1825:
1135:
4810:
7272:(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.
1820:{\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}}}
5976:
4480:
6409:
5569:
6076:
4805:{\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}}}
3940:
5971:{\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}}}
6404:{\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)}}}
92:(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
7477:
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
7473:
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
115:
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
7466:
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)
3657:
7016:
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
6996:. 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
6987:
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
5221:
4295:
6976:, 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.
2653:
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.
140:). 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.
4466:
6920:
1140:
2132:
6571:
6689:
112:‑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.
75:
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
6929:
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
4964:
4013:
4132:
3935:{\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}}}
5419:
5288:
2237:
7478:
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
5555:
5091:
3365:
4066:
7262:. 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.
4121:
5482:
193:
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
7482:. 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).
7265:
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.
7088:
4894:
5574:
4485:
3662:
3462:
237:
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
6465:
3621:
3416:
1877:
4332:
6741:
3243:
7190:
2862:
2598:
makes sampling difficult. A single iteration of the rejection algorithm requires sampling from the proposal distribution, drawing from a uniform distribution, and evaluating the
7461:
7423:
5080:
62:
7381:
7347:
7309:
7260:
7229:
7153:
7119:
7008:. (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.)
6746:
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
3569:
2164:
862:
2442:
955:
673:
5014:
3192:
3055:
2920:
2651:
602:
490:
5347:
5320:
2049:
6808:
6474:
5040:
4844:
4344:
2564:
1926:
803:
768:
576:
7349:
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.
6974:
6813:
2355:
6067:
3129:
1048:
987:
6029:
3650:
2596:
2500:
2471:
1123:
1079:
1016:
375:
326:
2001:
894:
6948:
6764:
6594:
6000:
3526:
3502:
3482:
3095:
3075:
3012:
2986:
2963:
2943:
2816:
2796:
2776:
2752:
2732:
2712:
2692:
2520:
2395:
2375:
2309:
2289:
2041:
2021:
1969:
1946:
1897:
733:
713:
693:
626:
530:
510:
435:
415:
395:
346:
294:
256:
231:
211:
188:
165:
134:
110:
2269:
3149:
2377:
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
7976:
Meyer, Renate; Cai, Bo; Perron, François (2008-03-15). "Adaptive rejection
Metropolis sampling using Lagrange interpolation polynomials of degree 2".
6599:
3508:(if it exists), also known as exponential tilting, provides a class of proposal distributions that can lower the computation complexity, the value of
3947:
4899:
2667:
7734:
7623:
7474:
random variable is straightforward. Just take the log of a uniform random variable (with appropriate interval and corresponding truncation).
3022:
Rejection sampling can be far more efficient compared with the naive methods in some situations. For example, given a problem as sampling
5358:
3464:, which could be close to infinity. Moreover, even when you apply the Rejection sampling method, it is always hard to optimize the bound
8030:
5216:{\textstyle \psi _{\theta }(\eta )=\psi (\theta +\eta )-\psi (\theta )=(\mu +\theta \sigma ^{2})\eta +{\frac {\sigma ^{2}\eta ^{2}}{2}}}
5233:
2169:
7543:
5487:
3250:
4018:
7615:
4071:
7018:
5429:
4290:{\displaystyle Z(x)={\frac {f(x)}{g_{\theta }(x)}}={\frac {f(x)}{e^{\theta x-\psi (\theta )}f(x)}}=e^{-\theta x+\psi (\theta )}}
7124:
Often, distributions that have algebraically messy density functions have reasonably simpler log density functions (i.e. when
7036:
If it helps, define your envelope distribution in log space (e.g. log-probability or log-density) instead. That is, work with
7501:
7039:
4849:
7197:
Instead of a single uniform envelope density function, use a piecewise linear density function as your envelope instead.
6984:
Rejection sampling requires knowing the target distribution (specifically, ability to evaluate target PDF at any point).
2043:
is the expected number of the iterations that are needed, as a measure of the computational complexity of the algorithm.
8025:
4335:
3505:
297:
89:
65:
7000:. In high dimensions, it is necessary to use a different approach, typically a Markov chain Monte Carlo method such as
3421:
7857:
Evans, M.; Swartz, T. (1998-12-01). "Random
Variable Generation Using Concavity Properties of Transformed Densities".
7491:
7280:
We can take even further advantage of the (log) concavity requirement, to potentially avoid the cost of evaluating
4124:
3152:
6418:
3574:
1830:
35:
or "accept-reject algorithm" and is a type of exact simulation method. The method works for any distribution in
7785:
Thomas, D. B.; Luk, W. (2007). "Non-uniform random number generation through piecewise linear approximations".
7275:
If not working in log space, a piecewise linear density function can also be sampled via triangle distributions
7269:
4302:
3378:
1093:
265:
28:
20:
4472:
It is easy to derive the cumulant-generation function of the proposal and therefore the proposal's cumulants.
2291:
is chosen closer to one, the unconditional acceptance probability is higher the less that ratio varies, since
6997:
6694:
3368:
3197:
1132:
The unconditional acceptance probability is the proportion of proposed samples which are accepted, which is
7866:
7821:
7158:
2821:
1972:
7428:
7390:
5049:
7318:
Just like we can construct a piecewise linear upper bound (the "envelope" function) using the values of
1052:
This means that, with enough replicates, the algorithm generates a sample from the desired distribution
605:
190:‑position, up to the maximum y-value of the probability density function of the proposal distribution.
38:
7565:
7355:
7321:
7283:
7234:
7203:
7127:
7093:
3623:
is the target distribution. Assume for simplicity, the density function can be explicitly written as
3539:
1126:
1082:
261:
2400:
2137:
811:
631:
7871:
7826:
7812:
Hörmann, Wolfgang (1995-06-01). "A Rejection
Technique for Sampling from T-concave Distributions".
7021:
density functions, which is in fact the case for most of the common distributions—even those whose
7001:
6926:
899:
4969:
4461:{\displaystyle \psi (\theta )=\log \mathbb {E} {\exp(tX)}|_{t=\theta }=\log M_{X}(t)|_{t=\theta }}
2601:
581:
440:
7958:
7884:
7767:
7668:
7506:
5325:
5298:
3162:
3025:
2870:
1096:
algorithms that also use a proxy distribution to achieve simulation from the target distribution
1089:
137:
6915:{\textstyle {\frac {1}{\mathbb {P} (X\geq b)}}=O(b\cdot e^{\frac {(b-\mu )^{2}}{2\sigma ^{2}}})}
2671:
6769:
5019:
4820:
2525:
1902:
773:
738:
535:
7941:; Tan, K. K. C. (1995-01-01). "Adaptive Rejection Metropolis Sampling within Gibbs Sampling".
7919:
7839:
7730:
7707:
7660:
7619:
7539:
7496:
6993:
7641:"Von Neumann's Comparison Method for Random Sampling from the Normal and Other Distributions"
7985:
7950:
7911:
7876:
7831:
7794:
7759:
7699:
7652:
7592:
7574:
7531:
6953:
2663:
7588:
6034:
2314:
1021:
960:
7596:
7584:
6005:
3626:
3100:
2572:
2476:
2447:
1099:
1055:
992:
351:
302:
72:
7032:
There are three basic ideas to this technique as ultimately introduced by Gilks in 1992:
3528:
and speed up the computations (see examples: working with
Natural Exponential Families).
1978:
7902:
Görür, Dilan; Teh, Yee Whye (2011-01-01). "Concave-Convex
Adaptive Rejection Sampling".
7727:
Essentials of Monte Carlo
Simulation: Statistical Methods for Building Simulation Models
7268:
A piecewise linear model of the proposal log distribution results in a set of piecewise
7005:
6933:
6749:
6579:
5985:
3511:
3487:
3467:
3080:
3060:
2997:
2971:
2948:
2928:
2801:
2781:
2761:
2737:
2717:
2697:
2677:
2505:
2380:
2360:
2294:
2274:
2026:
2006:
1954:
1931:
1882:
867:
718:
698:
678:
611:
515:
495:
420:
400:
380:
331:
279:
241:
216:
196:
173:
150:
119:
95:
2566:, i.e. that the target and proposal distributions are actually the same distribution.
2242:
8019:
7688:"Accounting for environmental change in continuous-time stochastic population models"
3504:
is large and the rejection rate is high, the algorithm can be very inefficient. The
3134:
2127:{\displaystyle M={\frac {1}{\mathbb {P} \left(U\leq {\frac {f(Y)}{Mg(Y)}}\right)}}}
276:
The rejection sampling method generates sampling values from a target distribution
6566:{\displaystyle U\leq {\frac {Z(x)}{M}}=e^{-\theta ^{*}(x-b)}\mathbb {I} (x\geq b)}
808:
The validation of this method is the envelope principle: when simulating the pair
7750:
Gilks, W. R.; Wild, P. (1992). "Adaptive
Rejection Sampling for Gibbs Sampling".
6684:{\textstyle X\sim _{i.i.d.}\mathrm {N} (\mu +\theta ^{*}\sigma ^{2},\sigma ^{2})}
7989:
7798:
7938:
7703:
7535:
7923:
7843:
7711:
7664:
7579:
7560:
7915:
7687:
88:
To visualize the motivation behind rejection sampling, imagine graphing the
4959:{\textstyle \psi (\theta )=\mu \theta +{\frac {\sigma ^{2}\theta ^{2}}{2}}}
4008:{\displaystyle \psi (\theta )=\log \left(\mathbb {E} \exp(\theta X)\right)}
7835:
5322:
for the proposal distribution. In this setup, the intuitive way to choose
6810:, while under the naive method, the expected number of the iterations is
7962:
7888:
7771:
7672:
7640:
260:. Thus, the algorithm can be used to sample from a distribution whose
5562:
Explicitly write out the target, the proposal and the likelihood ratio
7192:
may be easier to work with or, at least, closer to piecewise linear).
6992:), or with an appropriate change of variables with the method of the
7954:
7880:
7763:
7656:
5414:{\displaystyle \mathbb {E} _{\theta }(X)=\mu +\theta \sigma ^{2}=b}
7387:
if it will be accepted by comparing against the (ideally cheaper)
3375:
The problem is this sampling can be difficult and inefficient, if
1081:. There are a number of extensions to this algorithm, such as the
5283:{\displaystyle \mathrm {N} (\mu +\theta \sigma ^{2},\sigma ^{2})}
2569:
Rejection sampling is most often used in cases where the form of
2232:{\textstyle \mathbb {P} \left(U\leq {\frac {f(Y)}{Mg(Y)}}\right)}
7526:
Casella, George; Robert, Christian P.; Wells, Martin T. (2004).
7200:
Each time you have to reject a sample, you can use the value of
71:
Rejection sampling is based on the observation that to sample a
5550:{\displaystyle g_{\theta ^{*}}(x)=\mathrm {N} (b,\sigma ^{2})}
3360:{\displaystyle \{X_{1},X_{2},...,X_{N}:X_{i}\in A,i=1,...,N\}}
4061:{\displaystyle \Theta =\{\theta :\psi (\theta )<\infty \}}
116:
of some proposal distribution (not necessarily normalized to
4116:{\displaystyle \{F_{\theta }(\cdot )\}_{\theta \in \Theta }}
2239:
is a probability which can only take values in the interval
7231:
that you evaluated, to improve the piecewise approximation
4730:
4636:
7530:. Institute of Mathematical Statistics. pp. 342–347.
5477:{\displaystyle \theta ^{*}={\frac {b-\mu }{\sigma ^{2}}}.}
3151:
can be easily simulated, using the naive methods (e.g. by
27:
is a basic technique used to generate observations from a
864:, one produces a uniform simulation over the subgraph of
532:
here is a constant, finite bound on the likelihood ratio
377:. The idea is that one can generate a sample value from
7480:
Adaptive
Rejection Metropolis Sampling (ARMS) algorithms
7467:
this can make a sizable difference in ultimate runtime.
7729:(2013th ed.). New York, NY Heidelberg: Springer.
6816:
6697:
6602:
5094:
4902:
3381:
3165:
3137:
3103:
3028:
2873:
2317:
2172:
2140:
902:
870:
814:
136:) that we know how to sample from (for example, using
7686:
Legault, Geoffrey; Melbourne, Brett A. (2019-03-01).
7463:
in this case) squeezing function that have available.
7431:
7393:
7358:
7324:
7286:
7237:
7206:
7161:
7130:
7096:
7042:
6956:
6936:
6772:
6752:
6582:
6477:
6421:
6079:
6037:
6008:
5988:
5572:
5490:
5432:
5361:
5328:
5301:
5236:
5052:
5022:
4972:
4852:
4823:
4483:
4347:
4305:
4135:
4074:
4021:
3950:
3660:
3629:
3577:
3542:
3514:
3490:
3470:
3424:
3253:
3200:
3083:
3063:
3000:
2974:
2951:
2931:
2824:
2804:
2784:
2764:
2740:
2720:
2700:
2680:
2604:
2575:
2528:
2508:
2479:
2450:
2403:
2383:
2363:
2297:
2277:
2245:
2052:
2029:
2009:
1981:
1957:
1934:
1905:
1885:
1833:
1138:
1102:
1058:
1024:
995:
963:
776:
741:
721:
701:
681:
634:
614:
584:
538:
518:
498:
443:
423:
403:
383:
354:
334:
305:
282:
244:
219:
199:
176:
153:
122:
98:
41:
7610:
Bishop, Christopher (2006). "11.4: Slice sampling".
7083:{\displaystyle h\left(x\right)=\log g\left(x\right)}
4889:{\displaystyle X\sim \mathrm {N} (\mu ,\sigma ^{2})}
7455:
7417:
7375:
7341:
7303:
7254:
7223:
7184:
7147:
7113:
7082:
7025:functions are not concave themselves) is known as
6968:
6942:
6914:
6802:
6758:
6735:
6683:
6588:
6565:
6459:
6403:
6061:
6023:
5994:
5970:
5549:
5476:
5413:
5341:
5314:
5282:
5230:which further implies it is a normal distribution
5215:
5074:
5034:
5008:
4958:
4888:
4838:
4804:
4460:
4326:
4289:
4115:
4060:
4007:
3934:
3644:
3615:
3563:
3520:
3496:
3476:
3456:
3410:
3359:
3237:
3186:
3143:
3123:
3089:
3069:
3049:
3006:
2980:
2957:
2937:
2914:
2864:(the uniform distribution over the unit interval).
2856:
2810:
2790:
2770:
2746:
2726:
2706:
2686:
2645:
2590:
2558:
2514:
2494:
2465:
2436:
2389:
2369:
2349:
2303:
2283:
2263:
2231:
2158:
2126:
2035:
2015:
1995:
1963:
1940:
1920:
1899:each time is generated under the density function
1891:
1871:
1819:
1117:
1073:
1042:
1010:
981:
949:
888:
856:
797:
762:
727:
707:
687:
667:
620:
596:
570:
524:
504:
484:
429:
409:
389:
369:
340:
320:
288:
250:
225:
205:
182:
159:
128:
104:
56:
7904:Journal of Computational and Graphical Statistics
7859:Journal of Computational and Graphical Statistics
3457:{\displaystyle {\frac {1}{\mathbb {P} (X\in A)}}}
1462:
233:‑value is a sample from the desired distribution.
6989:
1546:
1125:. It forms the basis for algorithms such as the
7383:to see if your sample will be accepted, we may
3484:for the likelihood ratio. More often than not,
695:. Note that this requires that the support of
7352:Before evaluating (the potentially expensive)
8011:(Second ed.). New York: Springer-Verlag.
5046:Choose the form of the proposal distribution
3418:. The expected number of iterations would be
2397:as small as possible (while still satisfying
8:
7978:Computational Statistics & Data Analysis
4098:
4075:
4055:
4028:
3532:Rejection sampling using exponential tilting
3354:
3254:
3232:
3201:
3018:Advantages over sampling using naive methods
2522:cannot be equal to 1: such would imply that
2311:is the upper bound for the likelihood ratio
1088:This method relates to the general field of
3194:independently, and accept those satisfying
1971:to obtain an accepted value thus follows a
989:uniformly distributed over the subgraph of
7528:Generalized Accept-Reject sampling schemes
6460:{\displaystyle U\sim \mathrm {Unif} (0,1)}
3616:{\displaystyle F(x)=\mathbb {P} (X\leq x)}
3411:{\textstyle \mathbb {P} (X\in A)\approx 0}
1872:{\displaystyle U\sim \mathrm {Unif} (0,1)}
7870:
7825:
7578:
7436:
7430:
7398:
7392:
7357:
7323:
7285:
7236:
7205:
7160:
7129:
7095:
7041:
6955:
6935:
6899:
6884:
6865:
6824:
6823:
6817:
6815:
6771:
6751:
6704:
6696:
6672:
6659:
6649:
6631:
6610:
6601:
6581:
6544:
6543:
6520:
6512:
6484:
6476:
6428:
6420:
6375:
6352:
6343:
6342:
6326:
6311:
6292:
6275:
6250:
6249:
6233:
6218:
6199:
6182:
6157:
6156:
6142:
6120:
6101:
6078:
6036:
6007:
5987:
5942:
5941:
5920:
5919:
5907:
5885:
5866:
5843:
5838:
5807:
5803:
5796:
5761:
5739:
5690:
5685:
5655:
5654:
5633:
5632:
5617:
5585:
5581:
5573:
5571:
5538:
5520:
5500:
5495:
5489:
5463:
5446:
5437:
5431:
5399:
5368:
5364:
5363:
5360:
5333:
5327:
5306:
5300:
5271:
5258:
5237:
5235:
5201:
5191:
5184:
5169:
5099:
5093:
5057:
5051:
5021:
4976:
4971:
4944:
4934:
4927:
4901:
4877:
4859:
4851:
4822:
4786:
4770:
4749:
4739:
4732:
4706:
4695:
4678:
4648:
4638:
4612:
4608:
4607:
4531:
4527:
4526:
4492:
4484:
4482:
4446:
4441:
4425:
4400:
4395:
4374:
4370:
4369:
4346:
4327:{\displaystyle \psi (\theta )<\infty }
4304:
4260:
4214:
4193:
4172:
4151:
4134:
4101:
4082:
4073:
4020:
3978:
3977:
3949:
3892:
3867:
3841:
3791:
3781:
3773:
3735:
3734:
3692:
3691:
3669:
3661:
3659:
3628:
3594:
3593:
3576:
3541:
3513:
3489:
3469:
3432:
3431:
3425:
3423:
3383:
3382:
3380:
3312:
3299:
3274:
3261:
3252:
3220:
3199:
3164:
3136:
3107:
3102:
3082:
3062:
3027:
2999:
2973:
2950:
2930:
2892:
2872:
2825:
2823:
2803:
2783:
2763:
2739:
2719:
2699:
2679:
2617:
2603:
2574:
2527:
2507:
2478:
2449:
2402:
2382:
2362:
2330:
2316:
2296:
2276:
2244:
2189:
2174:
2173:
2171:
2139:
2081:
2066:
2065:
2059:
2051:
2028:
2008:
1985:
1980:
1956:
1933:
1904:
1884:
1840:
1832:
1802:
1794:
1779:
1762:
1723:
1709:
1692:
1645:
1618:
1581:
1573:
1541:
1497:
1461:
1460:
1425:
1410:
1409:
1380:
1359:
1317:
1305:
1300:
1230:
1218:
1213:
1159:
1144:
1143:
1139:
1137:
1101:
1057:
1023:
994:
962:
921:
901:
869:
813:
775:
740:
720:
700:
680:
633:
613:
583:
551:
537:
517:
497:
456:
442:
422:
402:
382:
353:
333:
304:
281:
243:
218:
198:
175:
152:
121:
97:
48:
44:
43:
40:
7943:Journal of the Royal Statistical Society
7752:Journal of the Royal Statistical Society
7612:Pattern Recognition and Machine Learning
1018:and thus, marginally, a simulation from
7518:
6736:{\textstyle U\sim \mathrm {Unif} (0,1)}
7787:IET Computers & Digital Techniques
5082:, with cumulant-generating function as
3238:{\displaystyle \{n\geq 1:X_{n}\in A\}}
2994:The algorithm will take an average of
213:‑value and return to step 1; else the
6031:, which is a decreasing function for
2674:, obtains a sample from distribution
167:‑axis from the proposal distribution.
143:Rejection sampling works as follows:
7:
7185:{\displaystyle \log f\left(x\right)}
4127:. Moreover, the likelihood ratio is
2857:{\displaystyle \mathrm {Unif} (0,1)}
1951:The number of samples required from
8007:Robert, C. P.; Casella, G. (2004).
7725:Thomopoulos, Nick T. (2012-12-19).
7456:{\displaystyle h_{l}\left(x\right)}
7418:{\displaystyle g_{l}\left(x\right)}
5075:{\displaystyle F_{\theta }(\cdot )}
4817:As a simple example, suppose under
6714:
6711:
6708:
6705:
6632:
6438:
6435:
6432:
6429:
6415:Rejection sampling criterion: for
6353:
6053:
5521:
5484:The proposal distribution is thus
5238:
4998:
4860:
4767:
4736:
4702:
4699:
4696:
4665:
4641:
4599:
4321:
4108:
4052:
4022:
3777:
2835:
2832:
2829:
2826:
2502:in some way. Note, however, that
2166:, due to the above formula, where
2153:
1850:
1847:
1844:
1841:
1398:
1290:
1206:
591:
16:Computational statistics technique
14:
7945:. Series C (Applied Statistics).
7754:. Series C (Applied Statistics).
7027:adaptive rejection sampling (ARS)
6922:, which is far more inefficient.
2662:The algorithm, which was used by
896:. Accepting only pairs such that
628:; in other words, M must satisfy
328:by using a proposal distribution
31:. It is also commonly called the
6596:; if not, continue sampling new
5042:. The analysis goes as follows:
2988:and return to the sampling step.
2714:using samples from distribution
1301:
1214:
57:{\displaystyle \mathbb {R} ^{m}}
8009:Monte Carlo Statistical Methods
7376:{\displaystyle f\left(x\right)}
7342:{\displaystyle h\left(x\right)}
7304:{\displaystyle f\left(x\right)}
7255:{\displaystyle h\left(x\right)}
7224:{\displaystyle f\left(x\right)}
7148:{\displaystyle f\left(x\right)}
7114:{\displaystyle g\left(x\right)}
3564:{\displaystyle X\sim F(\cdot )}
3014:iterations to obtain a sample.
2159:{\textstyle 1\leq M<\infty }
1804: includes support of
857:{\textstyle (x,v=u\cdot Mg(x))}
264:is unknown, which is common in
6909:
6881:
6868:
6852:
6840:
6828:
6797:
6791:
6782:
6776:
6730:
6718:
6678:
6636:
6560:
6548:
6538:
6526:
6496:
6490:
6454:
6442:
6369:
6357:
6308:
6295:
6266:
6254:
6215:
6202:
6173:
6161:
6151:
6148:
6135:
6110:
6095:
6089:
6056:
6044:
6018:
6012:
5958:
5946:
5936:
5924:
5916:
5913:
5900:
5875:
5857:
5851:
5829:
5823:
5808:
5786:
5780:
5770:
5767:
5754:
5732:
5723:
5717:
5704:
5698:
5671:
5659:
5649:
5637:
5629:
5623:
5607:
5601:
5586:
5544:
5525:
5514:
5508:
5380:
5374:
5277:
5242:
5175:
5153:
5147:
5141:
5132:
5120:
5111:
5105:
5069:
5063:
4977:
4912:
4906:
4883:
4864:
4833:
4827:
4761:
4755:
4718:
4712:
4660:
4654:
4624:
4618:
4593:
4587:
4578:
4566:
4552:
4543:
4504:
4498:
4442:
4437:
4431:
4396:
4390:
4381:
4357:
4351:
4315:
4309:
4282:
4276:
4247:
4241:
4233:
4227:
4205:
4199:
4184:
4178:
4163:
4157:
4145:
4139:
4094:
4088:
4046:
4040:
3997:
3988:
3960:
3954:
3925:
3919:
3911:
3905:
3882:
3876:
3853:
3847:
3824:
3818:
3810:
3804:
3751:
3739:
3731:
3728:
3722:
3707:
3681:
3675:
3639:
3633:
3610:
3598:
3587:
3581:
3558:
3552:
3448:
3436:
3399:
3387:
3181:
3175:
3108:
3044:
3038:
2909:
2903:
2889:
2883:
2851:
2839:
2640:
2637:
2631:
2622:
2614:
2608:
2585:
2579:
2553:
2547:
2538:
2532:
2489:
2483:
2460:
2454:
2437:{\displaystyle f(x)\leq Mg(x)}
2431:
2425:
2413:
2407:
2344:
2338:
2327:
2321:
2258:
2246:
2218:
2212:
2201:
2195:
2110:
2104:
2093:
2087:
1915:
1909:
1866:
1854:
1810:
1791:
1759:
1753:
1739:
1733:
1689:
1683:
1674:
1668:
1657:
1651:
1634:
1628:
1601:
1598:
1586:
1561:
1549:
1538:
1526:
1520:
1509:
1503:
1454:
1448:
1437:
1431:
1385:
1377:
1367:
1360:
1346:
1340:
1329:
1323:
1296:
1259:
1253:
1242:
1236:
1188:
1182:
1171:
1165:
1112:
1106:
1068:
1062:
1034:
1028:
1005:
999:
976:
964:
950:{\textstyle u<f(x)/(Mg(x))}
944:
941:
935:
926:
918:
912:
883:
877:
851:
848:
842:
815:
786:
780:
751:
745:
668:{\displaystyle f(x)\leq Mg(x)}
662:
656:
644:
638:
565:
559:
548:
542:
479:
476:
470:
461:
453:
447:
417:and accepting the sample from
364:
358:
315:
309:
1:
7502:Pseudo-random number sampling
1928:of the proposal distribution
170:Draw a vertical line at this
7639:Forsythe, George E. (1972).
5009:{\displaystyle X|X\in \left}
4336:cumulant-generation function
4334:implies that it is indeed a
3187:{\textstyle X\sim F(\cdot )}
3050:{\textstyle X\sim F(\cdot )}
2968:if not, reject the value of
2915:{\textstyle u<f(y)/Mg(y)}
2646:{\displaystyle f(x)/(Mg(x))}
2046:Rewrite the above equation,
715:must include the support of
597:{\displaystyle M<\infty }
512:until a value is accepted.
485:{\displaystyle f(x)/(Mg(x))}
298:probability density function
90:probability density function
7012:Adaptive rejection sampling
6990:adaptive rejection sampling
6576:holds, accept the value of
5342:{\displaystyle \theta ^{*}}
5315:{\displaystyle \theta ^{*}}
2357:. In practice, a value of
492:, repeating the draws from
33:acceptance-rejection method
8047:
8031:Non-uniform random numbers
7990:10.1016/j.csda.2008.01.005
7645:Mathematics of Computation
7492:Inverse transform sampling
4125:natural exponential family
3506:Natural Exponential Family
3153:inverse transform sampling
2473:should generally resemble
19:In numerical analysis and
7704:10.1007/s12080-018-0386-z
7559:Neal, Radford M. (2003).
7270:exponential distributions
6803:{\displaystyle M(b)=O(b)}
6002:for the likelihood ratio
5035:{\displaystyle b>\mu }
4839:{\displaystyle F(\cdot )}
3652:. Choose the proposal as
2559:{\displaystyle f(x)=g(x)}
1921:{\displaystyle g(\cdot )}
1583: is uniform on
798:{\displaystyle f(x)>0}
763:{\displaystyle g(x)>0}
571:{\displaystyle f(x)/g(x)}
397:by instead sampling from
348:with probability density
7799:10.1049/iet-cdt:20060188
4966:. The goal is to sample
3536:Given a random variable
1094:Markov chain Monte Carlo
266:computational statistics
21:computational statistics
7916:10.1198/jcgs.2011.09058
7536:10.1214/lnms/1196285403
6998:curse of dimensionality
5295:Decide the well chosen
3369:truncation (statistics)
2945:as a sample drawn from
1382:by tower property
7814:ACM Trans. Math. Softw
7580:10.1214/aos/1056562461
7457:
7419:
7377:
7343:
7305:
7256:
7225:
7186:
7149:
7115:
7084:
6970:
6969:{\displaystyle X\in A}
6944:
6916:
6804:
6760:
6737:
6685:
6590:
6567:
6461:
6405:
6063:
6025:
5996:
5972:
5551:
5478:
5415:
5343:
5316:
5284:
5217:
5076:
5036:
5010:
4960:
4890:
4840:
4806:
4462:
4328:
4291:
4117:
4062:
4009:
3936:
3646:
3617:
3565:
3522:
3498:
3478:
3458:
3412:
3361:
3239:
3188:
3145:
3125:
3091:
3071:
3051:
3008:
2982:
2959:
2939:
2925:If this holds, accept
2916:
2858:
2812:
2792:
2772:
2748:
2728:
2708:
2688:
2647:
2592:
2560:
2516:
2496:
2467:
2444:, which suggests that
2438:
2391:
2371:
2351:
2350:{\textstyle f(x)/g(x)}
2305:
2285:
2265:
2233:
2160:
2128:
2037:
2017:
1997:
1973:geometric distribution
1965:
1942:
1922:
1893:
1873:
1821:
1796:since support of
1119:
1092:techniques, including
1075:
1044:
1012:
983:
951:
890:
858:
799:
764:
729:
709:
689:
669:
622:
598:
572:
526:
506:
486:
431:
411:
391:
371:
342:
322:
290:
252:
227:
207:
184:
161:
147:Sample a point on the
130:
106:
80:-dimension functions.
58:
7836:10.1145/203082.203089
7458:
7420:
7378:
7344:
7306:
7257:
7226:
7187:
7150:
7116:
7085:
6971:
6945:
6917:
6805:
6761:
6738:
6686:
6591:
6568:
6462:
6406:
6064:
6062:{\displaystyle x\in }
6026:
5997:
5973:
5552:
5479:
5416:
5344:
5317:
5285:
5218:
5077:
5037:
5011:
4961:
4891:
4841:
4807:
4463:
4329:
4292:
4118:
4063:
4010:
3937:
3647:
3618:
3566:
3523:
3499:
3479:
3459:
3413:
3362:
3240:
3189:
3146:
3126:
3124:{\textstyle X|X\in A}
3092:
3072:
3052:
3009:
2983:
2960:
2940:
2917:
2859:
2813:
2793:
2773:
2749:
2729:
2709:
2689:
2648:
2593:
2561:
2517:
2497:
2468:
2439:
2392:
2372:
2352:
2306:
2286:
2266:
2234:
2161:
2129:
2038:
2018:
1998:
1966:
1943:
1923:
1894:
1874:
1822:
1120:
1076:
1045:
1043:{\displaystyle f(x).}
1013:
984:
982:{\displaystyle (x,v)}
952:
891:
859:
800:
765:
730:
710:
690:
670:
623:
599:
573:
527:
507:
487:
432:
412:
392:
372:
343:
323:
291:
253:
228:
208:
185:
162:
131:
107:
59:
7566:Annals of Statistics
7429:
7391:
7356:
7322:
7284:
7235:
7204:
7159:
7128:
7094:
7040:
6954:
6934:
6814:
6770:
6750:
6695:
6600:
6580:
6475:
6419:
6077:
6035:
6024:{\displaystyle Z(x)}
6006:
5986:
5570:
5488:
5430:
5359:
5326:
5299:
5234:
5092:
5050:
5020:
4970:
4900:
4850:
4821:
4481:
4345:
4303:
4133:
4072:
4019:
3948:
3658:
3645:{\displaystyle f(x)}
3627:
3575:
3540:
3512:
3488:
3468:
3422:
3379:
3251:
3198:
3163:
3135:
3101:
3081:
3061:
3026:
2998:
2972:
2949:
2929:
2871:
2822:
2802:
2782:
2762:
2738:
2718:
2698:
2678:
2602:
2591:{\displaystyle f(x)}
2573:
2526:
2506:
2495:{\displaystyle f(x)}
2477:
2466:{\displaystyle g(x)}
2448:
2401:
2381:
2361:
2315:
2295:
2275:
2243:
2170:
2138:
2050:
2027:
2007:
1979:
1955:
1932:
1903:
1883:
1831:
1136:
1127:Metropolis algorithm
1118:{\displaystyle f(x)}
1100:
1083:Metropolis algorithm
1074:{\displaystyle f(x)}
1056:
1022:
1011:{\displaystyle f(x)}
993:
961:
957:then produces pairs
900:
868:
812:
774:
739:
719:
699:
679:
632:
612:
582:
536:
516:
496:
441:
421:
401:
381:
370:{\displaystyle g(x)}
352:
332:
321:{\displaystyle f(x)}
303:
280:
262:normalizing constant
242:
217:
197:
174:
151:
120:
96:
39:
8026:Monte Carlo methods
7692:Theoretical Ecology
7002:Metropolis sampling
6927:exponential tilting
3875:
3786:
1996:{\displaystyle 1/M}
1879:, and the value of
7507:Ziggurat algorithm
7453:
7415:
7373:
7339:
7301:
7252:
7221:
7182:
7145:
7111:
7080:
6966:
6940:
6912:
6800:
6756:
6743:until acceptance.
6733:
6681:
6586:
6563:
6457:
6401:
6059:
6021:
5992:
5968:
5966:
5547:
5474:
5411:
5339:
5312:
5280:
5213:
5072:
5032:
5006:
4956:
4886:
4836:
4802:
4800:
4458:
4324:
4287:
4113:
4058:
4005:
3932:
3930:
3863:
3769:
3642:
3613:
3561:
3518:
3494:
3474:
3454:
3408:
3357:
3235:
3184:
3141:
3121:
3087:
3067:
3047:
3004:
2978:
2955:
2935:
2912:
2854:
2808:
2788:
2778:from distribution
2768:
2744:
2724:
2704:
2684:
2666:and dates back to
2643:
2588:
2556:
2512:
2492:
2463:
2434:
2387:
2367:
2347:
2301:
2281:
2261:
2229:
2156:
2124:
2033:
2013:
1993:
1961:
1938:
1918:
1889:
1869:
1817:
1815:
1749:
1644:
1115:
1071:
1040:
1008:
979:
947:
889:{\textstyle Mg(x)}
886:
854:
795:
760:
725:
705:
685:
675:for all values of
665:
618:
594:
568:
522:
502:
482:
427:
407:
387:
367:
338:
318:
296:with an arbitrary
286:
248:
223:
203:
180:
157:
138:inversion sampling
126:
102:
54:
25:rejection sampling
7736:978-1-4614-6021-3
7625:978-0-387-31073-2
7497:Ratio of uniforms
7311:when your sample
6994:ratio of uniforms
6950:conditionally on
6943:{\displaystyle X}
6906:
6844:
6759:{\displaystyle b}
6589:{\displaystyle X}
6503:
6399:
6391:
6333:
6270:
6240:
6177:
5995:{\displaystyle M}
5982:Derive the bound
5962:
5861:
5675:
5469:
5211:
4954:
4780:
4672:
4251:
4188:
3521:{\displaystyle M}
3497:{\displaystyle M}
3477:{\displaystyle M}
3452:
3090:{\displaystyle A}
3070:{\displaystyle X}
3057:conditionally on
3007:{\displaystyle M}
2981:{\displaystyle y}
2958:{\displaystyle f}
2938:{\displaystyle y}
2811:{\displaystyle u}
2791:{\displaystyle Y}
2771:{\displaystyle y}
2747:{\displaystyle g}
2727:{\displaystyle Y}
2707:{\displaystyle f}
2687:{\displaystyle X}
2515:{\displaystyle M}
2390:{\displaystyle M}
2370:{\displaystyle M}
2304:{\displaystyle M}
2284:{\displaystyle M}
2222:
2122:
2114:
2036:{\displaystyle M}
2016:{\displaystyle M}
2003:, which has mean
1975:with probability
1964:{\displaystyle Y}
1941:{\displaystyle Y}
1892:{\displaystyle y}
1805:
1797:
1787:
1719:
1717:
1678:
1614:
1584:
1576:
1544:
1530:
1458:
1383:
1350:
1263:
1192:
735:—in other words,
728:{\displaystyle X}
708:{\displaystyle Y}
688:{\displaystyle x}
621:{\displaystyle X}
525:{\displaystyle M}
505:{\displaystyle Y}
437:with probability
430:{\displaystyle Y}
410:{\displaystyle Y}
390:{\displaystyle X}
341:{\displaystyle Y}
289:{\displaystyle X}
251:{\displaystyle x}
226:{\displaystyle x}
206:{\displaystyle x}
183:{\displaystyle x}
160:{\displaystyle x}
129:{\displaystyle 1}
105:{\displaystyle x}
8038:
8012:
7994:
7993:
7984:(7): 3408–3423.
7973:
7967:
7966:
7934:
7928:
7927:
7899:
7893:
7892:
7874:
7854:
7848:
7847:
7829:
7809:
7803:
7802:
7782:
7776:
7775:
7747:
7741:
7740:
7722:
7716:
7715:
7683:
7677:
7676:
7651:(120): 817–826.
7636:
7630:
7629:
7607:
7601:
7600:
7582:
7561:"Slice Sampling"
7556:
7550:
7549:
7523:
7462:
7460:
7459:
7454:
7452:
7441:
7440:
7424:
7422:
7421:
7416:
7414:
7403:
7402:
7382:
7380:
7379:
7374:
7372:
7348:
7346:
7345:
7340:
7338:
7310:
7308:
7307:
7302:
7300:
7261:
7259:
7258:
7253:
7251:
7230:
7228:
7227:
7222:
7220:
7191:
7189:
7188:
7183:
7181:
7154:
7152:
7151:
7146:
7144:
7120:
7118:
7117:
7112:
7110:
7089:
7087:
7086:
7081:
7079:
7056:
6975:
6973:
6972:
6967:
6949:
6947:
6946:
6941:
6921:
6919:
6918:
6913:
6908:
6907:
6905:
6904:
6903:
6890:
6889:
6888:
6866:
6845:
6843:
6827:
6818:
6809:
6807:
6806:
6801:
6765:
6763:
6762:
6757:
6742:
6740:
6739:
6734:
6717:
6690:
6688:
6687:
6682:
6677:
6676:
6664:
6663:
6654:
6653:
6635:
6630:
6629:
6595:
6593:
6592:
6587:
6572:
6570:
6569:
6564:
6547:
6542:
6541:
6525:
6524:
6504:
6499:
6485:
6466:
6464:
6463:
6458:
6441:
6410:
6408:
6407:
6402:
6400:
6398:
6397:
6393:
6392:
6387:
6376:
6356:
6346:
6340:
6339:
6335:
6334:
6332:
6331:
6330:
6317:
6316:
6315:
6293:
6276:
6271:
6269:
6253:
6247:
6246:
6242:
6241:
6239:
6238:
6237:
6224:
6223:
6222:
6200:
6183:
6178:
6176:
6160:
6154:
6147:
6146:
6125:
6124:
6102:
6068:
6066:
6065:
6060:
6030:
6028:
6027:
6022:
6001:
5999:
5998:
5993:
5977:
5975:
5974:
5969:
5967:
5963:
5961:
5945:
5939:
5923:
5912:
5911:
5890:
5889:
5867:
5862:
5860:
5850:
5849:
5848:
5847:
5832:
5822:
5821:
5811:
5797:
5766:
5765:
5744:
5743:
5697:
5696:
5695:
5694:
5676:
5674:
5658:
5652:
5636:
5618:
5600:
5599:
5589:
5556:
5554:
5553:
5548:
5543:
5542:
5524:
5507:
5506:
5505:
5504:
5483:
5481:
5480:
5475:
5470:
5468:
5467:
5458:
5447:
5442:
5441:
5420:
5418:
5417:
5412:
5404:
5403:
5373:
5372:
5367:
5348:
5346:
5345:
5340:
5338:
5337:
5321:
5319:
5318:
5313:
5311:
5310:
5289:
5287:
5286:
5281:
5276:
5275:
5263:
5262:
5241:
5222:
5220:
5219:
5214:
5212:
5207:
5206:
5205:
5196:
5195:
5185:
5174:
5173:
5104:
5103:
5081:
5079:
5078:
5073:
5062:
5061:
5041:
5039:
5038:
5033:
5015:
5013:
5012:
5007:
5005:
5001:
4980:
4965:
4963:
4962:
4957:
4955:
4950:
4949:
4948:
4939:
4938:
4928:
4895:
4893:
4892:
4887:
4882:
4881:
4863:
4845:
4843:
4842:
4837:
4811:
4809:
4808:
4803:
4801:
4797:
4796:
4785:
4781:
4779:
4775:
4774:
4764:
4754:
4753:
4744:
4743:
4733:
4711:
4710:
4705:
4689:
4688:
4677:
4673:
4671:
4663:
4653:
4652:
4639:
4617:
4616:
4611:
4559:
4555:
4536:
4535:
4530:
4497:
4496:
4467:
4465:
4464:
4459:
4457:
4456:
4445:
4430:
4429:
4411:
4410:
4399:
4393:
4373:
4333:
4331:
4330:
4325:
4296:
4294:
4293:
4288:
4286:
4285:
4252:
4250:
4237:
4236:
4208:
4194:
4189:
4187:
4177:
4176:
4166:
4152:
4122:
4120:
4119:
4114:
4112:
4111:
4087:
4086:
4067:
4065:
4064:
4059:
4014:
4012:
4011:
4006:
4004:
4000:
3981:
3941:
3939:
3938:
3933:
3931:
3915:
3914:
3871:
3846:
3845:
3814:
3813:
3785:
3780:
3762:
3758:
3754:
3738:
3695:
3674:
3673:
3651:
3649:
3648:
3643:
3622:
3620:
3619:
3614:
3597:
3570:
3568:
3567:
3562:
3527:
3525:
3524:
3519:
3503:
3501:
3500:
3495:
3483:
3481:
3480:
3475:
3463:
3461:
3460:
3455:
3453:
3451:
3435:
3426:
3417:
3415:
3414:
3409:
3386:
3366:
3364:
3363:
3358:
3317:
3316:
3304:
3303:
3279:
3278:
3266:
3265:
3244:
3242:
3241:
3236:
3225:
3224:
3193:
3191:
3190:
3185:
3150:
3148:
3147:
3142:
3130:
3128:
3127:
3122:
3111:
3096:
3094:
3093:
3088:
3076:
3074:
3073:
3068:
3056:
3054:
3053:
3048:
3013:
3011:
3010:
3005:
2987:
2985:
2984:
2979:
2964:
2962:
2961:
2956:
2944:
2942:
2941:
2936:
2921:
2919:
2918:
2913:
2896:
2863:
2861:
2860:
2855:
2838:
2817:
2815:
2814:
2809:
2797:
2795:
2794:
2789:
2777:
2775:
2774:
2769:
2758:Obtain a sample
2753:
2751:
2750:
2745:
2733:
2731:
2730:
2725:
2713:
2711:
2710:
2705:
2693:
2691:
2690:
2685:
2664:John von Neumann
2652:
2650:
2649:
2644:
2621:
2597:
2595:
2594:
2589:
2565:
2563:
2562:
2557:
2521:
2519:
2518:
2513:
2501:
2499:
2498:
2493:
2472:
2470:
2469:
2464:
2443:
2441:
2440:
2435:
2396:
2394:
2393:
2388:
2376:
2374:
2373:
2368:
2356:
2354:
2353:
2348:
2334:
2310:
2308:
2307:
2302:
2290:
2288:
2287:
2282:
2270:
2268:
2267:
2264:{\displaystyle }
2262:
2238:
2236:
2235:
2230:
2228:
2224:
2223:
2221:
2204:
2190:
2177:
2165:
2163:
2162:
2157:
2133:
2131:
2130:
2125:
2123:
2121:
2120:
2116:
2115:
2113:
2096:
2082:
2069:
2060:
2042:
2040:
2039:
2034:
2023:. Intuitively,
2022:
2020:
2019:
2014:
2002:
2000:
1999:
1994:
1989:
1970:
1968:
1967:
1962:
1947:
1945:
1944:
1939:
1927:
1925:
1924:
1919:
1898:
1896:
1895:
1890:
1878:
1876:
1875:
1870:
1853:
1826:
1824:
1823:
1818:
1816:
1806:
1803:
1798:
1795:
1788:
1780:
1772:
1748:
1718:
1710:
1702:
1679:
1677:
1660:
1646:
1643:
1607:
1585:
1582:
1577:
1574:
1545:
1542:
1535:
1531:
1529:
1512:
1498:
1483:
1479:
1475:
1474:
1470:
1466:
1465:
1459:
1457:
1440:
1426:
1413:
1391:
1384:
1381:
1374:
1370:
1363:
1358:
1357:
1356:
1352:
1351:
1349:
1332:
1318:
1304:
1275:
1271:
1270:
1269:
1265:
1264:
1262:
1245:
1231:
1217:
1198:
1194:
1193:
1191:
1174:
1160:
1147:
1124:
1122:
1121:
1116:
1080:
1078:
1077:
1072:
1049:
1047:
1046:
1041:
1017:
1015:
1014:
1009:
988:
986:
985:
980:
956:
954:
953:
948:
925:
895:
893:
892:
887:
863:
861:
860:
855:
804:
802:
801:
796:
769:
767:
766:
761:
734:
732:
731:
726:
714:
712:
711:
706:
694:
692:
691:
686:
674:
672:
671:
666:
627:
625:
624:
619:
603:
601:
600:
595:
577:
575:
574:
569:
555:
531:
529:
528:
523:
511:
509:
508:
503:
491:
489:
488:
483:
460:
436:
434:
433:
428:
416:
414:
413:
408:
396:
394:
393:
388:
376:
374:
373:
368:
347:
345:
344:
339:
327:
325:
324:
319:
295:
293:
292:
287:
259:
257:
255:
254:
249:
232:
230:
229:
224:
212:
210:
209:
204:
189:
187:
186:
181:
166:
164:
163:
158:
135:
133:
132:
127:
111:
109:
108:
103:
63:
61:
60:
55:
53:
52:
47:
8046:
8045:
8041:
8040:
8039:
8037:
8036:
8035:
8016:
8015:
8006:
8003:
8001:Further reading
7998:
7997:
7975:
7974:
7970:
7955:10.2307/2986138
7936:
7935:
7931:
7901:
7900:
7896:
7881:10.2307/1390680
7856:
7855:
7851:
7811:
7810:
7806:
7784:
7783:
7779:
7764:10.2307/2347565
7749:
7748:
7744:
7737:
7724:
7723:
7719:
7685:
7684:
7680:
7657:10.2307/2005864
7638:
7637:
7633:
7626:
7609:
7608:
7604:
7558:
7557:
7553:
7546:
7525:
7524:
7520:
7515:
7488:
7442:
7432:
7427:
7426:
7404:
7394:
7389:
7388:
7362:
7354:
7353:
7328:
7320:
7319:
7290:
7282:
7281:
7241:
7233:
7232:
7210:
7202:
7201:
7171:
7157:
7156:
7134:
7126:
7125:
7100:
7092:
7091:
7069:
7046:
7038:
7037:
7014:
6982:
6952:
6951:
6932:
6931:
6895:
6891:
6880:
6867:
6861:
6822:
6812:
6811:
6768:
6767:
6748:
6747:
6693:
6692:
6668:
6655:
6645:
6606:
6598:
6597:
6578:
6577:
6516:
6508:
6486:
6473:
6472:
6417:
6416:
6377:
6351:
6347:
6341:
6322:
6318:
6307:
6294:
6288:
6284:
6277:
6248:
6229:
6225:
6214:
6201:
6195:
6191:
6184:
6155:
6138:
6116:
6103:
6075:
6074:
6033:
6032:
6004:
6003:
5984:
5983:
5965:
5964:
5940:
5903:
5881:
5868:
5839:
5834:
5833:
5799:
5798:
5789:
5774:
5773:
5757:
5735:
5707:
5686:
5681:
5678:
5677:
5653:
5619:
5610:
5577:
5568:
5567:
5534:
5496:
5491:
5486:
5485:
5459:
5448:
5433:
5428:
5427:
5395:
5362:
5357:
5356:
5329:
5324:
5323:
5302:
5297:
5296:
5267:
5254:
5232:
5231:
5197:
5187:
5186:
5165:
5095:
5090:
5089:
5053:
5048:
5047:
5018:
5017:
4991:
4987:
4968:
4967:
4940:
4930:
4929:
4898:
4897:
4873:
4848:
4847:
4819:
4818:
4799:
4798:
4766:
4765:
4745:
4735:
4734:
4729:
4728:
4721:
4694:
4691:
4690:
4664:
4644:
4640:
4635:
4634:
4627:
4606:
4603:
4602:
4525:
4524:
4520:
4507:
4488:
4479:
4478:
4440:
4421:
4394:
4343:
4342:
4301:
4300:
4256:
4210:
4209:
4195:
4168:
4167:
4153:
4131:
4130:
4097:
4078:
4070:
4069:
4017:
4016:
3976:
3972:
3946:
3945:
3929:
3928:
3888:
3856:
3837:
3834:
3833:
3787:
3760:
3759:
3700:
3696:
3684:
3665:
3656:
3655:
3625:
3624:
3573:
3572:
3538:
3537:
3534:
3510:
3509:
3486:
3485:
3466:
3465:
3430:
3420:
3419:
3377:
3376:
3308:
3295:
3270:
3257:
3249:
3248:
3216:
3196:
3195:
3161:
3160:
3133:
3132:
3099:
3098:
3079:
3078:
3059:
3058:
3024:
3023:
3020:
2996:
2995:
2970:
2969:
2947:
2946:
2927:
2926:
2869:
2868:
2820:
2819:
2800:
2799:
2780:
2779:
2760:
2759:
2736:
2735:
2716:
2715:
2696:
2695:
2676:
2675:
2660:
2600:
2599:
2571:
2570:
2524:
2523:
2504:
2503:
2475:
2474:
2446:
2445:
2399:
2398:
2379:
2378:
2359:
2358:
2313:
2312:
2293:
2292:
2273:
2272:
2241:
2240:
2205:
2191:
2182:
2178:
2168:
2167:
2136:
2135:
2097:
2083:
2074:
2070:
2064:
2048:
2047:
2025:
2024:
2005:
2004:
1977:
1976:
1953:
1952:
1930:
1929:
1901:
1900:
1881:
1880:
1829:
1828:
1814:
1813:
1789:
1770:
1769:
1700:
1699:
1661:
1647:
1605:
1604:
1536:
1513:
1499:
1493:
1481:
1480:
1441:
1427:
1418:
1414:
1408:
1404:
1389:
1388:
1375:
1333:
1319:
1310:
1306:
1299:
1289:
1285:
1273:
1272:
1246:
1232:
1223:
1219:
1212:
1199:
1175:
1161:
1152:
1148:
1134:
1133:
1098:
1097:
1054:
1053:
1020:
1019:
991:
990:
959:
958:
898:
897:
866:
865:
810:
809:
772:
771:
737:
736:
717:
716:
697:
696:
677:
676:
630:
629:
610:
609:
580:
579:
534:
533:
514:
513:
494:
493:
439:
438:
419:
418:
399:
398:
379:
378:
350:
349:
330:
329:
301:
300:
278:
277:
274:
240:
239:
238:
215:
214:
195:
194:
172:
171:
149:
148:
118:
117:
94:
93:
86:
73:random variable
42:
37:
36:
17:
12:
11:
5:
8044:
8042:
8034:
8033:
8028:
8018:
8017:
8014:
8013:
8002:
7999:
7996:
7995:
7968:
7949:(4): 455–472.
7937:Gilks, W. R.;
7929:
7910:(3): 670–691.
7894:
7872:10.1.1.53.9001
7865:(4): 514–528.
7849:
7827:10.1.1.56.6055
7820:(2): 182–193.
7804:
7793:(4): 312–321.
7777:
7758:(2): 337–348.
7742:
7735:
7717:
7678:
7631:
7624:
7602:
7573:(3): 705–767.
7551:
7544:
7517:
7516:
7514:
7511:
7510:
7509:
7504:
7499:
7494:
7487:
7484:
7471:
7470:
7469:
7468:
7464:
7451:
7448:
7445:
7439:
7435:
7413:
7410:
7407:
7401:
7397:
7371:
7368:
7365:
7361:
7350:
7337:
7334:
7331:
7327:
7299:
7296:
7293:
7289:
7278:
7277:
7276:
7273:
7266:
7263:
7250:
7247:
7244:
7240:
7219:
7216:
7213:
7209:
7195:
7194:
7193:
7180:
7177:
7174:
7170:
7167:
7164:
7143:
7140:
7137:
7133:
7109:
7106:
7103:
7099:
7078:
7075:
7072:
7068:
7065:
7062:
7059:
7055:
7052:
7049:
7045:
7013:
7010:
7006:Gibbs sampling
6981:
6978:
6965:
6962:
6959:
6939:
6911:
6902:
6898:
6894:
6887:
6883:
6879:
6876:
6873:
6870:
6864:
6860:
6857:
6854:
6851:
6848:
6842:
6839:
6836:
6833:
6830:
6826:
6821:
6799:
6796:
6793:
6790:
6787:
6784:
6781:
6778:
6775:
6755:
6732:
6729:
6726:
6723:
6720:
6716:
6713:
6710:
6707:
6703:
6700:
6680:
6675:
6671:
6667:
6662:
6658:
6652:
6648:
6644:
6641:
6638:
6634:
6628:
6625:
6622:
6619:
6616:
6613:
6609:
6605:
6585:
6574:
6573:
6562:
6559:
6556:
6553:
6550:
6546:
6540:
6537:
6534:
6531:
6528:
6523:
6519:
6515:
6511:
6507:
6502:
6498:
6495:
6492:
6489:
6483:
6480:
6469:
6468:
6456:
6453:
6450:
6447:
6444:
6440:
6437:
6434:
6431:
6427:
6424:
6412:
6411:
6396:
6390:
6386:
6383:
6380:
6374:
6371:
6368:
6365:
6362:
6359:
6355:
6350:
6345:
6338:
6329:
6325:
6321:
6314:
6310:
6306:
6303:
6300:
6297:
6291:
6287:
6283:
6280:
6274:
6268:
6265:
6262:
6259:
6256:
6252:
6245:
6236:
6232:
6228:
6221:
6217:
6213:
6210:
6207:
6204:
6198:
6194:
6190:
6187:
6181:
6175:
6172:
6169:
6166:
6163:
6159:
6153:
6150:
6145:
6141:
6137:
6134:
6131:
6128:
6123:
6119:
6115:
6112:
6109:
6106:
6100:
6097:
6094:
6091:
6088:
6085:
6082:
6071:
6070:
6058:
6055:
6052:
6049:
6046:
6043:
6040:
6020:
6017:
6014:
6011:
5991:
5979:
5978:
5960:
5957:
5954:
5951:
5948:
5944:
5938:
5935:
5932:
5929:
5926:
5922:
5918:
5915:
5910:
5906:
5902:
5899:
5896:
5893:
5888:
5884:
5880:
5877:
5874:
5871:
5865:
5859:
5856:
5853:
5846:
5842:
5837:
5831:
5828:
5825:
5820:
5817:
5814:
5810:
5806:
5802:
5795:
5792:
5790:
5788:
5785:
5782:
5779:
5776:
5775:
5772:
5769:
5764:
5760:
5756:
5753:
5750:
5747:
5742:
5738:
5734:
5731:
5728:
5725:
5722:
5719:
5716:
5713:
5710:
5708:
5706:
5703:
5700:
5693:
5689:
5684:
5680:
5679:
5673:
5670:
5667:
5664:
5661:
5657:
5651:
5648:
5645:
5642:
5639:
5635:
5631:
5628:
5625:
5622:
5616:
5613:
5611:
5609:
5606:
5603:
5598:
5595:
5592:
5588:
5584:
5580:
5576:
5575:
5564:
5563:
5559:
5558:
5546:
5541:
5537:
5533:
5530:
5527:
5523:
5519:
5516:
5513:
5510:
5503:
5499:
5494:
5473:
5466:
5462:
5457:
5454:
5451:
5445:
5440:
5436:
5424:
5423:
5422:
5410:
5407:
5402:
5398:
5394:
5391:
5388:
5385:
5382:
5379:
5376:
5371:
5366:
5351:
5350:
5336:
5332:
5309:
5305:
5292:
5291:
5279:
5274:
5270:
5266:
5261:
5257:
5253:
5250:
5247:
5244:
5240:
5227:
5226:
5225:
5224:
5210:
5204:
5200:
5194:
5190:
5183:
5180:
5177:
5172:
5168:
5164:
5161:
5158:
5155:
5152:
5149:
5146:
5143:
5140:
5137:
5134:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5102:
5098:
5084:
5083:
5071:
5068:
5065:
5060:
5056:
5031:
5028:
5025:
5004:
5000:
4997:
4994:
4990:
4986:
4983:
4979:
4975:
4953:
4947:
4943:
4937:
4933:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4885:
4880:
4876:
4872:
4869:
4866:
4862:
4858:
4855:
4835:
4832:
4829:
4826:
4815:
4814:
4813:
4812:
4795:
4792:
4789:
4784:
4778:
4773:
4769:
4763:
4760:
4757:
4752:
4748:
4742:
4738:
4731:
4727:
4724:
4722:
4720:
4717:
4714:
4709:
4704:
4701:
4698:
4693:
4692:
4687:
4684:
4681:
4676:
4670:
4667:
4662:
4659:
4656:
4651:
4647:
4643:
4637:
4633:
4630:
4628:
4626:
4623:
4620:
4615:
4610:
4605:
4604:
4601:
4598:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4558:
4554:
4551:
4548:
4545:
4542:
4539:
4534:
4529:
4523:
4519:
4516:
4513:
4510:
4508:
4506:
4503:
4500:
4495:
4491:
4487:
4486:
4470:
4469:
4455:
4452:
4449:
4444:
4439:
4436:
4433:
4428:
4424:
4420:
4417:
4414:
4409:
4406:
4403:
4398:
4392:
4389:
4386:
4383:
4380:
4377:
4372:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4323:
4320:
4317:
4314:
4311:
4308:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4259:
4255:
4249:
4246:
4243:
4240:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4213:
4207:
4204:
4201:
4198:
4192:
4186:
4183:
4180:
4175:
4171:
4165:
4162:
4159:
4156:
4150:
4147:
4144:
4141:
4138:
4110:
4107:
4104:
4100:
4096:
4093:
4090:
4085:
4081:
4077:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4003:
3999:
3996:
3993:
3990:
3987:
3984:
3980:
3975:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3927:
3924:
3921:
3918:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3891:
3887:
3884:
3881:
3878:
3874:
3870:
3866:
3862:
3859:
3857:
3855:
3852:
3849:
3844:
3840:
3836:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3790:
3784:
3779:
3776:
3772:
3768:
3765:
3763:
3761:
3757:
3753:
3750:
3747:
3744:
3741:
3737:
3733:
3730:
3727:
3724:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3699:
3694:
3690:
3687:
3685:
3683:
3680:
3677:
3672:
3668:
3664:
3663:
3641:
3638:
3635:
3632:
3612:
3609:
3606:
3603:
3600:
3596:
3592:
3589:
3586:
3583:
3580:
3560:
3557:
3554:
3551:
3548:
3545:
3533:
3530:
3517:
3493:
3473:
3450:
3447:
3444:
3441:
3438:
3434:
3429:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3385:
3373:
3372:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3315:
3311:
3307:
3302:
3298:
3294:
3291:
3288:
3285:
3282:
3277:
3273:
3269:
3264:
3260:
3256:
3245:
3234:
3231:
3228:
3223:
3219:
3215:
3212:
3209:
3206:
3203:
3183:
3180:
3177:
3174:
3171:
3168:
3144:{\textstyle X}
3140:
3120:
3117:
3114:
3110:
3106:
3086:
3077:given the set
3066:
3046:
3043:
3040:
3037:
3034:
3031:
3019:
3016:
3003:
2992:
2991:
2990:
2989:
2977:
2966:
2954:
2934:
2911:
2908:
2905:
2902:
2899:
2895:
2891:
2888:
2885:
2882:
2879:
2876:
2865:
2853:
2850:
2847:
2844:
2841:
2837:
2834:
2831:
2828:
2807:
2787:
2767:
2743:
2723:
2703:
2683:
2659:
2656:
2642:
2639:
2636:
2633:
2630:
2627:
2624:
2620:
2616:
2613:
2610:
2607:
2587:
2584:
2581:
2578:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2531:
2511:
2491:
2488:
2485:
2482:
2462:
2459:
2456:
2453:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2386:
2366:
2346:
2343:
2340:
2337:
2333:
2329:
2326:
2323:
2320:
2300:
2280:
2260:
2257:
2254:
2251:
2248:
2227:
2220:
2217:
2214:
2211:
2208:
2203:
2200:
2197:
2194:
2188:
2185:
2181:
2176:
2155:
2152:
2149:
2146:
2143:
2119:
2112:
2109:
2106:
2103:
2100:
2095:
2092:
2089:
2086:
2080:
2077:
2073:
2068:
2063:
2058:
2055:
2032:
2012:
1992:
1988:
1984:
1960:
1937:
1917:
1914:
1911:
1908:
1888:
1868:
1865:
1862:
1859:
1856:
1852:
1849:
1846:
1843:
1839:
1836:
1812:
1809:
1801:
1793:
1790:
1786:
1783:
1778:
1775:
1773:
1771:
1768:
1765:
1761:
1758:
1755:
1752:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1722:
1716:
1713:
1708:
1705:
1703:
1701:
1698:
1695:
1691:
1688:
1685:
1682:
1676:
1673:
1670:
1667:
1664:
1659:
1656:
1653:
1650:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1617:
1613:
1610:
1608:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1580:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1540:
1537:
1534:
1528:
1525:
1522:
1519:
1516:
1511:
1508:
1505:
1502:
1496:
1492:
1489:
1486:
1484:
1482:
1478:
1473:
1469:
1464:
1456:
1453:
1450:
1447:
1444:
1439:
1436:
1433:
1430:
1424:
1421:
1417:
1412:
1407:
1403:
1400:
1397:
1394:
1392:
1390:
1387:
1379:
1376:
1373:
1369:
1366:
1362:
1355:
1348:
1345:
1342:
1339:
1336:
1331:
1328:
1325:
1322:
1316:
1313:
1309:
1303:
1298:
1295:
1292:
1288:
1284:
1281:
1278:
1276:
1274:
1268:
1261:
1258:
1255:
1252:
1249:
1244:
1241:
1238:
1235:
1229:
1226:
1222:
1216:
1211:
1208:
1205:
1202:
1200:
1197:
1190:
1187:
1184:
1181:
1178:
1173:
1170:
1167:
1164:
1158:
1155:
1151:
1146:
1142:
1141:
1114:
1111:
1108:
1105:
1070:
1067:
1064:
1061:
1039:
1036:
1033:
1030:
1027:
1007:
1004:
1001:
998:
978:
975:
972:
969:
966:
946:
943:
940:
937:
934:
931:
928:
924:
920:
917:
914:
911:
908:
905:
885:
882:
879:
876:
873:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
794:
791:
788:
785:
782:
779:
759:
756:
753:
750:
747:
744:
724:
704:
684:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
617:
593:
590:
587:
567:
564:
561:
558:
554:
550:
547:
544:
541:
521:
501:
481:
478:
475:
472:
469:
466:
463:
459:
455:
452:
449:
446:
426:
406:
386:
366:
363:
360:
357:
337:
317:
314:
311:
308:
285:
273:
270:
247:
235:
234:
222:
202:
191:
179:
168:
156:
125:
101:
85:
82:
51:
46:
15:
13:
10:
9:
6:
4:
3:
2:
8043:
8032:
8029:
8027:
8024:
8023:
8021:
8010:
8005:
8004:
8000:
7991:
7987:
7983:
7979:
7972:
7969:
7964:
7960:
7956:
7952:
7948:
7944:
7940:
7933:
7930:
7925:
7921:
7917:
7913:
7909:
7905:
7898:
7895:
7890:
7886:
7882:
7878:
7873:
7868:
7864:
7860:
7853:
7850:
7845:
7841:
7837:
7833:
7828:
7823:
7819:
7815:
7808:
7805:
7800:
7796:
7792:
7788:
7781:
7778:
7773:
7769:
7765:
7761:
7757:
7753:
7746:
7743:
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7732:
7728:
7721:
7718:
7713:
7709:
7705:
7701:
7697:
7693:
7689:
7682:
7679:
7674:
7670:
7666:
7662:
7658:
7654:
7650:
7646:
7642:
7635:
7632:
7627:
7621:
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7613:
7606:
7603:
7598:
7594:
7590:
7586:
7581:
7576:
7572:
7568:
7567:
7562:
7555:
7552:
7547:
7545:9780940600614
7541:
7537:
7533:
7529:
7522:
7519:
7512:
7508:
7505:
7503:
7500:
7498:
7495:
7493:
7490:
7489:
7485:
7483:
7481:
7475:
7465:
7449:
7446:
7443:
7437:
7433:
7411:
7408:
7405:
7399:
7395:
7386:
7369:
7366:
7363:
7359:
7351:
7335:
7332:
7329:
7325:
7317:
7316:
7314:
7297:
7294:
7291:
7287:
7279:
7274:
7271:
7267:
7264:
7248:
7245:
7242:
7238:
7217:
7214:
7211:
7207:
7199:
7198:
7196:
7178:
7175:
7172:
7168:
7165:
7162:
7141:
7138:
7135:
7131:
7123:
7122:
7107:
7104:
7101:
7097:
7076:
7073:
7070:
7066:
7063:
7060:
7057:
7053:
7050:
7047:
7043:
7035:
7034:
7033:
7030:
7028:
7024:
7020:
7011:
7009:
7007:
7003:
6999:
6995:
6991:
6985:
6979:
6977:
6963:
6960:
6957:
6937:
6928:
6923:
6900:
6896:
6892:
6885:
6877:
6874:
6871:
6862:
6858:
6855:
6849:
6846:
6837:
6834:
6831:
6819:
6794:
6788:
6785:
6779:
6773:
6753:
6744:
6727:
6724:
6721:
6701:
6698:
6673:
6669:
6665:
6660:
6656:
6650:
6646:
6642:
6639:
6626:
6623:
6620:
6617:
6614:
6611:
6607:
6603:
6583:
6557:
6554:
6551:
6535:
6532:
6529:
6521:
6517:
6513:
6509:
6505:
6500:
6493:
6487:
6481:
6478:
6471:
6470:
6451:
6448:
6445:
6425:
6422:
6414:
6413:
6394:
6388:
6384:
6381:
6378:
6372:
6366:
6363:
6360:
6348:
6336:
6327:
6323:
6319:
6312:
6304:
6301:
6298:
6289:
6285:
6281:
6278:
6272:
6263:
6260:
6257:
6243:
6234:
6230:
6226:
6219:
6211:
6208:
6205:
6196:
6192:
6188:
6185:
6179:
6170:
6167:
6164:
6143:
6139:
6132:
6129:
6126:
6121:
6117:
6113:
6107:
6104:
6098:
6092:
6086:
6083:
6080:
6073:
6072:
6050:
6047:
6041:
6038:
6015:
6009:
5989:
5981:
5980:
5955:
5952:
5949:
5933:
5930:
5927:
5908:
5904:
5897:
5894:
5891:
5886:
5882:
5878:
5872:
5869:
5863:
5854:
5844:
5840:
5835:
5826:
5818:
5815:
5812:
5804:
5800:
5793:
5791:
5783:
5777:
5762:
5758:
5751:
5748:
5745:
5740:
5736:
5729:
5726:
5720:
5714:
5711:
5709:
5701:
5691:
5687:
5682:
5668:
5665:
5662:
5646:
5643:
5640:
5626:
5620:
5614:
5612:
5604:
5596:
5593:
5590:
5582:
5578:
5566:
5565:
5561:
5560:
5539:
5535:
5531:
5528:
5517:
5511:
5501:
5497:
5492:
5471:
5464:
5460:
5455:
5452:
5449:
5443:
5438:
5434:
5425:
5408:
5405:
5400:
5396:
5392:
5389:
5386:
5383:
5377:
5369:
5355:
5354:
5353:
5352:
5334:
5330:
5307:
5303:
5294:
5293:
5272:
5268:
5264:
5259:
5255:
5251:
5248:
5245:
5229:
5228:
5208:
5202:
5198:
5192:
5188:
5181:
5178:
5170:
5166:
5162:
5159:
5156:
5150:
5144:
5138:
5135:
5129:
5126:
5123:
5117:
5114:
5108:
5100:
5096:
5088:
5087:
5086:
5085:
5066:
5058:
5054:
5045:
5044:
5043:
5029:
5026:
5023:
5002:
4995:
4992:
4988:
4984:
4981:
4973:
4951:
4945:
4941:
4935:
4931:
4924:
4921:
4918:
4915:
4909:
4903:
4878:
4874:
4870:
4867:
4856:
4853:
4830:
4824:
4793:
4790:
4787:
4782:
4776:
4771:
4758:
4750:
4746:
4740:
4725:
4723:
4715:
4707:
4685:
4682:
4679:
4674:
4668:
4657:
4649:
4645:
4631:
4629:
4621:
4613:
4596:
4590:
4584:
4581:
4575:
4572:
4569:
4563:
4560:
4556:
4549:
4546:
4540:
4537:
4532:
4521:
4517:
4514:
4511:
4509:
4501:
4493:
4489:
4477:
4476:
4475:
4474:
4473:
4453:
4450:
4447:
4434:
4426:
4422:
4418:
4415:
4412:
4407:
4404:
4401:
4387:
4384:
4378:
4375:
4366:
4363:
4360:
4354:
4348:
4341:
4340:
4339:
4337:
4318:
4312:
4306:
4297:
4279:
4273:
4270:
4267:
4264:
4261:
4257:
4253:
4244:
4238:
4230:
4224:
4221:
4218:
4215:
4211:
4202:
4196:
4190:
4181:
4173:
4169:
4160:
4154:
4148:
4142:
4136:
4128:
4126:
4105:
4102:
4091:
4083:
4079:
4049:
4043:
4037:
4034:
4031:
4025:
4001:
3994:
3991:
3985:
3982:
3973:
3969:
3966:
3963:
3957:
3951:
3942:
3922:
3916:
3908:
3902:
3899:
3896:
3893:
3889:
3885:
3879:
3872:
3868:
3864:
3860:
3858:
3850:
3842:
3838:
3830:
3827:
3821:
3815:
3807:
3801:
3798:
3795:
3792:
3788:
3782:
3774:
3770:
3766:
3764:
3755:
3748:
3745:
3742:
3725:
3719:
3716:
3713:
3710:
3704:
3701:
3697:
3688:
3686:
3678:
3670:
3666:
3653:
3636:
3630:
3607:
3604:
3601:
3590:
3584:
3578:
3555:
3549:
3546:
3543:
3531:
3529:
3515:
3507:
3491:
3471:
3445:
3442:
3439:
3427:
3405:
3402:
3396:
3393:
3390:
3370:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3313:
3309:
3305:
3300:
3296:
3292:
3289:
3286:
3283:
3280:
3275:
3271:
3267:
3262:
3258:
3246:
3229:
3226:
3221:
3217:
3213:
3210:
3207:
3204:
3178:
3172:
3169:
3166:
3158:
3157:
3156:
3154:
3138:
3118:
3115:
3112:
3104:
3084:
3064:
3041:
3035:
3032:
3029:
3017:
3015:
3001:
2975:
2967:
2952:
2932:
2924:
2923:
2906:
2900:
2897:
2893:
2886:
2880:
2877:
2874:
2866:
2848:
2845:
2842:
2805:
2798:and a sample
2785:
2765:
2757:
2756:
2755:
2741:
2734:with density
2721:
2701:
2694:with density
2681:
2673:
2669:
2665:
2657:
2655:
2634:
2628:
2625:
2618:
2611:
2605:
2582:
2576:
2567:
2550:
2544:
2541:
2535:
2529:
2509:
2486:
2480:
2457:
2451:
2428:
2422:
2419:
2416:
2410:
2404:
2384:
2364:
2341:
2335:
2331:
2324:
2318:
2298:
2278:
2255:
2252:
2249:
2225:
2215:
2209:
2206:
2198:
2192:
2186:
2183:
2179:
2150:
2147:
2144:
2141:
2117:
2107:
2101:
2098:
2090:
2084:
2078:
2075:
2071:
2061:
2056:
2053:
2044:
2030:
2010:
1990:
1986:
1982:
1974:
1958:
1949:
1935:
1912:
1906:
1886:
1863:
1860:
1857:
1837:
1834:
1807:
1799:
1784:
1781:
1776:
1774:
1766:
1763:
1756:
1750:
1745:
1742:
1736:
1730:
1727:
1724:
1720:
1714:
1711:
1706:
1704:
1696:
1693:
1686:
1680:
1671:
1665:
1662:
1654:
1648:
1640:
1637:
1631:
1625:
1622:
1619:
1615:
1611:
1609:
1595:
1592:
1589:
1578:
1570:
1567:
1564:
1558:
1555:
1552:
1543:because
1532:
1523:
1517:
1514:
1506:
1500:
1494:
1490:
1487:
1485:
1476:
1471:
1467:
1451:
1445:
1442:
1434:
1428:
1422:
1419:
1415:
1405:
1401:
1395:
1393:
1371:
1364:
1353:
1343:
1337:
1334:
1326:
1320:
1314:
1311:
1307:
1293:
1286:
1282:
1279:
1277:
1266:
1256:
1250:
1247:
1239:
1233:
1227:
1224:
1220:
1209:
1203:
1201:
1195:
1185:
1179:
1176:
1168:
1162:
1156:
1153:
1149:
1130:
1128:
1109:
1103:
1095:
1091:
1086:
1084:
1065:
1059:
1050:
1037:
1031:
1025:
1002:
996:
973:
970:
967:
938:
932:
929:
922:
915:
909:
906:
903:
880:
874:
871:
845:
839:
836:
833:
830:
827:
824:
821:
818:
806:
792:
789:
783:
777:
757:
754:
748:
742:
722:
702:
682:
659:
653:
650:
647:
641:
635:
615:
607:
588:
585:
578:, satisfying
562:
556:
552:
545:
539:
519:
499:
473:
467:
464:
457:
450:
444:
424:
404:
384:
361:
355:
335:
312:
306:
299:
283:
271:
269:
267:
263:
245:
220:
200:
192:
177:
169:
154:
146:
145:
144:
141:
139:
123:
113:
99:
91:
83:
81:
79:
74:
69:
67:
49:
34:
30:
26:
22:
8008:
7981:
7977:
7971:
7946:
7942:
7932:
7907:
7903:
7897:
7862:
7858:
7852:
7817:
7813:
7807:
7790:
7786:
7780:
7755:
7751:
7745:
7726:
7720:
7698:(1): 31–48.
7695:
7691:
7681:
7648:
7644:
7634:
7611:
7605:
7570:
7564:
7554:
7527:
7521:
7479:
7476:
7472:
7385:already know
7384:
7312:
7031:
7026:
7022:
7015:
6986:
6983:
6925:In general,
6924:
6745:
6575:
4816:
4471:
4298:
4129:
4123:, is from a
3943:
3654:
3535:
3374:
3131:, sometimes
3021:
2993:
2754:as follows:
2661:
2568:
2045:
1950:
1131:
1087:
1051:
807:
275:
236:
142:
114:
87:
77:
70:
32:
29:distribution
24:
18:
7939:Best, N. G.
7090:instead of
7019:log-concave
6069:, therefore
4338:, that is,
4068:. Clearly,
1090:Monte Carlo
84:Description
8020:Categories
7597:1051.65007
7513:References
7315:accepted.
7155:is messy,
7121:directly.
6766:, that is
4299:Note that
3367:(see also
2672:his needle
2134:Note that
1575:when
258:‑positions
7924:1061-8600
7867:CiteSeerX
7844:0098-3500
7822:CiteSeerX
7712:1874-1746
7665:0025-5718
7166:
7064:
6980:Drawbacks
6961:∈
6897:σ
6878:μ
6875:−
6859:⋅
6835:≥
6702:∼
6670:σ
6657:σ
6651:∗
6647:θ
6640:μ
6608:∼
6555:≥
6533:−
6522:∗
6518:θ
6514:−
6482:≤
6426:∼
6389:σ
6385:μ
6382:−
6373:≥
6324:σ
6305:μ
6302:−
6290:−
6282:
6261:≥
6231:σ
6212:μ
6209:−
6197:−
6189:
6168:≥
6144:∗
6140:θ
6133:ψ
6122:∗
6118:θ
6114:−
6108:
6054:∞
6042:∈
5953:≥
5931:≥
5909:∗
5905:θ
5898:ψ
5887:∗
5883:θ
5879:−
5873:
5845:∗
5841:θ
5816:≥
5763:∗
5759:θ
5752:ψ
5749:−
5741:∗
5737:θ
5730:
5692:∗
5688:θ
5666:≥
5644:≥
5594:≥
5536:σ
5502:∗
5498:θ
5461:σ
5456:μ
5453:−
5439:∗
5435:θ
5397:σ
5393:θ
5387:μ
5370:θ
5349:is to set
5335:∗
5331:θ
5308:∗
5304:θ
5269:σ
5256:σ
5252:θ
5246:μ
5199:η
5189:σ
5179:η
5167:σ
5163:θ
5157:μ
5145:θ
5139:ψ
5136:−
5130:η
5124:θ
5118:ψ
5109:η
5101:θ
5097:ψ
5067:⋅
5059:θ
5030:μ
4999:∞
4985:∈
4942:θ
4932:σ
4922:θ
4919:μ
4910:θ
4904:ψ
4875:σ
4868:μ
4857:∼
4831:⋅
4788:η
4777:η
4768:∂
4759:η
4751:θ
4747:ψ
4737:∂
4708:θ
4680:η
4669:η
4666:∂
4658:η
4650:θ
4646:ψ
4642:∂
4614:θ
4600:∞
4591:θ
4585:ψ
4582:−
4576:η
4570:θ
4564:ψ
4547:η
4541:
4533:θ
4518:
4502:η
4494:θ
4490:ψ
4454:θ
4419:
4408:θ
4379:
4367:
4355:θ
4349:ψ
4322:∞
4313:θ
4307:ψ
4280:θ
4274:ψ
4265:θ
4262:−
4231:θ
4225:ψ
4222:−
4216:θ
4174:θ
4109:Θ
4106:∈
4103:θ
4092:⋅
4084:θ
4053:∞
4044:θ
4038:ψ
4032:θ
4023:Θ
3992:θ
3986:
3970:
3958:θ
3952:ψ
3909:θ
3903:ψ
3900:−
3894:θ
3869:θ
3843:θ
3808:θ
3802:ψ
3799:−
3793:θ
3778:∞
3775:−
3771:∫
3746:≤
3726:θ
3720:ψ
3717:−
3711:θ
3705:
3671:θ
3605:≤
3556:⋅
3547:∼
3443:∈
3403:≈
3394:∈
3319:∈
3227:∈
3208:≥
3179:⋅
3170:∼
3116:∈
3042:⋅
3033:∼
2867:Check if
2658:Algorithm
2417:≤
2187:≤
2154:∞
2145:≤
2079:≤
1913:⋅
1838:∼
1721:∫
1616:∫
1556:≤
1423:≤
1402:
1315:≤
1294:
1228:≤
1210:
1157:≤
834:⋅
770:whenever
648:≤
604:over the
592:∞
7616:Springer
7486:See also
6691:and new
5426:that is
5016:, where
3873:′
3247:Output:
3097:, i.e.,
7963:2986138
7889:1390680
7772:2347565
7673:2005864
7589:1994729
7023:density
4896:, with
3159:Sample
2271:. When
606:support
66:density
64:with a
7961:
7922:
7887:
7869:
7842:
7824:
7770:
7733:
7710:
7671:
7663:
7622:
7595:
7587:
7542:
3944:where
2668:Buffon
1827:where
272:Theory
7959:JSTOR
7885:JSTOR
7768:JSTOR
7669:JSTOR
7017:have
2818:from
7920:ISSN
7840:ISSN
7731:ISBN
7708:ISSN
7661:ISSN
7620:ISBN
7540:ISBN
7425:(or
6467:, if
5027:>
4597:<
4319:<
4050:<
4015:and
2878:<
2670:and
2151:<
1743:>
1638:>
907:<
790:>
755:>
589:<
7986:doi
7951:doi
7912:doi
7877:doi
7832:doi
7795:doi
7760:doi
7700:doi
7653:doi
7593:Zbl
7575:doi
7532:doi
7163:log
7061:log
7004:or
6279:exp
6186:exp
6105:exp
5870:exp
5727:exp
4538:exp
4515:log
4416:log
4376:exp
4364:log
3983:exp
3967:log
3702:exp
3155:):
608:of
8022::
7982:52
7980:.
7957:.
7947:44
7918:.
7908:20
7906:.
7883:.
7875:.
7861:.
7838:.
7830:.
7818:21
7816:.
7789:.
7766:.
7756:41
7706:.
7696:12
7694:.
7690:.
7667:.
7659:.
7649:26
7647:.
7643:.
7618:.
7614:.
7591:.
7585:MR
7583:.
7571:31
7569:.
7563:.
7538:.
7313:is
7029:.
4846:,
3571:,
2922:.
1948:.
1547:Pr
1129:.
1085:.
805:.
268:.
68:.
23:,
7992:.
7988::
7965:.
7953::
7926:.
7914::
7891:.
7879::
7863:7
7846:.
7834::
7801:.
7797::
7791:1
7774:.
7762::
7739:.
7714:.
7702::
7675:.
7655::
7628:.
7599:.
7577::
7548:.
7534::
7450:)
7447:x
7444:(
7438:l
7434:h
7412:)
7409:x
7406:(
7400:l
7396:g
7370:)
7367:x
7364:(
7360:f
7336:)
7333:x
7330:(
7326:h
7298:)
7295:x
7292:(
7288:f
7249:)
7246:x
7243:(
7239:h
7218:)
7215:x
7212:(
7208:f
7179:)
7176:x
7173:(
7169:f
7142:)
7139:x
7136:(
7132:f
7108:)
7105:x
7102:(
7098:g
7077:)
7074:x
7071:(
7067:g
7058:=
7054:)
7051:x
7048:(
7044:h
6964:A
6958:X
6938:X
6910:)
6901:2
6893:2
6886:2
6882:)
6872:b
6869:(
6863:e
6856:b
6853:(
6850:O
6847:=
6841:)
6838:b
6832:X
6829:(
6825:P
6820:1
6798:)
6795:b
6792:(
6789:O
6786:=
6783:)
6780:b
6777:(
6774:M
6754:b
6731:)
6728:1
6725:,
6722:0
6719:(
6715:f
6712:i
6709:n
6706:U
6699:U
6679:)
6674:2
6666:,
6661:2
6643:+
6637:(
6633:N
6627:.
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5587:|
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5557:.
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5512:x
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5365:E
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5070:)
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4884:)
4879:2
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4499:(
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4076:{
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4041:(
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3955:(
3926:)
3923:x
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3917:f
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3906:(
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3828:d
3825:)
3822:y
3819:(
3816:f
3811:)
3805:(
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3749:x
3743:X
3740:(
3736:I
3732:)
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3723:(
3714:X
3708:(
3698:[
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3679:x
3676:(
3667:F
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3637:x
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3631:f
3611:)
3608:x
3602:X
3599:(
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3585:x
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3559:)
3553:(
3550:F
3544:X
3516:M
3492:M
3472:M
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3433:P
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3406:0
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3384:P
3371:)
3355:}
3352:N
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3346:.
3343:.
3340:.
3337:,
3334:1
3331:=
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3325:,
3322:A
3314:i
3310:X
3306::
3301:N
3297:X
3293:,
3290:.
3287:.
3284:.
3281:,
3276:2
3272:X
3268:,
3263:1
3259:X
3255:{
3233:}
3230:A
3222:n
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3214::
3211:1
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3176:(
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3119:A
3113:X
3109:|
3105:X
3085:A
3065:X
3045:)
3039:(
3036:F
3030:X
3002:M
2976:y
2965:;
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2910:)
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2635:x
2632:(
2629:g
2626:M
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2619:/
2615:)
2612:x
2609:(
2606:f
2586:)
2583:x
2580:(
2577:f
2554:)
2551:x
2548:(
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2536:x
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2530:f
2510:M
2490:)
2487:x
2484:(
2481:f
2461:)
2458:x
2455:(
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2429:x
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2411:x
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2345:)
2342:x
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2279:M
2259:]
2256:1
2253:,
2250:0
2247:[
2226:)
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2102:g
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2091:Y
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2072:(
2067:P
2062:1
2057:=
2054:M
2031:M
2011:M
1991:M
1987:/
1983:1
1959:Y
1936:Y
1916:)
1910:(
1907:g
1887:y
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1864:1
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1858:0
1855:(
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1835:U
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1800:Y
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1785:M
1782:1
1777:=
1767:y
1764:d
1760:)
1757:y
1754:(
1751:f
1746:0
1740:)
1737:y
1734:(
1731:g
1728::
1725:y
1715:M
1712:1
1707:=
1697:y
1694:d
1690:)
1687:y
1684:(
1681:g
1675:)
1672:y
1669:(
1666:g
1663:M
1658:)
1655:y
1652:(
1649:f
1641:0
1635:)
1632:y
1629:(
1626:g
1623::
1620:y
1612:=
1602:)
1599:)
1596:1
1593:,
1590:0
1587:(
1579:U
1571:,
1568:u
1565:=
1562:)
1559:u
1553:U
1550:(
1539:(
1533:]
1527:)
1524:Y
1521:(
1518:g
1515:M
1510:)
1507:Y
1504:(
1501:f
1495:[
1491:E
1488:=
1477:]
1472:)
1468:Y
1463:|
1455:)
1452:Y
1449:(
1446:g
1443:M
1438:)
1435:Y
1432:(
1429:f
1420:U
1416:(
1411:P
1406:[
1399:E
1396:=
1386:)
1378:(
1372:]
1368:]
1365:Y
1361:|
1354:]
1347:)
1344:Y
1341:(
1338:g
1335:M
1330:)
1327:Y
1324:(
1321:f
1312:U
1308:[
1302:1
1297:[
1291:E
1287:[
1283:E
1280:=
1267:]
1260:)
1257:Y
1254:(
1251:g
1248:M
1243:)
1240:Y
1237:(
1234:f
1225:U
1221:[
1215:1
1207:E
1204:=
1196:)
1189:)
1186:Y
1183:(
1180:g
1177:M
1172:)
1169:Y
1166:(
1163:f
1154:U
1150:(
1145:P
1113:)
1110:x
1107:(
1104:f
1069:)
1066:x
1063:(
1060:f
1038:.
1035:)
1032:x
1029:(
1026:f
1006:)
1003:x
1000:(
997:f
977:)
974:v
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968:x
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933:g
930:M
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923:/
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916:x
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910:f
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884:)
881:x
878:(
875:g
872:M
852:)
849:)
846:x
843:(
840:g
837:M
831:u
828:=
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822:,
819:x
816:(
793:0
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784:x
781:(
778:f
758:0
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749:x
746:(
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723:X
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683:x
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657:(
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642:x
639:(
636:f
616:X
586:M
566:)
563:x
560:(
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549:)
546:x
543:(
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500:Y
480:)
477:)
474:x
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462:(
458:/
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451:x
448:(
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425:Y
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385:X
365:)
362:x
359:(
356:g
336:Y
316:)
313:x
310:(
307:f
284:X
246:x
221:x
201:x
178:x
155:x
124:1
100:x
78:N
50:m
45:R
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