=0. For the sake of clarity, curves are normalized to 1 at C/L=1. Curves from top to bottom are for values N=1, 2, 4, 8, 16, 32, 64.

Fig. 5
Fig. 5

Posterior pdf’s [relation (13)] of (a) the standard deviation σ and (b) the variance V of a noisy signal with standard deviation σ=L after the ideal observer has received N samples si of the signal, when the observed statistic SN=si2 is equal to its expected value NL2. Prior pdf’s (N=0) used by the ideal observer are assumed to be flat for (a) the standard deviation and (b) the variance. For the sake of clarity, the peak value for each of the curves has been normalized to 1. Curves in order of decreasing width, from top to bottom, are for values N=1, 2, 4, 8, 16, 32, 64.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

I(t)=I0[1+C(t)sin(2πft+ϕ)].
E(t)=[M(t)-M0]/M0.
E(t)=(t0/t)γ.
PN({si}|C)=i=1N12πCexp-si22C2=1(2πC)Nexp-i=1Nsi2/2C2.
PN(C|{si})=P0(C) PN({si}|C)PN({si})P0(C)CNexp-i=1Nsi2/2C2.
C^Nn(SN)=0CnPN(C|SN)dC0PN(C|SN)dC.
C^n(N)=0C^Nn(SN)PN(SN|C)dSN.
PN(C|SN=0)P0(C)CN.
P0(C)=CmforCL,P0(C)=0forC<L.
C^n(N)=LnN-m-1N-n-m-1=Ln1+nN-n-m-1.
si=si+ni.
PN({si}|σ)=1(2πσ)Nexp-i=1Nsi2/2σ2.
PN(σ|{si})1σN-mexp-i=1Nsi2/2σ2.
σˆ(N)=L[1+1/(N-2)].
Vˆ(N)=L2[1+4/(N-4)].
P0(C)=12[δ(C-L)+δ(C-H)].
C^n(N)=C^Nn(SN)=1/LN-n+1/HN-n1/LN+1/HN=Ln1+1/hN-n1+1/hN,
C^n(N)Ln1+1hN-n-1hN=Ln[1+(hn-1)/hN]=Ln[1+(hn-1)exp(-N ln h)]
Pσ=H|i=1Nsi2=NL2Pσ=L|i=1Nsi2=NL2
=1hNexp-12h2-12N
=exp{-[ln h-(h2-1)/2h2]N},
σ^Nn(Sn)=0σnPN(σ|SN)dσ0PN(σ|SN)dσ=01σN-n-mexp(-SN/2σ2)dσ01σN-mexp(-SN/2σ2)dσ,
=(SN)n/20u(N-n-m-3)/2exp(-u/2)du0u(N-m-3)/2exp(-u/2)du,
P(SN|σ=L)=(SN)(N-2)/2exp(-SN/2L2)0(SN)(N-2)/2exp(-SN/2L2)dSN.
σ^n(N)=0σ^Nn(SN)P(SN|σ=L)dSN=0(SN)(N+n-2)/2exp(-SN/2L2)dSN0(SN)(N-2)/2exp(-SN/2L2)dSN×0u(N-n-m-3)/2exp(-u/2)du0u(N-m-3)/2exp(-u/2)du=Ln0ν(N+n-2)/2exp(-ν/2)dν0ν(N-2)/2exp(-ν/2)dν×0u(N-n-m-3)/2exp(-u/2)du0u(N-m-3)/2exp(-u/2)du=LnΓ[(N+n)/2]Γ[N/2]Γ[(N-n-m-1)/2]Γ[(N-m-1)/2],
Γ(α)0yα-1exp(-y)dy=2-α0wα-1exp(-w/2)dw.

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