1836:
1146:
4821:
7283:(i.e. segments of one or more exponential distributions, attached end to end). Exponential distributions are well behaved and well understood. The logarithm of an exponential distribution is a straight line, and hence this method essentially involves enclosing the logarithm of the density in a series of line segments. This is the source of the log-concave restriction: if a distribution is log-concave, then its logarithm is concave (shaped like an upside-down U), meaning that a line segment tangent to the curve will always pass over the curve.
1831:{\displaystyle {\begin{aligned}\mathbb {P} \left(U\leq {\frac {f(Y)}{Mg(Y)}}\right)&=\operatorname {E} \mathbf {1} _{\left}\\&=E\left}|Y]\right]&({\text{by tower property }})\\&=\operatorname {E} \left\\&=E\left&({\text{because }}\Pr(U\leq u)=u,{\text{when }}U{\text{ is uniform on }}(0,1))\\&=\int \limits _{y:g(y)>0}{\frac {f(y)}{Mg(y)}}g(y)\,dy\\&={\frac {1}{M}}\int \limits _{y:g(y)>0}f(y)\,dy\\&={\frac {1}{M}}&({\text{since support of }}Y{\text{ includes support of }}X)\end{aligned}}}
5987:
4491:
6420:
5580:
6087:
4816:{\displaystyle {\begin{aligned}\psi _{\theta }(\eta )&=\log \left(\mathbb {E} _{\theta }\exp(\eta X)\right)=\psi (\theta +\eta )-\psi (\theta )<\infty \\\mathbb {E} _{\theta }(X)&=\left.{\frac {\partial \psi _{\theta }(\eta )}{\partial \eta }}\right|_{\eta =0}\\\mathrm {Var} _{\theta }(X)&=\left.{\frac {\partial ^{2}\psi _{\theta }(\eta )}{\partial ^{2}\eta }}\right|_{\eta =0}\end{aligned}}}
3951:
5982:{\displaystyle {\begin{aligned}f_{X|X\geq b}(x)&={\frac {f(x)\mathbb {I} (x\geq b)}{\mathbb {P} (X\geq b)}}\\g_{\theta ^{*}}(x)&=f(x)\exp(\theta ^{*}x-\psi (\theta ^{*}))\\Z(x)&={\frac {f_{X|X\geq b}(x)}{g_{\theta ^{*}}(x)}}={\frac {\exp(-\theta ^{*}x+\psi (\theta ^{*}))\mathbb {I} (x\geq b)}{\mathbb {P} (X\geq b)}}\end{aligned}}}
6415:{\displaystyle M=Z(b)={\frac {\exp(-\theta ^{*}b+\psi (\theta ^{*}))}{\mathbb {P} (X\geq b)}}={\frac {\exp \left(-{\frac {(b-\mu )^{2}}{2\sigma ^{2}}}\right)}{\mathbb {P} (X\geq b)}}={\frac {\exp \left(-{\frac {(b-\mu )^{2}}{2\sigma ^{2}}}\right)}{\mathbb {P} \left(\mathrm {N} (0,1)\geq {\frac {b-\mu }{\sigma }}\right)}}}
103:(PDF) of a random variable onto a large rectangular board and throwing darts at it. Assume that the darts are uniformly distributed around the board. Now remove all of the darts that are outside the area under the curve. The remaining darts will be distributed uniformly within the area under the curve, and the
7488:
Unfortunately, ARS can only be applied for sampling from log-concave target densities. For this reason, several extensions of ARS have been proposed in literature for tackling non-log-concave target distributions. Furthermore, different combinations of ARS and the
Metropolis-Hastings method have been
7484:
The method essentially involves successively determining an envelope of straight-line segments that approximates the logarithm better and better while still remaining above the curve, starting with a fixed number of segments (possibly just a single tangent line). Sampling from a truncated exponential
126:
The visualization just described is equivalent to a particular form of rejection sampling where the "proposal distribution" is uniform. Hence its graph is a rectangle. The general form of rejection sampling assumes that the board is not necessarily rectangular but is shaped according to the density
7477:
This squeezing step is optional, even when suggested by Gilks. At best it saves you from only one extra evaluation of your (messy and/or expensive) target density. However, presumably for particularly expensive density functions (and assuming the rapid convergence of the rejection rate toward zero)
3668:
7027:
For many distributions, finding a proposal distribution that includes the given distribution without a lot of wasted space is difficult. An extension of rejection sampling that can be used to overcome this difficulty and efficiently sample from a wide variety of distributions (provided that they
7007:. In addition, as the dimensions of the problem get larger, the ratio of the embedded volume to the "corners" of the embedding volume tends towards zero, thus a lot of rejections can take place before a useful sample is generated, thus making the algorithm inefficient and impractical. See
6998:
Rejection sampling can lead to a lot of unwanted samples being taken if the function being sampled is highly concentrated in a certain region, for example a function that has a spike at some location. For many distributions, this problem can be solved using an adaptive extension (see
5232:
4306:
6987:, among the class of simple distributions, the trick is to use natural exponential family, which helps to gain some control over the complexity and considerably speed up the computation. Indeed, there are deep mathematical reasons for using natural exponential family.
2664:
expression. Rejection sampling is thus more efficient than some other method whenever M times the cost of these operations—which is the expected cost of obtaining a sample with rejection sampling—is lower than the cost of obtaining a sample using the other method.
151:). Its shape must be at least as high at every point as the distribution we want to sample from, so that the former completely encloses the latter. Otherwise, there would be parts of the curved area we want to sample from that could never be reached.
4477:
6931:
1151:
2143:
6582:
6700:
123:‑positions of these darts will be distributed according to the random variable's density. This is because there is the most room for the darts to land where the curve is highest and thus the probability density is greatest.
86:
in one dimension, one can perform a uniformly random sampling of the two-dimensional
Cartesian graph, and keep the samples in the region under the graph of its density function. Note that this property can be extended to
6940:
a parametric class of proposal distribution, solves the optimization problems conveniently, with its useful properties that directly characterize the distribution of the proposal. For this type of problem, to simulate
4975:
4024:
4143:
3946:{\displaystyle {\begin{aligned}F_{\theta }(x)&=\mathbb {E} \left\\&=\int _{-\infty }^{x}e^{\theta y-\psi (\theta )}f(y)dy\\g_{\theta }(x)&=F'_{\theta }(x)=e^{\theta x-\psi (\theta )}f(x)\end{aligned}}}
5430:
5299:
2248:
7489:
designed in order to obtain a universal sampler that builds a self-tuning proposal densities (i.e., a proposal automatically constructed and adapted to the target). This class of methods are often called as
5566:
5102:
3376:
4077:
7273:. This therefore reduces the chance that your next attempt will be rejected. Asymptotically, the probability of needing to reject your sample should converge to zero, and in practice, often very rapidly.
4132:
5493:
204:
Sample uniformly along this line from 0 to the maximum of the probability density function. If the sampled value is greater than the value of the desired distribution at this vertical line, reject the
7493:. The resulting adaptive techniques can be always applied but the generated samples are correlated in this case (although the correlation vanishes quickly to zero as the number of iterations grows).
7276:
As proposed, any time we choose a point that is rejected, we tighten the envelope with another line segment that is tangent to the curve at the point with the same x-coordinate as the chosen point.
7099:
4905:
5585:
4496:
3673:
3473:
248:
This algorithm can be used to sample from the area under any curve, regardless of whether the function integrates to 1. In fact, scaling a function by a constant has no effect on the sampled
6476:
3632:
3427:
1888:
4343:
6752:
3254:
7201:
2873:
2609:
makes sampling difficult. A single iteration of the rejection algorithm requires sampling from the proposal distribution, drawing from a uniform distribution, and evaluating the
7472:
7434:
5091:
73:
7392:
7358:
7320:
7271:
7240:
7164:
7130:
7019:. (However, Gibbs sampling, which breaks down a multi-dimensional sampling problem into a series of low-dimensional samples, may use rejection sampling as one of its steps.)
6757:
For the above example, as the measurement of the efficiency, the expected number of the iterations the natural exponential family based rejection sampling method is of order
3580:
2175:
873:
2453:
966:
684:
5025:
3203:
3066:
2931:
2662:
613:
501:
5358:
5331:
2060:
6819:
6485:
5051:
4855:
4355:
2575:
1937:
814:
779:
587:
7360:
that we had to evaluate in the current chain of rejections, we can also construct a piecewise linear lower bound (the "squeezing" function) using these values as well.
6985:
6824:
2366:
6078:
3140:
1059:
998:
6040:
3661:
2607:
2511:
2482:
1134:
1090:
1027:
386:
337:
2012:
905:
6959:
6775:
6605:
6011:
3537:
3513:
3493:
3106:
3086:
3023:
2997:
2974:
2954:
2827:
2807:
2787:
2763:
2743:
2723:
2703:
2531:
2406:
2386:
2320:
2300:
2052:
2032:
1980:
1957:
1908:
744:
724:
704:
637:
541:
521:
446:
426:
406:
357:
305:
267:
242:
222:
199:
176:
145:
121:
2280:
3160:
2388:
closer to 1 is preferred as it implies fewer rejected samples, on average, and thus fewer iterations of the algorithm. In this sense, one prefers to have
7987:
Meyer, Renate; Cai, Bo; Perron, François (2008-03-15). "Adaptive rejection
Metropolis sampling using Lagrange interpolation polynomials of degree 2".
6610:
3519:(if it exists), also known as exponential tilting, provides a class of proposal distributions that can lower the computation complexity, the value of
3958:
4910:
2678:
7745:
7634:
7485:
random variable is straightforward. Just take the log of a uniform random variable (with appropriate interval and corresponding truncation).
3033:
Rejection sampling can be far more efficient compared with the naive methods in some situations. For example, given a problem as sampling
5369:
3475:, which could be close to infinity. Moreover, even when you apply the Rejection sampling method, it is always hard to optimize the bound
8041:
5227:{\textstyle \psi _{\theta }(\eta )=\psi (\theta +\eta )-\psi (\theta )=(\mu +\theta \sigma ^{2})\eta +{\frac {\sigma ^{2}\eta ^{2}}{2}}}
5244:
2180:
7554:
5498:
3261:
4029:
7626:
4082:
7029:
5440:
4301:{\displaystyle Z(x)={\frac {f(x)}{g_{\theta }(x)}}={\frac {f(x)}{e^{\theta x-\psi (\theta )}f(x)}}=e^{-\theta x+\psi (\theta )}}
7135:
Often, distributions that have algebraically messy density functions have reasonably simpler log density functions (i.e. when
7047:
If it helps, define your envelope distribution in log space (e.g. log-probability or log-density) instead. That is, work with
7512:
7050:
4860:
7208:
Instead of a single uniform envelope density function, use a piecewise linear density function as your envelope instead.
6995:
Rejection sampling requires knowing the target distribution (specifically, ability to evaluate target PDF at any point).
2054:
is the expected number of the iterations that are needed, as a measure of the computational complexity of the algorithm.
8036:
4346:
3516:
308:
100:
76:
7011:. In high dimensions, it is necessary to use a different approach, typically a Markov chain Monte Carlo method such as
3432:
7868:
Evans, M.; Swartz, T. (1998-12-01). "Random
Variable Generation Using Concavity Properties of Transformed Densities".
7502:
7291:
We can take even further advantage of the (log) concavity requirement, to potentially avoid the cost of evaluating
4135:
3163:
6429:
3585:
1841:
46:
or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in
7796:
Thomas, D. B.; Luk, W. (2007). "Non-uniform random number generation through piecewise linear approximations".
7286:
If not working in log space, a piecewise linear density function can also be sampled via triangle distributions
7280:
4313:
3389:
1104:
276:
39:
31:
4483:
It is easy to derive the cumulant-generation function of the proposal and therefore the proposal's cumulants.
2302:
is chosen closer to one, the unconditional acceptance probability is higher the less that ratio varies, since
7008:
6705:
3379:
3208:
1143:
The unconditional acceptance probability is the proportion of proposed samples which are accepted, which is
7877:
7832:
7169:
2832:
1983:
7439:
7401:
5060:
7329:
Just like we can construct a piecewise linear upper bound (the "envelope" function) using the values of
1063:
This means that, with enough replicates, the algorithm generates a sample from the desired distribution
616:
201:‑position, up to the maximum y-value of the probability density function of the proposal distribution.
49:
7576:
7366:
7332:
7294:
7245:
7214:
7138:
7104:
3634:
is the target distribution. Assume for simplicity, the density function can be explicitly written as
3550:
1137:
1093:
272:
2411:
2148:
822:
642:
7882:
7837:
7823:
Hörmann, Wolfgang (1995-06-01). "A Rejection
Technique for Sampling from T-concave Distributions".
7032:
density functions, which is in fact the case for most of the common distributions—even those whose
7012:
6937:
910:
4980:
4472:{\displaystyle \psi (\theta )=\log \mathbb {E} {\exp(tX)}|_{t=\theta }=\log M_{X}(t)|_{t=\theta }}
2612:
592:
451:
7969:
7895:
7778:
7679:
7517:
5336:
5309:
3173:
3036:
2881:
1107:
algorithms that also use a proxy distribution to achieve simulation from the target distribution
1100:
148:
6926:{\textstyle {\frac {1}{\mathbb {P} (X\geq b)}}=O(b\cdot e^{\frac {(b-\mu )^{2}}{2\sigma ^{2}}})}
2682:
6780:
5030:
4831:
2536:
1913:
784:
749:
546:
7952:; Tan, K. K. C. (1995-01-01). "Adaptive Rejection Metropolis Sampling within Gibbs Sampling".
7930:
7850:
7741:
7718:
7671:
7630:
7550:
7507:
7004:
7652:"Von Neumann's Comparison Method for Random Sampling from the Normal and Other Distributions"
7996:
7961:
7922:
7887:
7842:
7805:
7770:
7710:
7663:
7603:
7585:
7542:
6964:
2674:
7599:
6045:
2325:
1032:
971:
17:
7607:
7595:
6016:
3637:
3111:
2583:
2487:
2458:
1110:
1066:
1003:
362:
313:
83:
7043:
There are three basic ideas to this technique as ultimately introduced by Gilks in 1992:
3539:
and speed up the computations (see examples: working with
Natural Exponential Families).
1989:
7913:
Görür, Dilan; Teh, Yee Whye (2011-01-01). "Concave-Convex
Adaptive Rejection Sampling".
7738:
Essentials of Monte Carlo
Simulation: Statistical Methods for Building Simulation Models
7279:
A piecewise linear model of the proposal log distribution results in a set of piecewise
7016:
6944:
6760:
6590:
5996:
3522:
3498:
3478:
3091:
3071:
3008:
2982:
2959:
2939:
2812:
2792:
2772:
2748:
2728:
2708:
2688:
2516:
2391:
2371:
2305:
2285:
2037:
2017:
1965:
1942:
1893:
878:
729:
709:
689:
622:
526:
506:
431:
411:
391:
342:
290:
252:
227:
207:
184:
161:
130:
106:
2577:, i.e. that the target and proposal distributions are actually the same distribution.
2253:
8030:
7699:"Accounting for environmental change in continuous-time stochastic population models"
3515:
is large and the rejection rate is high, the algorithm can be very inefficient. The
3145:
2138:{\displaystyle M={\frac {1}{\mathbb {P} \left(U\leq {\frac {f(Y)}{Mg(Y)}}\right)}}}
287:
The rejection sampling method generates sampling values from a target distribution
6577:{\displaystyle U\leq {\frac {Z(x)}{M}}=e^{-\theta ^{*}(x-b)}\mathbb {I} (x\geq b)}
819:
The validation of this method is the envelope principle: when simulating the pair
7761:
Gilks, W. R.; Wild, P. (1992). "Adaptive
Rejection Sampling for Gibbs Sampling".
6695:{\textstyle X\sim _{i.i.d.}\mathrm {N} (\mu +\theta ^{*}\sigma ^{2},\sigma ^{2})}
8000:
7809:
7949:
7714:
7546:
7934:
7854:
7722:
7675:
7590:
7571:
7926:
7698:
99:
To visualize the motivation behind rejection sampling, imagine graphing the
4970:{\textstyle \psi (\theta )=\mu \theta +{\frac {\sigma ^{2}\theta ^{2}}{2}}}
4019:{\displaystyle \psi (\theta )=\log \left(\mathbb {E} \exp(\theta X)\right)}
7846:
5333:
for the proposal distribution. In this setup, the intuitive way to choose
6821:, while under the naive method, the expected number of the iterations is
7973:
7899:
7782:
7683:
7651:
271:. Thus, the algorithm can be used to sample from a distribution whose
5573:
Explicitly write out the target, the proposal and the likelihood ratio
7203:
may be easier to work with or, at least, closer to piecewise linear).
7003:), or with an appropriate change of variables with the method of the
7965:
7891:
7774:
7667:
5425:{\displaystyle \mathbb {E} _{\theta }(X)=\mu +\theta \sigma ^{2}=b}
7398:
if it will be accepted by comparing against the (ideally cheaper)
3386:
The problem is this sampling can be difficult and inefficient, if
1092:. There are a number of extensions to this algorithm, such as the
5294:{\displaystyle \mathrm {N} (\mu +\theta \sigma ^{2},\sigma ^{2})}
2580:
Rejection sampling is most often used in cases where the form of
2243:{\textstyle \mathbb {P} \left(U\leq {\frac {f(Y)}{Mg(Y)}}\right)}
7537:
Casella, George; Robert, Christian P.; Wells, Martin T. (2004).
7211:
Each time you have to reject a sample, you can use the value of
82:
Rejection sampling is based on the observation that to sample a
5561:{\displaystyle g_{\theta ^{*}}(x)=\mathrm {N} (b,\sigma ^{2})}
3371:{\displaystyle \{X_{1},X_{2},...,X_{N}:X_{i}\in A,i=1,...,N\}}
4072:{\displaystyle \Theta =\{\theta :\psi (\theta )<\infty \}}
127:
of some proposal distribution (not necessarily normalized to
4127:{\displaystyle \{F_{\theta }(\cdot )\}_{\theta \in \Theta }}
2250:
is a probability which can only take values in the interval
7242:
that you evaluated, to improve the piecewise approximation
4741:
4647:
7541:. Institute of Mathematical Statistics. pp. 342–347.
5488:{\displaystyle \theta ^{*}={\frac {b-\mu }{\sigma ^{2}}}.}
3162:
can be easily simulated, using the naive methods (e.g. by
38:
is a basic technique used to generate observations from a
875:, one produces a uniform simulation over the subgraph of
543:
here is a constant, finite bound on the likelihood ratio
388:. The idea is that one can generate a sample value from
7491:
Adaptive
Rejection Metropolis Sampling (ARMS) algorithms
7478:
this can make a sizable difference in ultimate runtime.
7740:(2013th ed.). New York, NY Heidelberg: Springer.
6827:
6708:
6613:
5105:
4913:
3392:
3176:
3148:
3114:
3039:
2884:
2328:
2183:
2151:
913:
881:
825:
147:) that we know how to sample from (for example, using
7697:
Legault, Geoffrey; Melbourne, Brett A. (2019-03-01).
7474:
in this case) squeezing function that have available.
7442:
7404:
7369:
7335:
7297:
7248:
7217:
7172:
7141:
7107:
7053:
6967:
6947:
6783:
6763:
6593:
6488:
6432:
6090:
6048:
6019:
5999:
5583:
5501:
5443:
5372:
5339:
5312:
5247:
5063:
5033:
4983:
4863:
4834:
4494:
4358:
4316:
4146:
4085:
4032:
3961:
3671:
3640:
3588:
3553:
3525:
3501:
3481:
3435:
3264:
3211:
3094:
3074:
3011:
2985:
2962:
2942:
2835:
2815:
2795:
2775:
2751:
2731:
2711:
2691:
2615:
2586:
2539:
2519:
2490:
2461:
2414:
2394:
2374:
2308:
2288:
2256:
2063:
2040:
2020:
1992:
1968:
1945:
1916:
1896:
1844:
1149:
1113:
1069:
1035:
1006:
974:
787:
752:
732:
712:
692:
645:
625:
595:
549:
529:
509:
454:
434:
414:
394:
365:
345:
316:
293:
255:
230:
210:
187:
164:
133:
109:
52:
7621:
Bishop, Christopher (2006). "11.4: Slice sampling".
7094:{\displaystyle h\left(x\right)=\log g\left(x\right)}
4900:{\displaystyle X\sim \mathrm {N} (\mu ,\sigma ^{2})}
7466:
7428:
7386:
7352:
7314:
7265:
7234:
7195:
7158:
7124:
7093:
7036:functions are not concave themselves) is known as
6979:
6953:
6925:
6813:
6769:
6746:
6694:
6599:
6576:
6470:
6414:
6072:
6034:
6005:
5981:
5560:
5487:
5424:
5352:
5325:
5293:
5241:which further implies it is a normal distribution
5226:
5085:
5045:
5019:
4969:
4899:
4849:
4815:
4471:
4337:
4300:
4126:
4071:
4018:
3945:
3655:
3626:
3574:
3531:
3507:
3487:
3467:
3421:
3370:
3248:
3197:
3154:
3134:
3100:
3080:
3060:
3017:
2991:
2968:
2948:
2925:
2875:(the uniform distribution over the unit interval).
2867:
2821:
2801:
2781:
2757:
2737:
2717:
2697:
2656:
2601:
2569:
2525:
2505:
2476:
2447:
2400:
2380:
2360:
2314:
2294:
2274:
2242:
2169:
2137:
2046:
2026:
2006:
1974:
1951:
1931:
1910:each time is generated under the density function
1902:
1882:
1830:
1128:
1084:
1053:
1021:
992:
960:
899:
867:
808:
773:
738:
718:
698:
678:
631:
607:
581:
535:
515:
495:
440:
420:
400:
380:
351:
331:
299:
261:
236:
216:
193:
170:
139:
115:
67:
7915:Journal of Computational and Graphical Statistics
7870:Journal of Computational and Graphical Statistics
3468:{\displaystyle {\frac {1}{\mathbb {P} (X\in A)}}}
1473:
244:‑value is a sample from the desired distribution.
7000:
1557:
1136:. It forms the basis for algorithms such as the
7394:to see if your sample will be accepted, we may
3495:for the likelihood ratio. More often than not,
706:. Note that this requires that the support of
7363:Before evaluating (the potentially expensive)
8022:(Second ed.). New York: Springer-Verlag.
5057:Choose the form of the proposal distribution
3429:. The expected number of iterations would be
2408:as small as possible (while still satisfying
8:
7989:Computational Statistics & Data Analysis
4109:
4086:
4066:
4039:
3543:Rejection sampling using exponential tilting
3365:
3265:
3243:
3212:
3029:Advantages over sampling using naive methods
2533:cannot be equal to 1: such would imply that
2322:is the upper bound for the likelihood ratio
1099:This method relates to the general field of
3205:independently, and accept those satisfying
1982:to obtain an accepted value thus follows a
1000:uniformly distributed over the subgraph of
7539:Generalized Accept-Reject sampling schemes
6471:{\displaystyle U\sim \mathrm {Unif} (0,1)}
3627:{\displaystyle F(x)=\mathbb {P} (X\leq x)}
3422:{\textstyle \mathbb {P} (X\in A)\approx 0}
1883:{\displaystyle U\sim \mathrm {Unif} (0,1)}
7881:
7836:
7589:
7447:
7441:
7409:
7403:
7368:
7334:
7296:
7247:
7216:
7171:
7140:
7106:
7052:
6966:
6946:
6910:
6895:
6876:
6835:
6834:
6828:
6826:
6782:
6762:
6715:
6707:
6683:
6670:
6660:
6642:
6621:
6612:
6592:
6555:
6554:
6531:
6523:
6495:
6487:
6439:
6431:
6386:
6363:
6354:
6353:
6337:
6322:
6303:
6286:
6261:
6260:
6244:
6229:
6210:
6193:
6168:
6167:
6153:
6131:
6112:
6089:
6047:
6018:
5998:
5953:
5952:
5931:
5930:
5918:
5896:
5877:
5854:
5849:
5818:
5814:
5807:
5772:
5750:
5701:
5696:
5666:
5665:
5644:
5643:
5628:
5596:
5592:
5584:
5582:
5549:
5531:
5511:
5506:
5500:
5474:
5457:
5448:
5442:
5410:
5379:
5375:
5374:
5371:
5344:
5338:
5317:
5311:
5282:
5269:
5248:
5246:
5212:
5202:
5195:
5180:
5110:
5104:
5068:
5062:
5032:
4987:
4982:
4955:
4945:
4938:
4912:
4888:
4870:
4862:
4833:
4797:
4781:
4760:
4750:
4743:
4717:
4706:
4689:
4659:
4649:
4623:
4619:
4618:
4542:
4538:
4537:
4503:
4495:
4493:
4457:
4452:
4436:
4411:
4406:
4385:
4381:
4380:
4357:
4338:{\displaystyle \psi (\theta )<\infty }
4315:
4271:
4225:
4204:
4183:
4162:
4145:
4112:
4093:
4084:
4031:
3989:
3988:
3960:
3903:
3878:
3852:
3802:
3792:
3784:
3746:
3745:
3703:
3702:
3680:
3672:
3670:
3639:
3605:
3604:
3587:
3552:
3524:
3500:
3480:
3443:
3442:
3436:
3434:
3394:
3393:
3391:
3323:
3310:
3285:
3272:
3263:
3231:
3210:
3175:
3147:
3118:
3113:
3093:
3073:
3038:
3010:
2984:
2961:
2941:
2903:
2883:
2836:
2834:
2814:
2794:
2774:
2750:
2730:
2710:
2690:
2628:
2614:
2585:
2538:
2518:
2489:
2460:
2413:
2393:
2373:
2341:
2327:
2307:
2287:
2255:
2200:
2185:
2184:
2182:
2150:
2092:
2077:
2076:
2070:
2062:
2039:
2019:
1996:
1991:
1967:
1944:
1915:
1895:
1851:
1843:
1813:
1805:
1790:
1773:
1734:
1720:
1703:
1656:
1629:
1592:
1584:
1552:
1508:
1472:
1471:
1436:
1421:
1420:
1391:
1370:
1328:
1316:
1311:
1241:
1229:
1224:
1170:
1155:
1154:
1150:
1148:
1112:
1068:
1034:
1005:
973:
932:
912:
880:
824:
786:
751:
731:
711:
691:
644:
624:
594:
562:
548:
528:
508:
467:
453:
433:
413:
393:
364:
344:
315:
292:
254:
229:
209:
186:
163:
132:
108:
59:
55:
54:
51:
7954:Journal of the Royal Statistical Society
7763:Journal of the Royal Statistical Society
7623:Pattern Recognition and Machine Learning
1029:and thus, marginally, a simulation from
7529:
6747:{\textstyle U\sim \mathrm {Unif} (0,1)}
7798:IET Computers & Digital Techniques
5093:, with cumulant-generating function as
3249:{\displaystyle \{n\geq 1:X_{n}\in A\}}
3005:The algorithm will take an average of
224:‑value and return to step 1; else the
6042:, which is a decreasing function for
2685:, obtains a sample from distribution
178:‑axis from the proposal distribution.
154:Rejection sampling works as follows:
7:
7196:{\displaystyle \log f\left(x\right)}
4138:. Moreover, the likelihood ratio is
2868:{\displaystyle \mathrm {Unif} (0,1)}
1962:The number of samples required from
8018:Robert, C. P.; Casella, G. (2004).
7736:Thomopoulos, Nick T. (2012-12-19).
7467:{\displaystyle h_{l}\left(x\right)}
7429:{\displaystyle g_{l}\left(x\right)}
5086:{\displaystyle F_{\theta }(\cdot )}
4828:As a simple example, suppose under
6725:
6722:
6719:
6716:
6643:
6449:
6446:
6443:
6440:
6426:Rejection sampling criterion: for
6364:
6064:
5532:
5495:The proposal distribution is thus
5249:
5009:
4871:
4778:
4747:
4713:
4710:
4707:
4676:
4652:
4610:
4332:
4119:
4063:
4033:
3788:
2846:
2843:
2840:
2837:
2513:in some way. Note, however, that
2177:, due to the above formula, where
2164:
1861:
1858:
1855:
1852:
1409:
1301:
1217:
602:
27:Computational statistics technique
25:
7956:. Series C (Applied Statistics).
7765:. Series C (Applied Statistics).
7038:adaptive rejection sampling (ARS)
6933:, which is far more inefficient.
2673:The algorithm, which was used by
907:. Accepting only pairs such that
639:; in other words, M must satisfy
339:by using a proposal distribution
42:. It is also commonly called the
6607:; if not, continue sampling new
5053:. The analysis goes as follows:
2999:and return to the sampling step.
2725:using samples from distribution
1312:
1225:
68:{\displaystyle \mathbb {R} ^{m}}
8020:Monte Carlo Statistical Methods
7387:{\displaystyle f\left(x\right)}
7353:{\displaystyle h\left(x\right)}
7315:{\displaystyle f\left(x\right)}
7266:{\displaystyle h\left(x\right)}
7235:{\displaystyle f\left(x\right)}
7159:{\displaystyle f\left(x\right)}
7125:{\displaystyle g\left(x\right)}
3575:{\displaystyle X\sim F(\cdot )}
3025:iterations to obtain a sample.
2170:{\textstyle 1\leq M<\infty }
1815: includes support of
868:{\textstyle (x,v=u\cdot Mg(x))}
275:is unknown, which is common in
6920:
6892:
6879:
6863:
6851:
6839:
6808:
6802:
6793:
6787:
6741:
6729:
6689:
6647:
6571:
6559:
6549:
6537:
6507:
6501:
6465:
6453:
6380:
6368:
6319:
6306:
6277:
6265:
6226:
6213:
6184:
6172:
6162:
6159:
6146:
6121:
6106:
6100:
6067:
6055:
6029:
6023:
5969:
5957:
5947:
5935:
5927:
5924:
5911:
5886:
5868:
5862:
5840:
5834:
5819:
5797:
5791:
5781:
5778:
5765:
5743:
5734:
5728:
5715:
5709:
5682:
5670:
5660:
5648:
5640:
5634:
5618:
5612:
5597:
5555:
5536:
5525:
5519:
5391:
5385:
5288:
5253:
5186:
5164:
5158:
5152:
5143:
5131:
5122:
5116:
5080:
5074:
4988:
4923:
4917:
4894:
4875:
4844:
4838:
4772:
4766:
4729:
4723:
4671:
4665:
4635:
4629:
4604:
4598:
4589:
4577:
4563:
4554:
4515:
4509:
4453:
4448:
4442:
4407:
4401:
4392:
4368:
4362:
4326:
4320:
4293:
4287:
4258:
4252:
4244:
4238:
4216:
4210:
4195:
4189:
4174:
4168:
4156:
4150:
4105:
4099:
4057:
4051:
4008:
3999:
3971:
3965:
3936:
3930:
3922:
3916:
3893:
3887:
3864:
3858:
3835:
3829:
3821:
3815:
3762:
3750:
3742:
3739:
3733:
3718:
3692:
3686:
3650:
3644:
3621:
3609:
3598:
3592:
3569:
3563:
3459:
3447:
3410:
3398:
3192:
3186:
3119:
3055:
3049:
2920:
2914:
2900:
2894:
2862:
2850:
2651:
2648:
2642:
2633:
2625:
2619:
2596:
2590:
2564:
2558:
2549:
2543:
2500:
2494:
2471:
2465:
2448:{\displaystyle f(x)\leq Mg(x)}
2442:
2436:
2424:
2418:
2355:
2349:
2338:
2332:
2269:
2257:
2229:
2223:
2212:
2206:
2121:
2115:
2104:
2098:
1926:
1920:
1877:
1865:
1821:
1802:
1770:
1764:
1750:
1744:
1700:
1694:
1685:
1679:
1668:
1662:
1645:
1639:
1612:
1609:
1597:
1572:
1560:
1549:
1537:
1531:
1520:
1514:
1465:
1459:
1448:
1442:
1396:
1388:
1378:
1371:
1357:
1351:
1340:
1334:
1307:
1270:
1264:
1253:
1247:
1199:
1193:
1182:
1176:
1123:
1117:
1079:
1073:
1045:
1039:
1016:
1010:
987:
975:
961:{\textstyle u<f(x)/(Mg(x))}
955:
952:
946:
937:
929:
923:
894:
888:
862:
859:
853:
826:
797:
791:
762:
756:
679:{\displaystyle f(x)\leq Mg(x)}
673:
667:
655:
649:
576:
570:
559:
553:
490:
487:
481:
472:
464:
458:
428:and accepting the sample from
375:
369:
326:
320:
1:
7513:Pseudo-random number sampling
1939:of the proposal distribution
181:Draw a vertical line at this
7650:Forsythe, George E. (1972).
5020:{\displaystyle X|X\in \left}
4347:cumulant-generation function
4345:implies that it is indeed a
3198:{\textstyle X\sim F(\cdot )}
3061:{\textstyle X\sim F(\cdot )}
2979:if not, reject the value of
2926:{\textstyle u<f(y)/Mg(y)}
2657:{\displaystyle f(x)/(Mg(x))}
2057:Rewrite the above equation,
726:must include the support of
608:{\displaystyle M<\infty }
523:until a value is accepted.
496:{\displaystyle f(x)/(Mg(x))}
309:probability density function
101:probability density function
7023:Adaptive rejection sampling
7001:adaptive rejection sampling
6587:holds, accept the value of
5353:{\displaystyle \theta ^{*}}
5326:{\displaystyle \theta ^{*}}
2368:. In practice, a value of
503:, repeating the draws from
44:acceptance-rejection method
18:Adaptive rejection sampling
8058:
8042:Non-uniform random numbers
8001:10.1016/j.csda.2008.01.005
7656:Mathematics of Computation
7503:Inverse transform sampling
4136:natural exponential family
3517:Natural Exponential Family
3164:inverse transform sampling
2484:should generally resemble
30:In numerical analysis and
7715:10.1007/s12080-018-0386-z
7570:Neal, Radford M. (2003).
7281:exponential distributions
6814:{\displaystyle M(b)=O(b)}
6013:for the likelihood ratio
5046:{\displaystyle b>\mu }
4850:{\displaystyle F(\cdot )}
3663:. Choose the proposal as
2570:{\displaystyle f(x)=g(x)}
1932:{\displaystyle g(\cdot )}
1594: is uniform on
809:{\displaystyle f(x)>0}
774:{\displaystyle g(x)>0}
582:{\displaystyle f(x)/g(x)}
408:by instead sampling from
359:with probability density
7810:10.1049/iet-cdt:20060188
4977:. The goal is to sample
3547:Given a random variable
1105:Markov chain Monte Carlo
277:computational statistics
32:computational statistics
7927:10.1198/jcgs.2011.09058
7547:10.1214/lnms/1196285403
7009:curse of dimensionality
5306:Decide the well chosen
3380:truncation (statistics)
2956:as a sample drawn from
1393:by tower property
7825:ACM Trans. Math. Softw
7591:10.1214/aos/1056562461
7468:
7430:
7388:
7354:
7316:
7267:
7236:
7197:
7160:
7126:
7095:
6981:
6980:{\displaystyle X\in A}
6955:
6927:
6815:
6771:
6748:
6696:
6601:
6578:
6472:
6416:
6074:
6036:
6007:
5983:
5562:
5489:
5426:
5354:
5327:
5295:
5228:
5087:
5047:
5021:
4971:
4901:
4851:
4817:
4473:
4339:
4302:
4128:
4073:
4020:
3947:
3657:
3628:
3576:
3533:
3509:
3489:
3469:
3423:
3372:
3250:
3199:
3156:
3136:
3102:
3082:
3062:
3019:
2993:
2970:
2950:
2936:If this holds, accept
2927:
2869:
2823:
2803:
2783:
2759:
2739:
2719:
2699:
2658:
2603:
2571:
2527:
2507:
2478:
2455:, which suggests that
2449:
2402:
2382:
2362:
2361:{\textstyle f(x)/g(x)}
2316:
2296:
2276:
2244:
2171:
2139:
2048:
2028:
2008:
1984:geometric distribution
1976:
1953:
1933:
1904:
1884:
1832:
1807:since support of
1130:
1103:techniques, including
1086:
1055:
1023:
994:
962:
901:
869:
810:
775:
740:
720:
700:
680:
633:
609:
583:
537:
517:
497:
442:
422:
402:
382:
353:
333:
301:
263:
238:
218:
195:
172:
158:Sample a point on the
141:
117:
91:-dimension functions.
69:
7847:10.1145/203082.203089
7469:
7431:
7389:
7355:
7317:
7268:
7237:
7198:
7161:
7127:
7096:
6982:
6956:
6928:
6816:
6772:
6749:
6697:
6602:
6579:
6473:
6417:
6075:
6073:{\displaystyle x\in }
6037:
6008:
5984:
5563:
5490:
5427:
5355:
5328:
5296:
5229:
5088:
5048:
5022:
4972:
4902:
4852:
4818:
4474:
4340:
4303:
4129:
4074:
4021:
3948:
3658:
3629:
3577:
3534:
3510:
3490:
3470:
3424:
3373:
3251:
3200:
3157:
3137:
3135:{\textstyle X|X\in A}
3103:
3083:
3063:
3020:
2994:
2971:
2951:
2928:
2870:
2824:
2804:
2784:
2760:
2740:
2720:
2700:
2659:
2604:
2572:
2528:
2508:
2479:
2450:
2403:
2383:
2363:
2317:
2297:
2277:
2245:
2172:
2140:
2049:
2029:
2009:
1977:
1954:
1934:
1905:
1885:
1833:
1131:
1087:
1056:
1054:{\displaystyle f(x).}
1024:
995:
993:{\displaystyle (x,v)}
963:
902:
870:
811:
776:
741:
721:
701:
681:
634:
610:
584:
538:
518:
498:
443:
423:
403:
383:
354:
334:
302:
264:
239:
219:
196:
173:
142:
118:
70:
7577:Annals of Statistics
7440:
7402:
7367:
7333:
7295:
7246:
7215:
7170:
7139:
7105:
7051:
6965:
6945:
6825:
6781:
6761:
6706:
6611:
6591:
6486:
6430:
6088:
6046:
6035:{\displaystyle Z(x)}
6017:
5997:
5581:
5499:
5441:
5370:
5337:
5310:
5245:
5103:
5061:
5031:
4981:
4911:
4861:
4832:
4492:
4356:
4314:
4144:
4083:
4030:
3959:
3669:
3656:{\displaystyle f(x)}
3638:
3586:
3551:
3523:
3499:
3479:
3433:
3390:
3262:
3209:
3174:
3146:
3112:
3092:
3072:
3037:
3009:
2983:
2960:
2940:
2882:
2833:
2813:
2793:
2773:
2749:
2729:
2709:
2689:
2613:
2602:{\displaystyle f(x)}
2584:
2537:
2517:
2506:{\displaystyle f(x)}
2488:
2477:{\displaystyle g(x)}
2459:
2412:
2392:
2372:
2326:
2306:
2286:
2254:
2181:
2149:
2061:
2038:
2018:
1990:
1966:
1943:
1914:
1894:
1842:
1147:
1138:Metropolis algorithm
1129:{\displaystyle f(x)}
1111:
1094:Metropolis algorithm
1085:{\displaystyle f(x)}
1067:
1033:
1022:{\displaystyle f(x)}
1004:
972:
968:then produces pairs
911:
879:
823:
785:
750:
730:
710:
690:
643:
623:
593:
547:
527:
507:
452:
432:
412:
392:
381:{\displaystyle g(x)}
363:
343:
332:{\displaystyle f(x)}
314:
291:
273:normalizing constant
253:
228:
208:
185:
162:
131:
107:
50:
8037:Monte Carlo methods
7703:Theoretical Ecology
7013:Metropolis sampling
6938:exponential tilting
3886:
3797:
2007:{\displaystyle 1/M}
1890:, and the value of
7518:Ziggurat algorithm
7464:
7426:
7384:
7350:
7312:
7263:
7232:
7193:
7156:
7122:
7091:
6977:
6951:
6923:
6811:
6767:
6754:until acceptance.
6744:
6692:
6597:
6574:
6468:
6412:
6070:
6032:
6003:
5979:
5977:
5558:
5485:
5422:
5350:
5323:
5291:
5224:
5083:
5043:
5017:
4967:
4897:
4847:
4813:
4811:
4469:
4335:
4298:
4124:
4069:
4016:
3943:
3941:
3874:
3780:
3653:
3624:
3572:
3529:
3505:
3485:
3465:
3419:
3368:
3246:
3195:
3152:
3132:
3098:
3078:
3058:
3015:
2989:
2966:
2946:
2923:
2865:
2819:
2799:
2789:from distribution
2779:
2755:
2735:
2715:
2695:
2677:and dates back to
2654:
2599:
2567:
2523:
2503:
2474:
2445:
2398:
2378:
2358:
2312:
2292:
2272:
2240:
2167:
2135:
2044:
2024:
2004:
1972:
1949:
1929:
1900:
1880:
1828:
1826:
1760:
1655:
1126:
1082:
1051:
1019:
990:
958:
900:{\textstyle Mg(x)}
897:
865:
806:
771:
736:
716:
696:
686:for all values of
676:
629:
605:
579:
533:
513:
493:
438:
418:
398:
378:
349:
329:
307:with an arbitrary
297:
259:
234:
214:
191:
168:
149:inversion sampling
137:
113:
65:
36:rejection sampling
7747:978-1-4614-6021-3
7636:978-0-387-31073-2
7508:Ratio of uniforms
7322:when your sample
7005:ratio of uniforms
6961:conditionally on
6954:{\displaystyle X}
6917:
6855:
6770:{\displaystyle b}
6600:{\displaystyle X}
6514:
6410:
6402:
6344:
6281:
6251:
6188:
6006:{\displaystyle M}
5993:Derive the bound
5973:
5872:
5686:
5480:
5222:
4965:
4791:
4683:
4262:
4199:
3532:{\displaystyle M}
3508:{\displaystyle M}
3488:{\displaystyle M}
3463:
3101:{\displaystyle A}
3081:{\displaystyle X}
3068:conditionally on
3018:{\displaystyle M}
2992:{\displaystyle y}
2969:{\displaystyle f}
2949:{\displaystyle y}
2822:{\displaystyle u}
2802:{\displaystyle Y}
2782:{\displaystyle y}
2758:{\displaystyle g}
2738:{\displaystyle Y}
2718:{\displaystyle f}
2698:{\displaystyle X}
2526:{\displaystyle M}
2401:{\displaystyle M}
2381:{\displaystyle M}
2315:{\displaystyle M}
2295:{\displaystyle M}
2233:
2133:
2125:
2047:{\displaystyle M}
2027:{\displaystyle M}
2014:, which has mean
1986:with probability
1975:{\displaystyle Y}
1952:{\displaystyle Y}
1903:{\displaystyle y}
1816:
1808:
1798:
1730:
1728:
1689:
1625:
1595:
1587:
1555:
1541:
1469:
1394:
1361:
1274:
1203:
746:—in other words,
739:{\displaystyle X}
719:{\displaystyle Y}
699:{\displaystyle x}
632:{\displaystyle X}
536:{\displaystyle M}
516:{\displaystyle Y}
448:with probability
441:{\displaystyle Y}
421:{\displaystyle Y}
401:{\displaystyle X}
352:{\displaystyle Y}
300:{\displaystyle X}
262:{\displaystyle x}
237:{\displaystyle x}
217:{\displaystyle x}
194:{\displaystyle x}
171:{\displaystyle x}
140:{\displaystyle 1}
116:{\displaystyle x}
16:(Redirected from
8049:
8023:
8005:
8004:
7995:(7): 3408–3423.
7984:
7978:
7977:
7945:
7939:
7938:
7910:
7904:
7903:
7885:
7865:
7859:
7858:
7840:
7820:
7814:
7813:
7793:
7787:
7786:
7758:
7752:
7751:
7733:
7727:
7726:
7694:
7688:
7687:
7662:(120): 817–826.
7647:
7641:
7640:
7618:
7612:
7611:
7593:
7572:"Slice Sampling"
7567:
7561:
7560:
7534:
7473:
7471:
7470:
7465:
7463:
7452:
7451:
7435:
7433:
7432:
7427:
7425:
7414:
7413:
7393:
7391:
7390:
7385:
7383:
7359:
7357:
7356:
7351:
7349:
7321:
7319:
7318:
7313:
7311:
7272:
7270:
7269:
7264:
7262:
7241:
7239:
7238:
7233:
7231:
7202:
7200:
7199:
7194:
7192:
7165:
7163:
7162:
7157:
7155:
7131:
7129:
7128:
7123:
7121:
7100:
7098:
7097:
7092:
7090:
7067:
6986:
6984:
6983:
6978:
6960:
6958:
6957:
6952:
6932:
6930:
6929:
6924:
6919:
6918:
6916:
6915:
6914:
6901:
6900:
6899:
6877:
6856:
6854:
6838:
6829:
6820:
6818:
6817:
6812:
6776:
6774:
6773:
6768:
6753:
6751:
6750:
6745:
6728:
6701:
6699:
6698:
6693:
6688:
6687:
6675:
6674:
6665:
6664:
6646:
6641:
6640:
6606:
6604:
6603:
6598:
6583:
6581:
6580:
6575:
6558:
6553:
6552:
6536:
6535:
6515:
6510:
6496:
6477:
6475:
6474:
6469:
6452:
6421:
6419:
6418:
6413:
6411:
6409:
6408:
6404:
6403:
6398:
6387:
6367:
6357:
6351:
6350:
6346:
6345:
6343:
6342:
6341:
6328:
6327:
6326:
6304:
6287:
6282:
6280:
6264:
6258:
6257:
6253:
6252:
6250:
6249:
6248:
6235:
6234:
6233:
6211:
6194:
6189:
6187:
6171:
6165:
6158:
6157:
6136:
6135:
6113:
6079:
6077:
6076:
6071:
6041:
6039:
6038:
6033:
6012:
6010:
6009:
6004:
5988:
5986:
5985:
5980:
5978:
5974:
5972:
5956:
5950:
5934:
5923:
5922:
5901:
5900:
5878:
5873:
5871:
5861:
5860:
5859:
5858:
5843:
5833:
5832:
5822:
5808:
5777:
5776:
5755:
5754:
5708:
5707:
5706:
5705:
5687:
5685:
5669:
5663:
5647:
5629:
5611:
5610:
5600:
5567:
5565:
5564:
5559:
5554:
5553:
5535:
5518:
5517:
5516:
5515:
5494:
5492:
5491:
5486:
5481:
5479:
5478:
5469:
5458:
5453:
5452:
5431:
5429:
5428:
5423:
5415:
5414:
5384:
5383:
5378:
5359:
5357:
5356:
5351:
5349:
5348:
5332:
5330:
5329:
5324:
5322:
5321:
5300:
5298:
5297:
5292:
5287:
5286:
5274:
5273:
5252:
5233:
5231:
5230:
5225:
5223:
5218:
5217:
5216:
5207:
5206:
5196:
5185:
5184:
5115:
5114:
5092:
5090:
5089:
5084:
5073:
5072:
5052:
5050:
5049:
5044:
5026:
5024:
5023:
5018:
5016:
5012:
4991:
4976:
4974:
4973:
4968:
4966:
4961:
4960:
4959:
4950:
4949:
4939:
4906:
4904:
4903:
4898:
4893:
4892:
4874:
4856:
4854:
4853:
4848:
4822:
4820:
4819:
4814:
4812:
4808:
4807:
4796:
4792:
4790:
4786:
4785:
4775:
4765:
4764:
4755:
4754:
4744:
4722:
4721:
4716:
4700:
4699:
4688:
4684:
4682:
4674:
4664:
4663:
4650:
4628:
4627:
4622:
4570:
4566:
4547:
4546:
4541:
4508:
4507:
4478:
4476:
4475:
4470:
4468:
4467:
4456:
4441:
4440:
4422:
4421:
4410:
4404:
4384:
4344:
4342:
4341:
4336:
4307:
4305:
4304:
4299:
4297:
4296:
4263:
4261:
4248:
4247:
4219:
4205:
4200:
4198:
4188:
4187:
4177:
4163:
4133:
4131:
4130:
4125:
4123:
4122:
4098:
4097:
4078:
4076:
4075:
4070:
4025:
4023:
4022:
4017:
4015:
4011:
3992:
3952:
3950:
3949:
3944:
3942:
3926:
3925:
3882:
3857:
3856:
3825:
3824:
3796:
3791:
3773:
3769:
3765:
3749:
3706:
3685:
3684:
3662:
3660:
3659:
3654:
3633:
3631:
3630:
3625:
3608:
3581:
3579:
3578:
3573:
3538:
3536:
3535:
3530:
3514:
3512:
3511:
3506:
3494:
3492:
3491:
3486:
3474:
3472:
3471:
3466:
3464:
3462:
3446:
3437:
3428:
3426:
3425:
3420:
3397:
3377:
3375:
3374:
3369:
3328:
3327:
3315:
3314:
3290:
3289:
3277:
3276:
3255:
3253:
3252:
3247:
3236:
3235:
3204:
3202:
3201:
3196:
3161:
3159:
3158:
3153:
3141:
3139:
3138:
3133:
3122:
3107:
3105:
3104:
3099:
3087:
3085:
3084:
3079:
3067:
3065:
3064:
3059:
3024:
3022:
3021:
3016:
2998:
2996:
2995:
2990:
2975:
2973:
2972:
2967:
2955:
2953:
2952:
2947:
2932:
2930:
2929:
2924:
2907:
2874:
2872:
2871:
2866:
2849:
2828:
2826:
2825:
2820:
2808:
2806:
2805:
2800:
2788:
2786:
2785:
2780:
2769:Obtain a sample
2764:
2762:
2761:
2756:
2744:
2742:
2741:
2736:
2724:
2722:
2721:
2716:
2704:
2702:
2701:
2696:
2675:John von Neumann
2663:
2661:
2660:
2655:
2632:
2608:
2606:
2605:
2600:
2576:
2574:
2573:
2568:
2532:
2530:
2529:
2524:
2512:
2510:
2509:
2504:
2483:
2481:
2480:
2475:
2454:
2452:
2451:
2446:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2367:
2365:
2364:
2359:
2345:
2321:
2319:
2318:
2313:
2301:
2299:
2298:
2293:
2281:
2279:
2278:
2275:{\displaystyle }
2273:
2249:
2247:
2246:
2241:
2239:
2235:
2234:
2232:
2215:
2201:
2188:
2176:
2174:
2173:
2168:
2144:
2142:
2141:
2136:
2134:
2132:
2131:
2127:
2126:
2124:
2107:
2093:
2080:
2071:
2053:
2051:
2050:
2045:
2034:. Intuitively,
2033:
2031:
2030:
2025:
2013:
2011:
2010:
2005:
2000:
1981:
1979:
1978:
1973:
1958:
1956:
1955:
1950:
1938:
1936:
1935:
1930:
1909:
1907:
1906:
1901:
1889:
1887:
1886:
1881:
1864:
1837:
1835:
1834:
1829:
1827:
1817:
1814:
1809:
1806:
1799:
1791:
1783:
1759:
1729:
1721:
1713:
1690:
1688:
1671:
1657:
1654:
1618:
1596:
1593:
1588:
1585:
1556:
1553:
1546:
1542:
1540:
1523:
1509:
1494:
1490:
1486:
1485:
1481:
1477:
1476:
1470:
1468:
1451:
1437:
1424:
1402:
1395:
1392:
1385:
1381:
1374:
1369:
1368:
1367:
1363:
1362:
1360:
1343:
1329:
1315:
1286:
1282:
1281:
1280:
1276:
1275:
1273:
1256:
1242:
1228:
1209:
1205:
1204:
1202:
1185:
1171:
1158:
1135:
1133:
1132:
1127:
1091:
1089:
1088:
1083:
1060:
1058:
1057:
1052:
1028:
1026:
1025:
1020:
999:
997:
996:
991:
967:
965:
964:
959:
936:
906:
904:
903:
898:
874:
872:
871:
866:
815:
813:
812:
807:
780:
778:
777:
772:
745:
743:
742:
737:
725:
723:
722:
717:
705:
703:
702:
697:
685:
683:
682:
677:
638:
636:
635:
630:
614:
612:
611:
606:
588:
586:
585:
580:
566:
542:
540:
539:
534:
522:
520:
519:
514:
502:
500:
499:
494:
471:
447:
445:
444:
439:
427:
425:
424:
419:
407:
405:
404:
399:
387:
385:
384:
379:
358:
356:
355:
350:
338:
336:
335:
330:
306:
304:
303:
298:
270:
268:
266:
265:
260:
243:
241:
240:
235:
223:
221:
220:
215:
200:
198:
197:
192:
177:
175:
174:
169:
146:
144:
143:
138:
122:
120:
119:
114:
74:
72:
71:
66:
64:
63:
58:
21:
8057:
8056:
8052:
8051:
8050:
8048:
8047:
8046:
8027:
8026:
8017:
8014:
8012:Further reading
8009:
8008:
7986:
7985:
7981:
7966:10.2307/2986138
7947:
7946:
7942:
7912:
7911:
7907:
7892:10.2307/1390680
7867:
7866:
7862:
7822:
7821:
7817:
7795:
7794:
7790:
7775:10.2307/2347565
7760:
7759:
7755:
7748:
7735:
7734:
7730:
7696:
7695:
7691:
7668:10.2307/2005864
7649:
7648:
7644:
7637:
7620:
7619:
7615:
7569:
7568:
7564:
7557:
7536:
7535:
7531:
7526:
7499:
7453:
7443:
7438:
7437:
7415:
7405:
7400:
7399:
7373:
7365:
7364:
7339:
7331:
7330:
7301:
7293:
7292:
7252:
7244:
7243:
7221:
7213:
7212:
7182:
7168:
7167:
7145:
7137:
7136:
7111:
7103:
7102:
7080:
7057:
7049:
7048:
7025:
6993:
6963:
6962:
6943:
6942:
6906:
6902:
6891:
6878:
6872:
6833:
6823:
6822:
6779:
6778:
6759:
6758:
6704:
6703:
6679:
6666:
6656:
6617:
6609:
6608:
6589:
6588:
6527:
6519:
6497:
6484:
6483:
6428:
6427:
6388:
6362:
6358:
6352:
6333:
6329:
6318:
6305:
6299:
6295:
6288:
6259:
6240:
6236:
6225:
6212:
6206:
6202:
6195:
6166:
6149:
6127:
6114:
6086:
6085:
6044:
6043:
6015:
6014:
5995:
5994:
5976:
5975:
5951:
5914:
5892:
5879:
5850:
5845:
5844:
5810:
5809:
5800:
5785:
5784:
5768:
5746:
5718:
5697:
5692:
5689:
5688:
5664:
5630:
5621:
5588:
5579:
5578:
5545:
5507:
5502:
5497:
5496:
5470:
5459:
5444:
5439:
5438:
5406:
5373:
5368:
5367:
5340:
5335:
5334:
5313:
5308:
5307:
5278:
5265:
5243:
5242:
5208:
5198:
5197:
5176:
5106:
5101:
5100:
5064:
5059:
5058:
5029:
5028:
5002:
4998:
4979:
4978:
4951:
4941:
4940:
4909:
4908:
4884:
4859:
4858:
4830:
4829:
4810:
4809:
4777:
4776:
4756:
4746:
4745:
4740:
4739:
4732:
4705:
4702:
4701:
4675:
4655:
4651:
4646:
4645:
4638:
4617:
4614:
4613:
4536:
4535:
4531:
4518:
4499:
4490:
4489:
4451:
4432:
4405:
4354:
4353:
4312:
4311:
4267:
4221:
4220:
4206:
4179:
4178:
4164:
4142:
4141:
4108:
4089:
4081:
4080:
4028:
4027:
3987:
3983:
3957:
3956:
3940:
3939:
3899:
3867:
3848:
3845:
3844:
3798:
3771:
3770:
3711:
3707:
3695:
3676:
3667:
3666:
3636:
3635:
3584:
3583:
3549:
3548:
3545:
3521:
3520:
3497:
3496:
3477:
3476:
3441:
3431:
3430:
3388:
3387:
3319:
3306:
3281:
3268:
3260:
3259:
3227:
3207:
3206:
3172:
3171:
3144:
3143:
3110:
3109:
3090:
3089:
3070:
3069:
3035:
3034:
3031:
3007:
3006:
2981:
2980:
2958:
2957:
2938:
2937:
2880:
2879:
2831:
2830:
2811:
2810:
2791:
2790:
2771:
2770:
2747:
2746:
2727:
2726:
2707:
2706:
2687:
2686:
2671:
2611:
2610:
2582:
2581:
2535:
2534:
2515:
2514:
2486:
2485:
2457:
2456:
2410:
2409:
2390:
2389:
2370:
2369:
2324:
2323:
2304:
2303:
2284:
2283:
2252:
2251:
2216:
2202:
2193:
2189:
2179:
2178:
2147:
2146:
2108:
2094:
2085:
2081:
2075:
2059:
2058:
2036:
2035:
2016:
2015:
1988:
1987:
1964:
1963:
1941:
1940:
1912:
1911:
1892:
1891:
1840:
1839:
1825:
1824:
1800:
1781:
1780:
1711:
1710:
1672:
1658:
1616:
1615:
1547:
1524:
1510:
1504:
1492:
1491:
1452:
1438:
1429:
1425:
1419:
1415:
1400:
1399:
1386:
1344:
1330:
1321:
1317:
1310:
1300:
1296:
1284:
1283:
1257:
1243:
1234:
1230:
1223:
1210:
1186:
1172:
1163:
1159:
1145:
1144:
1109:
1108:
1065:
1064:
1031:
1030:
1002:
1001:
970:
969:
909:
908:
877:
876:
821:
820:
783:
782:
748:
747:
728:
727:
708:
707:
688:
687:
641:
640:
621:
620:
591:
590:
545:
544:
525:
524:
505:
504:
450:
449:
430:
429:
410:
409:
390:
389:
361:
360:
341:
340:
312:
311:
289:
288:
285:
251:
250:
249:
226:
225:
206:
205:
183:
182:
160:
159:
129:
128:
105:
104:
97:
84:random variable
53:
48:
47:
28:
23:
22:
15:
12:
11:
5:
8055:
8053:
8045:
8044:
8039:
8029:
8028:
8025:
8024:
8013:
8010:
8007:
8006:
7979:
7960:(4): 455–472.
7948:Gilks, W. R.;
7940:
7921:(3): 670–691.
7905:
7883:10.1.1.53.9001
7876:(4): 514–528.
7860:
7838:10.1.1.56.6055
7831:(2): 182–193.
7815:
7804:(4): 312–321.
7788:
7769:(2): 337–348.
7753:
7746:
7728:
7689:
7642:
7635:
7613:
7584:(3): 705–767.
7562:
7555:
7528:
7527:
7525:
7522:
7521:
7520:
7515:
7510:
7505:
7498:
7495:
7482:
7481:
7480:
7479:
7475:
7462:
7459:
7456:
7450:
7446:
7424:
7421:
7418:
7412:
7408:
7382:
7379:
7376:
7372:
7361:
7348:
7345:
7342:
7338:
7310:
7307:
7304:
7300:
7289:
7288:
7287:
7284:
7277:
7274:
7261:
7258:
7255:
7251:
7230:
7227:
7224:
7220:
7206:
7205:
7204:
7191:
7188:
7185:
7181:
7178:
7175:
7154:
7151:
7148:
7144:
7120:
7117:
7114:
7110:
7089:
7086:
7083:
7079:
7076:
7073:
7070:
7066:
7063:
7060:
7056:
7024:
7021:
7017:Gibbs sampling
6992:
6989:
6976:
6973:
6970:
6950:
6922:
6913:
6909:
6905:
6898:
6894:
6890:
6887:
6884:
6881:
6875:
6871:
6868:
6865:
6862:
6859:
6853:
6850:
6847:
6844:
6841:
6837:
6832:
6810:
6807:
6804:
6801:
6798:
6795:
6792:
6789:
6786:
6766:
6743:
6740:
6737:
6734:
6731:
6727:
6724:
6721:
6718:
6714:
6711:
6691:
6686:
6682:
6678:
6673:
6669:
6663:
6659:
6655:
6652:
6649:
6645:
6639:
6636:
6633:
6630:
6627:
6624:
6620:
6616:
6596:
6585:
6584:
6573:
6570:
6567:
6564:
6561:
6557:
6551:
6548:
6545:
6542:
6539:
6534:
6530:
6526:
6522:
6518:
6513:
6509:
6506:
6503:
6500:
6494:
6491:
6480:
6479:
6467:
6464:
6461:
6458:
6455:
6451:
6448:
6445:
6442:
6438:
6435:
6423:
6422:
6407:
6401:
6397:
6394:
6391:
6385:
6382:
6379:
6376:
6373:
6370:
6366:
6361:
6356:
6349:
6340:
6336:
6332:
6325:
6321:
6317:
6314:
6311:
6308:
6302:
6298:
6294:
6291:
6285:
6279:
6276:
6273:
6270:
6267:
6263:
6256:
6247:
6243:
6239:
6232:
6228:
6224:
6221:
6218:
6215:
6209:
6205:
6201:
6198:
6192:
6186:
6183:
6180:
6177:
6174:
6170:
6164:
6161:
6156:
6152:
6148:
6145:
6142:
6139:
6134:
6130:
6126:
6123:
6120:
6117:
6111:
6108:
6105:
6102:
6099:
6096:
6093:
6082:
6081:
6069:
6066:
6063:
6060:
6057:
6054:
6051:
6031:
6028:
6025:
6022:
6002:
5990:
5989:
5971:
5968:
5965:
5962:
5959:
5955:
5949:
5946:
5943:
5940:
5937:
5933:
5929:
5926:
5921:
5917:
5913:
5910:
5907:
5904:
5899:
5895:
5891:
5888:
5885:
5882:
5876:
5870:
5867:
5864:
5857:
5853:
5848:
5842:
5839:
5836:
5831:
5828:
5825:
5821:
5817:
5813:
5806:
5803:
5801:
5799:
5796:
5793:
5790:
5787:
5786:
5783:
5780:
5775:
5771:
5767:
5764:
5761:
5758:
5753:
5749:
5745:
5742:
5739:
5736:
5733:
5730:
5727:
5724:
5721:
5719:
5717:
5714:
5711:
5704:
5700:
5695:
5691:
5690:
5684:
5681:
5678:
5675:
5672:
5668:
5662:
5659:
5656:
5653:
5650:
5646:
5642:
5639:
5636:
5633:
5627:
5624:
5622:
5620:
5617:
5614:
5609:
5606:
5603:
5599:
5595:
5591:
5587:
5586:
5575:
5574:
5570:
5569:
5557:
5552:
5548:
5544:
5541:
5538:
5534:
5530:
5527:
5524:
5521:
5514:
5510:
5505:
5484:
5477:
5473:
5468:
5465:
5462:
5456:
5451:
5447:
5435:
5434:
5433:
5421:
5418:
5413:
5409:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5382:
5377:
5362:
5361:
5347:
5343:
5320:
5316:
5303:
5302:
5290:
5285:
5281:
5277:
5272:
5268:
5264:
5261:
5258:
5255:
5251:
5238:
5237:
5236:
5235:
5221:
5215:
5211:
5205:
5201:
5194:
5191:
5188:
5183:
5179:
5175:
5172:
5169:
5166:
5163:
5160:
5157:
5154:
5151:
5148:
5145:
5142:
5139:
5136:
5133:
5130:
5127:
5124:
5121:
5118:
5113:
5109:
5095:
5094:
5082:
5079:
5076:
5071:
5067:
5042:
5039:
5036:
5015:
5011:
5008:
5005:
5001:
4997:
4994:
4990:
4986:
4964:
4958:
4954:
4948:
4944:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4896:
4891:
4887:
4883:
4880:
4877:
4873:
4869:
4866:
4846:
4843:
4840:
4837:
4826:
4825:
4824:
4823:
4806:
4803:
4800:
4795:
4789:
4784:
4780:
4774:
4771:
4768:
4763:
4759:
4753:
4749:
4742:
4738:
4735:
4733:
4731:
4728:
4725:
4720:
4715:
4712:
4709:
4704:
4703:
4698:
4695:
4692:
4687:
4681:
4678:
4673:
4670:
4667:
4662:
4658:
4654:
4648:
4644:
4641:
4639:
4637:
4634:
4631:
4626:
4621:
4616:
4615:
4612:
4609:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4582:
4579:
4576:
4573:
4569:
4565:
4562:
4559:
4556:
4553:
4550:
4545:
4540:
4534:
4530:
4527:
4524:
4521:
4519:
4517:
4514:
4511:
4506:
4502:
4498:
4497:
4481:
4480:
4466:
4463:
4460:
4455:
4450:
4447:
4444:
4439:
4435:
4431:
4428:
4425:
4420:
4417:
4414:
4409:
4403:
4400:
4397:
4394:
4391:
4388:
4383:
4379:
4376:
4373:
4370:
4367:
4364:
4361:
4334:
4331:
4328:
4325:
4322:
4319:
4295:
4292:
4289:
4286:
4283:
4280:
4277:
4274:
4270:
4266:
4260:
4257:
4254:
4251:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4224:
4218:
4215:
4212:
4209:
4203:
4197:
4194:
4191:
4186:
4182:
4176:
4173:
4170:
4167:
4161:
4158:
4155:
4152:
4149:
4121:
4118:
4115:
4111:
4107:
4104:
4101:
4096:
4092:
4088:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4014:
4010:
4007:
4004:
4001:
3998:
3995:
3991:
3986:
3982:
3979:
3976:
3973:
3970:
3967:
3964:
3938:
3935:
3932:
3929:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3902:
3898:
3895:
3892:
3889:
3885:
3881:
3877:
3873:
3870:
3868:
3866:
3863:
3860:
3855:
3851:
3847:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3801:
3795:
3790:
3787:
3783:
3779:
3776:
3774:
3772:
3768:
3764:
3761:
3758:
3755:
3752:
3748:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3720:
3717:
3714:
3710:
3705:
3701:
3698:
3696:
3694:
3691:
3688:
3683:
3679:
3675:
3674:
3652:
3649:
3646:
3643:
3623:
3620:
3617:
3614:
3611:
3607:
3603:
3600:
3597:
3594:
3591:
3571:
3568:
3565:
3562:
3559:
3556:
3544:
3541:
3528:
3504:
3484:
3461:
3458:
3455:
3452:
3449:
3445:
3440:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3396:
3384:
3383:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3326:
3322:
3318:
3313:
3309:
3305:
3302:
3299:
3296:
3293:
3288:
3284:
3280:
3275:
3271:
3267:
3256:
3245:
3242:
3239:
3234:
3230:
3226:
3223:
3220:
3217:
3214:
3194:
3191:
3188:
3185:
3182:
3179:
3155:{\textstyle X}
3151:
3131:
3128:
3125:
3121:
3117:
3097:
3088:given the set
3077:
3057:
3054:
3051:
3048:
3045:
3042:
3030:
3027:
3014:
3003:
3002:
3001:
3000:
2988:
2977:
2965:
2945:
2922:
2919:
2916:
2913:
2910:
2906:
2902:
2899:
2896:
2893:
2890:
2887:
2876:
2864:
2861:
2858:
2855:
2852:
2848:
2845:
2842:
2839:
2818:
2798:
2778:
2754:
2734:
2714:
2694:
2670:
2667:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2631:
2627:
2624:
2621:
2618:
2598:
2595:
2592:
2589:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2522:
2502:
2499:
2496:
2493:
2473:
2470:
2467:
2464:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2397:
2377:
2357:
2354:
2351:
2348:
2344:
2340:
2337:
2334:
2331:
2311:
2291:
2271:
2268:
2265:
2262:
2259:
2238:
2231:
2228:
2225:
2222:
2219:
2214:
2211:
2208:
2205:
2199:
2196:
2192:
2187:
2166:
2163:
2160:
2157:
2154:
2130:
2123:
2120:
2117:
2114:
2111:
2106:
2103:
2100:
2097:
2091:
2088:
2084:
2079:
2074:
2069:
2066:
2043:
2023:
2003:
1999:
1995:
1971:
1948:
1928:
1925:
1922:
1919:
1899:
1879:
1876:
1873:
1870:
1867:
1863:
1860:
1857:
1854:
1850:
1847:
1823:
1820:
1812:
1804:
1801:
1797:
1794:
1789:
1786:
1784:
1782:
1779:
1776:
1772:
1769:
1766:
1763:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1733:
1727:
1724:
1719:
1716:
1714:
1712:
1709:
1706:
1702:
1699:
1696:
1693:
1687:
1684:
1681:
1678:
1675:
1670:
1667:
1664:
1661:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1628:
1624:
1621:
1619:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1591:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1551:
1548:
1545:
1539:
1536:
1533:
1530:
1527:
1522:
1519:
1516:
1513:
1507:
1503:
1500:
1497:
1495:
1493:
1489:
1484:
1480:
1475:
1467:
1464:
1461:
1458:
1455:
1450:
1447:
1444:
1441:
1435:
1432:
1428:
1423:
1418:
1414:
1411:
1408:
1405:
1403:
1401:
1398:
1390:
1387:
1384:
1380:
1377:
1373:
1366:
1359:
1356:
1353:
1350:
1347:
1342:
1339:
1336:
1333:
1327:
1324:
1320:
1314:
1309:
1306:
1303:
1299:
1295:
1292:
1289:
1287:
1285:
1279:
1272:
1269:
1266:
1263:
1260:
1255:
1252:
1249:
1246:
1240:
1237:
1233:
1227:
1222:
1219:
1216:
1213:
1211:
1208:
1201:
1198:
1195:
1192:
1189:
1184:
1181:
1178:
1175:
1169:
1166:
1162:
1157:
1153:
1152:
1125:
1122:
1119:
1116:
1081:
1078:
1075:
1072:
1050:
1047:
1044:
1041:
1038:
1018:
1015:
1012:
1009:
989:
986:
983:
980:
977:
957:
954:
951:
948:
945:
942:
939:
935:
931:
928:
925:
922:
919:
916:
896:
893:
890:
887:
884:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
805:
802:
799:
796:
793:
790:
770:
767:
764:
761:
758:
755:
735:
715:
695:
675:
672:
669:
666:
663:
660:
657:
654:
651:
648:
628:
604:
601:
598:
578:
575:
572:
569:
565:
561:
558:
555:
552:
532:
512:
492:
489:
486:
483:
480:
477:
474:
470:
466:
463:
460:
457:
437:
417:
397:
377:
374:
371:
368:
348:
328:
325:
322:
319:
296:
284:
281:
258:
246:
245:
233:
213:
202:
190:
179:
167:
136:
112:
96:
93:
62:
57:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8054:
8043:
8040:
8038:
8035:
8034:
8032:
8021:
8016:
8015:
8011:
8002:
7998:
7994:
7990:
7983:
7980:
7975:
7971:
7967:
7963:
7959:
7955:
7951:
7944:
7941:
7936:
7932:
7928:
7924:
7920:
7916:
7909:
7906:
7901:
7897:
7893:
7889:
7884:
7879:
7875:
7871:
7864:
7861:
7856:
7852:
7848:
7844:
7839:
7834:
7830:
7826:
7819:
7816:
7811:
7807:
7803:
7799:
7792:
7789:
7784:
7780:
7776:
7772:
7768:
7764:
7757:
7754:
7749:
7743:
7739:
7732:
7729:
7724:
7720:
7716:
7712:
7708:
7704:
7700:
7693:
7690:
7685:
7681:
7677:
7673:
7669:
7665:
7661:
7657:
7653:
7646:
7643:
7638:
7632:
7628:
7624:
7617:
7614:
7609:
7605:
7601:
7597:
7592:
7587:
7583:
7579:
7578:
7573:
7566:
7563:
7558:
7556:9780940600614
7552:
7548:
7544:
7540:
7533:
7530:
7523:
7519:
7516:
7514:
7511:
7509:
7506:
7504:
7501:
7500:
7496:
7494:
7492:
7486:
7476:
7460:
7457:
7454:
7448:
7444:
7422:
7419:
7416:
7410:
7406:
7397:
7380:
7377:
7374:
7370:
7362:
7346:
7343:
7340:
7336:
7328:
7327:
7325:
7308:
7305:
7302:
7298:
7290:
7285:
7282:
7278:
7275:
7259:
7256:
7253:
7249:
7228:
7225:
7222:
7218:
7210:
7209:
7207:
7189:
7186:
7183:
7179:
7176:
7173:
7152:
7149:
7146:
7142:
7134:
7133:
7118:
7115:
7112:
7108:
7087:
7084:
7081:
7077:
7074:
7071:
7068:
7064:
7061:
7058:
7054:
7046:
7045:
7044:
7041:
7039:
7035:
7031:
7022:
7020:
7018:
7014:
7010:
7006:
7002:
6996:
6990:
6988:
6974:
6971:
6968:
6948:
6939:
6934:
6911:
6907:
6903:
6896:
6888:
6885:
6882:
6873:
6869:
6866:
6860:
6857:
6848:
6845:
6842:
6830:
6805:
6799:
6796:
6790:
6784:
6764:
6755:
6738:
6735:
6732:
6712:
6709:
6684:
6680:
6676:
6671:
6667:
6661:
6657:
6653:
6650:
6637:
6634:
6631:
6628:
6625:
6622:
6618:
6614:
6594:
6568:
6565:
6562:
6546:
6543:
6540:
6532:
6528:
6524:
6520:
6516:
6511:
6504:
6498:
6492:
6489:
6482:
6481:
6462:
6459:
6456:
6436:
6433:
6425:
6424:
6405:
6399:
6395:
6392:
6389:
6383:
6377:
6374:
6371:
6359:
6347:
6338:
6334:
6330:
6323:
6315:
6312:
6309:
6300:
6296:
6292:
6289:
6283:
6274:
6271:
6268:
6254:
6245:
6241:
6237:
6230:
6222:
6219:
6216:
6207:
6203:
6199:
6196:
6190:
6181:
6178:
6175:
6154:
6150:
6143:
6140:
6137:
6132:
6128:
6124:
6118:
6115:
6109:
6103:
6097:
6094:
6091:
6084:
6083:
6061:
6058:
6052:
6049:
6026:
6020:
6000:
5992:
5991:
5966:
5963:
5960:
5944:
5941:
5938:
5919:
5915:
5908:
5905:
5902:
5897:
5893:
5889:
5883:
5880:
5874:
5865:
5855:
5851:
5846:
5837:
5829:
5826:
5823:
5815:
5811:
5804:
5802:
5794:
5788:
5773:
5769:
5762:
5759:
5756:
5751:
5747:
5740:
5737:
5731:
5725:
5722:
5720:
5712:
5702:
5698:
5693:
5679:
5676:
5673:
5657:
5654:
5651:
5637:
5631:
5625:
5623:
5615:
5607:
5604:
5601:
5593:
5589:
5577:
5576:
5572:
5571:
5550:
5546:
5542:
5539:
5528:
5522:
5512:
5508:
5503:
5482:
5475:
5471:
5466:
5463:
5460:
5454:
5449:
5445:
5436:
5419:
5416:
5411:
5407:
5403:
5400:
5397:
5394:
5388:
5380:
5366:
5365:
5364:
5363:
5345:
5341:
5318:
5314:
5305:
5304:
5283:
5279:
5275:
5270:
5266:
5262:
5259:
5256:
5240:
5239:
5219:
5213:
5209:
5203:
5199:
5192:
5189:
5181:
5177:
5173:
5170:
5167:
5161:
5155:
5149:
5146:
5140:
5137:
5134:
5128:
5125:
5119:
5111:
5107:
5099:
5098:
5097:
5096:
5077:
5069:
5065:
5056:
5055:
5054:
5040:
5037:
5034:
5013:
5006:
5003:
4999:
4995:
4992:
4984:
4962:
4956:
4952:
4946:
4942:
4935:
4932:
4929:
4926:
4920:
4914:
4889:
4885:
4881:
4878:
4867:
4864:
4841:
4835:
4804:
4801:
4798:
4793:
4787:
4782:
4769:
4761:
4757:
4751:
4736:
4734:
4726:
4718:
4696:
4693:
4690:
4685:
4679:
4668:
4660:
4656:
4642:
4640:
4632:
4624:
4607:
4601:
4595:
4592:
4586:
4583:
4580:
4574:
4571:
4567:
4560:
4557:
4551:
4548:
4543:
4532:
4528:
4525:
4522:
4520:
4512:
4504:
4500:
4488:
4487:
4486:
4485:
4484:
4464:
4461:
4458:
4445:
4437:
4433:
4429:
4426:
4423:
4418:
4415:
4412:
4398:
4395:
4389:
4386:
4377:
4374:
4371:
4365:
4359:
4352:
4351:
4350:
4348:
4329:
4323:
4317:
4308:
4290:
4284:
4281:
4278:
4275:
4272:
4268:
4264:
4255:
4249:
4241:
4235:
4232:
4229:
4226:
4222:
4213:
4207:
4201:
4192:
4184:
4180:
4171:
4165:
4159:
4153:
4147:
4139:
4137:
4116:
4113:
4102:
4094:
4090:
4060:
4054:
4048:
4045:
4042:
4036:
4012:
4005:
4002:
3996:
3993:
3984:
3980:
3977:
3974:
3968:
3962:
3953:
3933:
3927:
3919:
3913:
3910:
3907:
3904:
3900:
3896:
3890:
3883:
3879:
3875:
3871:
3869:
3861:
3853:
3849:
3841:
3838:
3832:
3826:
3818:
3812:
3809:
3806:
3803:
3799:
3793:
3785:
3781:
3777:
3775:
3766:
3759:
3756:
3753:
3736:
3730:
3727:
3724:
3721:
3715:
3712:
3708:
3699:
3697:
3689:
3681:
3677:
3664:
3647:
3641:
3618:
3615:
3612:
3601:
3595:
3589:
3566:
3560:
3557:
3554:
3542:
3540:
3526:
3518:
3502:
3482:
3456:
3453:
3450:
3438:
3416:
3413:
3407:
3404:
3401:
3381:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3324:
3320:
3316:
3311:
3307:
3303:
3300:
3297:
3294:
3291:
3286:
3282:
3278:
3273:
3269:
3257:
3240:
3237:
3232:
3228:
3224:
3221:
3218:
3215:
3189:
3183:
3180:
3177:
3169:
3168:
3167:
3165:
3149:
3129:
3126:
3123:
3115:
3095:
3075:
3052:
3046:
3043:
3040:
3028:
3026:
3012:
2986:
2978:
2963:
2943:
2935:
2934:
2917:
2911:
2908:
2904:
2897:
2891:
2888:
2885:
2877:
2859:
2856:
2853:
2816:
2809:and a sample
2796:
2776:
2768:
2767:
2766:
2752:
2745:with density
2732:
2712:
2705:with density
2692:
2684:
2680:
2676:
2668:
2666:
2645:
2639:
2636:
2629:
2622:
2616:
2593:
2587:
2578:
2561:
2555:
2552:
2546:
2540:
2520:
2497:
2491:
2468:
2462:
2439:
2433:
2430:
2427:
2421:
2415:
2395:
2375:
2352:
2346:
2342:
2335:
2329:
2309:
2289:
2266:
2263:
2260:
2236:
2226:
2220:
2217:
2209:
2203:
2197:
2194:
2190:
2161:
2158:
2155:
2152:
2128:
2118:
2112:
2109:
2101:
2095:
2089:
2086:
2082:
2072:
2067:
2064:
2055:
2041:
2021:
2001:
1997:
1993:
1985:
1969:
1960:
1946:
1923:
1917:
1897:
1874:
1871:
1868:
1848:
1845:
1818:
1810:
1795:
1792:
1787:
1785:
1777:
1774:
1767:
1761:
1756:
1753:
1747:
1741:
1738:
1735:
1731:
1725:
1722:
1717:
1715:
1707:
1704:
1697:
1691:
1682:
1676:
1673:
1665:
1659:
1651:
1648:
1642:
1636:
1633:
1630:
1626:
1622:
1620:
1606:
1603:
1600:
1589:
1581:
1578:
1575:
1569:
1566:
1563:
1554:because
1543:
1534:
1528:
1525:
1517:
1511:
1505:
1501:
1498:
1496:
1487:
1482:
1478:
1462:
1456:
1453:
1445:
1439:
1433:
1430:
1426:
1416:
1412:
1406:
1404:
1382:
1375:
1364:
1354:
1348:
1345:
1337:
1331:
1325:
1322:
1318:
1304:
1297:
1293:
1290:
1288:
1277:
1267:
1261:
1258:
1250:
1244:
1238:
1235:
1231:
1220:
1214:
1212:
1206:
1196:
1190:
1187:
1179:
1173:
1167:
1164:
1160:
1141:
1139:
1120:
1114:
1106:
1102:
1097:
1095:
1076:
1070:
1061:
1048:
1042:
1036:
1013:
1007:
984:
981:
978:
949:
943:
940:
933:
926:
920:
917:
914:
891:
885:
882:
856:
850:
847:
844:
841:
838:
835:
832:
829:
817:
803:
800:
794:
788:
768:
765:
759:
753:
733:
713:
693:
670:
664:
661:
658:
652:
646:
626:
618:
599:
596:
589:, satisfying
573:
567:
563:
556:
550:
530:
510:
484:
478:
475:
468:
461:
455:
435:
415:
395:
372:
366:
346:
323:
317:
310:
294:
282:
280:
278:
274:
256:
231:
211:
203:
188:
180:
165:
157:
156:
155:
152:
150:
134:
124:
110:
102:
94:
92:
90:
85:
80:
78:
60:
45:
41:
37:
33:
19:
8019:
7992:
7988:
7982:
7957:
7953:
7943:
7918:
7914:
7908:
7873:
7869:
7863:
7828:
7824:
7818:
7801:
7797:
7791:
7766:
7762:
7756:
7737:
7731:
7709:(1): 31–48.
7706:
7702:
7692:
7659:
7655:
7645:
7622:
7616:
7581:
7575:
7565:
7538:
7532:
7490:
7487:
7483:
7396:already know
7395:
7323:
7042:
7037:
7033:
7026:
6997:
6994:
6936:In general,
6935:
6756:
6586:
4827:
4482:
4309:
4140:
4134:, is from a
3954:
3665:
3546:
3385:
3142:, sometimes
3032:
3004:
2765:as follows:
2672:
2579:
2056:
1961:
1142:
1098:
1062:
818:
286:
247:
153:
125:
98:
88:
81:
43:
40:distribution
35:
29:
7950:Best, N. G.
7101:instead of
7030:log-concave
6080:, therefore
4349:, that is,
4079:. Clearly,
1101:Monte Carlo
95:Description
8031:Categories
7608:1051.65007
7524:References
7326:accepted.
7166:is messy,
7132:directly.
6777:, that is
4310:Note that
3378:(see also
2683:his needle
2145:Note that
1586:when
269:‑positions
7935:1061-8600
7878:CiteSeerX
7855:0098-3500
7833:CiteSeerX
7723:1874-1746
7676:0025-5718
7177:
7075:
6991:Drawbacks
6972:∈
6908:σ
6889:μ
6886:−
6870:⋅
6846:≥
6713:∼
6681:σ
6668:σ
6662:∗
6658:θ
6651:μ
6619:∼
6566:≥
6544:−
6533:∗
6529:θ
6525:−
6493:≤
6437:∼
6400:σ
6396:μ
6393:−
6384:≥
6335:σ
6316:μ
6313:−
6301:−
6293:
6272:≥
6242:σ
6223:μ
6220:−
6208:−
6200:
6179:≥
6155:∗
6151:θ
6144:ψ
6133:∗
6129:θ
6125:−
6119:
6065:∞
6053:∈
5964:≥
5942:≥
5920:∗
5916:θ
5909:ψ
5898:∗
5894:θ
5890:−
5884:
5856:∗
5852:θ
5827:≥
5774:∗
5770:θ
5763:ψ
5760:−
5752:∗
5748:θ
5741:
5703:∗
5699:θ
5677:≥
5655:≥
5605:≥
5547:σ
5513:∗
5509:θ
5472:σ
5467:μ
5464:−
5450:∗
5446:θ
5408:σ
5404:θ
5398:μ
5381:θ
5360:is to set
5346:∗
5342:θ
5319:∗
5315:θ
5280:σ
5267:σ
5263:θ
5257:μ
5210:η
5200:σ
5190:η
5178:σ
5174:θ
5168:μ
5156:θ
5150:ψ
5147:−
5141:η
5135:θ
5129:ψ
5120:η
5112:θ
5108:ψ
5078:⋅
5070:θ
5041:μ
5010:∞
4996:∈
4953:θ
4943:σ
4933:θ
4930:μ
4921:θ
4915:ψ
4886:σ
4879:μ
4868:∼
4842:⋅
4799:η
4788:η
4779:∂
4770:η
4762:θ
4758:ψ
4748:∂
4719:θ
4691:η
4680:η
4677:∂
4669:η
4661:θ
4657:ψ
4653:∂
4625:θ
4611:∞
4602:θ
4596:ψ
4593:−
4587:η
4581:θ
4575:ψ
4558:η
4552:
4544:θ
4529:
4513:η
4505:θ
4501:ψ
4465:θ
4430:
4419:θ
4390:
4378:
4366:θ
4360:ψ
4333:∞
4324:θ
4318:ψ
4291:θ
4285:ψ
4276:θ
4273:−
4242:θ
4236:ψ
4233:−
4227:θ
4185:θ
4120:Θ
4117:∈
4114:θ
4103:⋅
4095:θ
4064:∞
4055:θ
4049:ψ
4043:θ
4034:Θ
4003:θ
3997:
3981:
3969:θ
3963:ψ
3920:θ
3914:ψ
3911:−
3905:θ
3880:θ
3854:θ
3819:θ
3813:ψ
3810:−
3804:θ
3789:∞
3786:−
3782:∫
3757:≤
3737:θ
3731:ψ
3728:−
3722:θ
3716:
3682:θ
3616:≤
3567:⋅
3558:∼
3454:∈
3414:≈
3405:∈
3330:∈
3238:∈
3219:≥
3190:⋅
3181:∼
3127:∈
3053:⋅
3044:∼
2878:Check if
2669:Algorithm
2428:≤
2198:≤
2165:∞
2156:≤
2090:≤
1924:⋅
1849:∼
1732:∫
1627:∫
1567:≤
1434:≤
1413:
1326:≤
1305:
1239:≤
1221:
1168:≤
845:⋅
781:whenever
659:≤
615:over the
603:∞
7627:Springer
7497:See also
6702:and new
5437:that is
5027:, where
3884:′
3258:Output:
3108:, i.e.,
7974:2986138
7900:1390680
7783:2347565
7684:2005864
7600:1994729
7034:density
4907:, with
3170:Sample
2282:. When
617:support
77:density
75:with a
7972:
7933:
7898:
7880:
7853:
7835:
7781:
7744:
7721:
7682:
7674:
7633:
7606:
7598:
7553:
3955:where
2679:Buffon
1838:where
283:Theory
7970:JSTOR
7896:JSTOR
7779:JSTOR
7680:JSTOR
7028:have
2829:from
7931:ISSN
7851:ISSN
7742:ISBN
7719:ISSN
7672:ISSN
7631:ISBN
7551:ISBN
7436:(or
6478:, if
5038:>
4608:<
4330:<
4061:<
4026:and
2889:<
2681:and
2162:<
1754:>
1649:>
918:<
801:>
766:>
600:<
7997:doi
7962:doi
7923:doi
7888:doi
7843:doi
7806:doi
7771:doi
7711:doi
7664:doi
7604:Zbl
7586:doi
7543:doi
7174:log
7072:log
7015:or
6290:exp
6197:exp
6116:exp
5881:exp
5738:exp
4549:exp
4526:log
4427:log
4387:exp
4375:log
3994:exp
3978:log
3713:exp
3166:):
619:of
8033::
7993:52
7991:.
7968:.
7958:44
7929:.
7919:20
7917:.
7894:.
7886:.
7872:.
7849:.
7841:.
7829:21
7827:.
7800:.
7777:.
7767:41
7717:.
7707:12
7705:.
7701:.
7678:.
7670:.
7660:26
7658:.
7654:.
7629:.
7625:.
7602:.
7596:MR
7594:.
7582:31
7580:.
7574:.
7549:.
7324:is
7040:.
4857:,
3582:,
2933:.
1959:.
1558:Pr
1140:.
1096:.
816:.
279:.
79:.
34:,
8003:.
7999::
7976:.
7964::
7937:.
7925::
7902:.
7890::
7874:7
7857:.
7845::
7812:.
7808::
7802:1
7785:.
7773::
7750:.
7725:.
7713::
7686:.
7666::
7639:.
7610:.
7588::
7559:.
7545::
7461:)
7458:x
7455:(
7449:l
7445:h
7423:)
7420:x
7417:(
7411:l
7407:g
7381:)
7378:x
7375:(
7371:f
7347:)
7344:x
7341:(
7337:h
7309:)
7306:x
7303:(
7299:f
7260:)
7257:x
7254:(
7250:h
7229:)
7226:x
7223:(
7219:f
7190:)
7187:x
7184:(
7180:f
7153:)
7150:x
7147:(
7143:f
7119:)
7116:x
7113:(
7109:g
7088:)
7085:x
7082:(
7078:g
7069:=
7065:)
7062:x
7059:(
7055:h
6975:A
6969:X
6949:X
6921:)
6912:2
6904:2
6897:2
6893:)
6883:b
6880:(
6874:e
6867:b
6864:(
6861:O
6858:=
6852:)
6849:b
6843:X
6840:(
6836:P
6831:1
6809:)
6806:b
6803:(
6800:O
6797:=
6794:)
6791:b
6788:(
6785:M
6765:b
6742:)
6739:1
6736:,
6733:0
6730:(
6726:f
6723:i
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