Rule of succession - Knowledge (XXG)

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sensible generalisation. The full derivation can be found in Jaynes' book, but it does admit an easier to understand alternative derivation, once the solution is known. Another point to emphasise is that the prior state of knowledge described by the rule of succession is given as an enumeration of the possibilities, with the additional information that it is possible to observe each category. This can be equivalently stated as observing each category once prior to gathering the data. To denote that this is the knowledge used, an

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not some "universal" set. In fact Larry Bretthorst shows that including the possibility of "something else" into the hypothesis space makes no difference to the relative probabilities of the other hypothesis—it simply renormalises them to add up to a value less than 1. Until "something else" is specified, the likelihood function conditional on this "something else" is indeterminate, for how is one to determine

50: 5488: 4110: 4451: 2624:. This puts the information contained in the rule of succession in greater light: it can be thought of as expressing the prior assumption that if sampling was continued indefinitely, we would eventually observe at least one success, and at least one failure in the sample. The prior expressing total ignorance does not assume this knowledge. 5201: 6262:

no account of an observation previously believed to have zero probability—it is still declared impossible. However, only considering a fixed set of the possibilities is an acceptable route, one just needs to remember that the results are conditional on (or restricted to) the set being considered, and

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Laplace knew this well, and he wrote to conclude the sunrise example: "But this number is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at the present moment can arrest the course of it." Yet Laplace was ridiculed for

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may need to be very large. They are not always small, and thereby soon outweighed by actual observations, as is often assumed. However, although a last resort, for everyday purposes, prior knowledge is usually vital. So most decisions must be subjective to some extent (dependent upon the analyst and

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region. Also important when there are many observations, where it is believed that the expectation should be heavily weighted towards the prior estimates, in spite of many observations to the contrary, such as for a roulette wheel in a well-respected casino. In the latter case, at least some of the

6211:. This indicates that mere knowledge of more than two outcomes we know are possible is relevant information when collapsing these categories down to just two. This illustrates the subtlety in describing the prior information, and why it is important to specify which prior information one is using. 6384:

Prior probabilities are only worth spending significant effort estimating when likely to have significant effect. They may be important when there are few observations — especially when so few that there have been few, if any, observations of some possibilities – such as a rare animal, in a given

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Note: The actual probability needs to use the length of blue arcs divided by the length of all arcs. However, when k points are uniformly randomly distributed on a circle, the distance from a point to the next point is 1/k. So on average each arc is of the same length and ratio of lengths becomes

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The rule of succession has many different intuitive interpretations, and depending on which intuition one uses, the generalisation may be different. Thus, the way to proceed from here is very carefully, and to re-derive the results from first principles, rather than to introduce an intuitively

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Laplace used the rule of succession to calculate the probability that the Sun will rise tomorrow, given that it has risen every day for the past 5000 years. One obtains a very large factor of approximately 5000 × 365.25, which gives odds of about 1,826,200 to 1 in favour of the Sun rising

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known possible outcomes prior to observing any data, only then does the rule of succession apply. If the rule of succession is applied in problems where this does not accurately describe the prior state of knowledge, then it may give counter-intuitive results. This is not because the rule of

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However, it is sometimes debatable whether prior knowledge should affect the relative probabilities, or also the total weight of the prior knowledge compared to actual observations. This does not have a clear cut answer, for it depends on what prior knowledge one is considering. In fact, an

5810:{\displaystyle f(p_{1},\ldots ,p_{m}\mid n_{1},\ldots ,n_{m},I)={\begin{cases}{\displaystyle {\frac {\Gamma \left(\sum _{i=1}^{m}(n_{i}+1)\right)}{\prod _{i=1}^{m}\Gamma (n_{i}+1)}}p_{1}^{n_{1}}\cdots p_{m}^{n_{m}}},\quad &\sum _{i=1}^{m}p_{i}=1\\\\0&{\text{otherwise.}}\end{cases}}} 4440: 3782: 351:+ 1 successes. Although this may seem the simplest and most reasonable assumption, which also happens to be true, it still requires a proof. Indeed, assuming a pseudocount of one per possibility is one way to generalise the binary result, but has unexpected consequences — see 5419:

The rule of succession comes from setting a binomial likelihood, and a uniform prior distribution. Thus a straightforward generalisation is just the multivariate extensions of these two distributions: 1) Setting a uniform prior over the initial m categories, and 2) using the

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However, as the mathematical details below show, the basic assumption for using the rule of succession would be that we have no prior knowledge about the question whether the Sun will or will not rise tomorrow, except that it can do either. This is not the case for sunrises.

4910:{\displaystyle {\begin{aligned}\sum _{R=1}^{N-n}{\prod _{j=1}^{n-1}(N-R-j) \over R}&\approx \int _{1}^{N-n}{(N-R)^{n-1} \over R}\,dR\\&=N\int _{1}^{N-n}{(N-R)^{n-2} \over R}\,dR-\int _{1}^{N-n}(N-R)^{n-2}\,dR\\&=N^{n-1}\left\approx N^{n-1}\ln(N)\end{aligned}}} 3332: 1287: 6321:

potential categories, but I am sure that only one of them is possible prior to observing the data. However, I do not know which particular category this is." A mathematical way to describe this prior is the Dirichlet distribution with all parameters equal to

3738: 5354:(for the simpler analytic properties) we are "throwing away" a piece of very important information. Note that this ignorance relationship only holds as long as only no successes are observed. It is correspondingly revised back to the observed frequency rule 6234:

One of the most difficult aspects of the rule of succession is not the mathematical formulas, but answering the question: When does the rule of succession apply? In the generalisation section, it was noted very explicitly by adding the prior information

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Since we have the prior knowledge that we are looking at an experiment for which both success and failure are possible, our estimate is as if we had observed one success and one failure for sure before we even started the experiments. In a sense we made

3531: 6254:), no possibility should have its probability (or its pseudocount) set to zero, since nothing in the physical world should be assumed strictly impossible (though it may be)—even if contrary to all observations and current theories. Indeed, 1936: 759:

defines the first non-blue/failure arc. Since the next point is a uniformly random point, if it falls in any of the blue arcs then the trial succeeds while if it falls in any of the other arcs, then it fails. So the probability of success

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The rule of succession can be interpreted in an intuitive manner by considering points randomly distributed on a circle rather than counting the number "success"/"failures" in an experiment. To mimic the behavior of the proportion

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points uniformly distributed on the circle; any point in the "success" fraction is a success and a failure otherwise. This provides an exact mapping from success/failure experiments with probability of success

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Given a good model, it is best to make as many observations as practicable, depending on the expected reliability of prior knowledge, cost of observations, time and resources available, and accuracy required.

4105:{\displaystyle E\left({S \over N}|n,s=0,N\right)={1 \over N}\sum _{S=1}^{N-n}SP(S|N,n=1,s=0)={1 \over N}{\sum _{S=1}^{N-n}\prod _{j=1}^{n-1}(N-S-j) \over \sum _{R=1}^{N-n}{\prod _{j=1}^{n-1}(N-R-j) \over R}}} 4216: 942: 6311: 6227:: Although we have a huge number of samples of the sun rising, there are far better models of the sun than assuming it has a certain probability of rising each day, e.g., simply having a half-life. 2576:

Thus, with the prior specifying total ignorance, the probability of success is governed by the observed frequency of success. However, the posterior distribution that led to this result is the Beta(

4456: 3091: 162:. The formula is still used, particularly to estimate underlying probabilities when there are few observations or events that have not been observed to occur at all in (finite) sample data. 1073: 3542: 647:

is precisely the number of blue arcs divided by the total number of arcs. If we let the first clockwise point of an arc define it, then every point on the circle defines one arc with

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Adding in the normalising constant, which is always finite (because there are no singularities in the range of the posterior, and there are a finite number of terms) gives:

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is assigned a uniform distribution to describe the uncertainty about its true value. (This proportion is not random, but uncertain. We assign a probability distribution to

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as the likelihood function (which is the multivariate generalisation of the binomial distribution). It can be shown that the uniform distribution is a special case of the

3359: 1760: 577: 5331: = 5 (tens of thousands), the expected proportion rises to approximately 0.86%, and so on. Similarly, if the number of observations is smaller, so that 529: 503: 5196:{\displaystyle E\left({S \over N}|n,s=0,N\right)\approx {1 \over N}{{N^{n} \over n} \over N^{n-1}\ln(N)}={1 \over n}={\log _{10}(e) \over n}={0.434294 \over n}} 3083: 3063: 3043: 3023: 3003: 2854: 2834: 552: 1597:{\displaystyle f(p\mid X_{1}=x_{1},\ldots ,X_{n}=x_{n})={L(p)f(p) \over \int _{0}^{1}L(r)f(r)\,dr}={p^{s}(1-p)^{n-s} \over \int _{0}^{1}r^{s}(1-r)^{n-s}\,dr}} 5220: 728:

to uniformly random points on the circle. In the figure the success fraction is colored blue to differentiate it from the rest of the circle and the points

2160:{\displaystyle \operatorname {E} (p\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)=\int _{0}^{1}pf(p\mid X_{1}=x_{1},\ldots ,X_{n}=x_{n})\,dp={s+1 \over n+2}.} 2455: 2419:{\displaystyle P(X_{n+1}=1\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)=\operatorname {E} (p\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)={s+1 \over n+2}.} 37:"Laplace–Bayes estimator" redirects here. For statistical estimators that maximize posterior expected utility or minimize posterior expected loss, see 5323: = 10 results without success, then the expected proportion in the population is approximately 0.43%. If the population is smaller, so that 6069: 4435:{\displaystyle \sum _{S=1}^{N-n}\prod _{j=1}^{n-1}(N-S-j)\approx \int _{1}^{N-n}(N-S)^{n-1}\,dS={(N-1)^{n}-n^{n} \over n}\approx {N^{n} \over n}} 1617: 751:

points such that the portion from a point on the circle to the next point (moving clockwise) is one arc associated with the first point. Thus,

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get more and more similar, which is intuitively clear: the more data we have, the less importance should be assigned to our prior information.

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for the multinomial distribution, which means that the posterior distribution is also a Dirichlet distribution with different parameters. Let

2612:). This means that we cannot use this form of the posterior distribution to calculate the probability of the next observation succeeding when 6448: 5889: 215: 368: 5350:. This means that the probability depends on the size of the population from which one is sampling. In passing to the limit of infinite 67: 6011:

This solution reduces to the probability that would be assigned using the principle of indifference before any observations made (i.e.

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on the circle, we will color the circle in two colors and the fraction of the circle colored in the "success" color will be equal to

930: 133: 6015: = 0), consistent with the original rule of succession. It also contains the rule of succession as a special case, when 4125: 607:, and developed measures of degree of confirmation, which he considered as alternatives to Laplace's rule of succession. See also 6266: 114: 4445:

The same procedure is followed for the denominator, but the process is a bit more tricky, as the integral is harder to evaluate

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with all of its parameters equal to 1 (just as the uniform is Beta(1,1) in the binary case). The Dirichlet distribution is the

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This article is about the rule of succession in probability theory. For monarchical and presidential rules of succession, see

2437:, including ignorance with regard to the question whether the experiment can succeed, or can fail. This improper prior is 1/( 71: 3327:{\displaystyle P(S|N,n,s)\propto {1 \over S(N-S)}{S \choose s}{N-S \choose n-s}\propto {S!(N-S)! \over S(N-S)(S-s)!(N-S-)!}} 6247:

succession is defective, but that it is effectively answering a different question, based on different prior information.

93: 6627: 1282:{\displaystyle L(p)=P(X_{1}=x_{1},\ldots ,X_{n}=x_{n}\mid p)=\prod _{i=1}^{n}p^{x_{i}}(1-p)^{1-x_{i}}=p^{s}(1-p)^{n-s}} 2640: 3733:{\displaystyle P(S|N,n,s=0)={\prod _{j=1}^{n-1}(N-S-j) \over S\sum _{R=1}^{N-n}{\prod _{j=1}^{n-1}(N-R-j) \over R}}} 3349: = 0, then one of the factorials in the numerator cancels exactly with one in the denominator. Taking the 100: 6179:

which is different from the original rule of succession. But note that the original rule of succession is based on

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This probability has no positive lower bound, and can be made arbitrarily small for larger and larger choices of

6313:? Thus no updating of the prior probability for "something else" can occur until it is more accurately defined. 5421: 2193: 2177: 886: 203: 82: 6044:

probabilities that correspond to "success" to get the probability of success. Supposing that this aggregates

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be the total number of observations made. The result, using the properties of the Dirichlet distribution is:

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tells us that the expected probability of success in the next experiment is just the expected value of

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this calculation; his opponents gave no heed to that sentence, or failed to understand its importance.

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A good model is essential (i.e., a good compromise between accuracy and practicality). To paraphrase

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has been used in the final answer for ease of calculation. For instance if the population is of size

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known from the start that both success and failure are possible, then we would have had to assign

6527: 6484: 6399: 1931:{\displaystyle f(p\mid X_{1}=x_{1},\ldots ,X_{n}=x_{n})={(n+1)! \over s!(n-s)!}p^{s}(1-p)^{n-s}.} 147: 6251: 6063:

values that have been termed "success". The probability of "success" at the next trial is then:

107: 6444: 4921: 2430: 1942: 922: 897: 2449: ≤ 1 and 0 otherwise. If the calculation above is repeated with this prior, we get 691:

A fraction is chosen by selecting two uniformly random points on the circle. The first point

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To get the generalised rule of succession, note that the probability of observing category

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into the calculations. Thus, when all that is known about a phenomenon is that there are

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as the data actually observed). Putting it all together, we can calculate the posterior:

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Laplace, Pierre-Simon (1814). Essai philosophique sur les probabilités. Paris: Courcier.

557: 5305:{\displaystyle E\left({S \over N}\mid n,s=0,N=10^{k}\right)\approx {0.434294 \over nk}} 3068: 3048: 3028: 3008: 2988: 2839: 2819: 1946: 815:

points. Substituting the values with number of successes gives the rule of succession.

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to each category. This gives a slightly different probability in the binary case of

5466:) actually was observed. Then the joint posterior distribution of the probabilities 2566:{\displaystyle P'(X_{n+1}=1\mid X_{i}=x_{i}{\text{ for }}i=1,\dots ,n)={s \over n}.} 209:

that each can assume the value 0 or 1, then, if we know nothing more about them,

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as soon as one success is observed. The corresponding results are found for the

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So for example, if the population be on the order of tens of billions, so that

6255: 6169:{\displaystyle P({\text{success}}|n_{1},\ldots ,n_{m},I_{m})={s+c \over n+m},} 2176:

tells us the probability of success in any experiment, and each experiment is

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failures, then what is the probability that the next repetition will succeed?

479:, below, for an analysis of its validity. In particular it is not valid when 6317:

alternative prior state of knowledge could be of the form "I have specified

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case by switching labels, and then subtracting the probability from 1.

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If we repeat an experiment that we know can result in a success or failure,

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is the total number of arcs. Note that there is one more blue arc (that of

2180:, the conditional probability for success in the next experiment is just 1737:{\displaystyle \int _{0}^{1}r^{s}(1-r)^{n-s}\,dr={s!(n-s)! \over (n+1)!}} 6531: 6220: 6001:{\displaystyle P(A_{i}|n_{1},\ldots ,n_{m},I_{m})={n_{i}+1 \over n+m}.} 323:{\displaystyle P(X_{n+1}=1\mid X_{1}+\cdots +X_{n}=s)={s+1 \over n+2}.} 6439:

Part II Section 18.6 of Jaynes, E. T. & Bretthorst, G. L. (2003).

6523: 465:{\displaystyle P'(X_{n+1}=1\mid X_{1}+\cdots +X_{n}=s)={s \over n}.} 6480: 5339: = 10, the proportion rise to approximately 0.86% again. 4221:

and then replacing the summation in the numerator with an integral

671:) more than total number of trials which is the rule of succession. 841:. But this amounts, mathematically, to the same thing as treating 618: 352: 5416:

is put as part of the conditions in the probability assignments.

2600: = 0 (i.e. the normalisation constant is infinite when 4119:

is given by first making the approximation to the product term:

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to express our uncertainty, not to attribute randomness to

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within . In terms of the circle the fraction of the circle from

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corresponds to the zero in the interval and the second point

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This section gives a heuristic derivation similar to that in

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then probability of success on the next sample is given by:

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plugging in these approximations into the expectation gives

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as the number of successes in the total population, of size

2700:. This is the approach taken in Jaynes (2003). The binomial 743:

is the fraction colored blue. Let us divide the circle into

6306:{\displaystyle Pr({\text{data}}|{\text{something else}},I)} 5803: 1043: 5851:

denote the event that the next observation is in category

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The posterior probability density function is therefore

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http://www.stats.org.uk/priors/noninformative/Smith.pdf

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are mutually exclusive, it is possible to collapse the

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is the point such that the fraction of the circle from

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to find the conditional probability distribution of

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Bayesian Spectrum Analysis and parameter estimation

74:. Unsourced material may be challenged and removed. 6548:"An elegant proof of Laplace's rule of succession" 6373: 6305: 6168: 6000: 5809: 5379: 5304: 5195: 4909: 4434: 4210: 4104: 3768: 3732: 3525: 3326: 3077: 3057: 3037: 3017: 2997: 2977: 2939: 2895: 2848: 2828: 2808: 2775: 2743: 2692: 2565: 2418: 2159: 1930: 1736: 1596: 1281: 1049: 783: 571: 546: 523: 497: 464: 322: 3211: 3182: 3173: 3160: 803:) than success point and two more arcs (those of 684:. To express the uncertainty about the value of 6195:. This means that the information contained in 4115:An approximate analytical expression for large 2627:To evaluate the "complete ignorance" case when 2429:The same calculation can be performed with the 688:, we need to select a fraction of the circle. 154:is a formula introduced in the 18th century by 1316:is the number of trials (we are using capital 5842:, we simply require its expectation. Letting 5395:Generalization to any number of possibilities 583:Historical application to the sunrise problem 353:Generalization to any number of possibilities 8: 6587:, Cambridge, UK, Cambridge University Press. 6019: = 2, as a generalisation should. 5824:on the next observation, conditional on the 2639:can be dealt with, we first go back to the 1320:to denote a random variable and lower-case 603:investigated a probability-based theory of 6035:categories into 2. Simply add up the 925:" (i.e., marginal) probability measure of 655:defining a non-blue arc. The estimate for 6441:Probability Theory: The Logic of Science. 6345: 6343: 6289: 6284: 6279: 6268: 6186:, whereas the generalisation is based on 6137: 6125: 6112: 6093: 6084: 6079: 6071: 5969: 5962: 5950: 5937: 5918: 5909: 5903: 5891: 5795: 5771: 5761: 5750: 5730: 5725: 5720: 5705: 5700: 5695: 5673: 5657: 5646: 5620: 5607: 5596: 5581: 5579: 5571: 5553: 5534: 5521: 5502: 5490: 5367: 5359: 5287: 5273: 5232: 5222: 5169: 5153: 5126: 5096: 5089: 5053: 5020: 5004: 4998: 4996: 4986: 4952: 4942: 4932: 4876: 4850: 4822: 4804: 4792: 4787: 4766: 4745: 4733: 4705: 4700: 4686: 4668: 4649: 4637: 4632: 4608: 4590: 4571: 4559: 4554: 4504: 4493: 4486: 4474: 4463: 4455: 4453: 4421: 4415: 4400: 4387: 4368: 4358: 4346: 4318: 4313: 4273: 4262: 4246: 4235: 4229: 4196: 4144: 4133: 4127: 4060: 4049: 4042: 4030: 4019: 3980: 3969: 3953: 3942: 3935: 3925: 3887: 3863: 3852: 3838: 3804: 3794: 3784: 3756: 3748: 3688: 3677: 3670: 3658: 3647: 3605: 3594: 3587: 3555: 3544: 3484: 3473: 3466: 3404: 3372: 3361: 3220: 3210: 3181: 3179: 3172: 3159: 3157: 3130: 3104: 3093: 3070: 3050: 3030: 3010: 2990: 2952: 2910: 2908: 2866: 2864: 2841: 2821: 2796: 2788: 2756: 2751:can be derived as a limiting form, where 2724: 2707: 2705: 2667: 2650: 2648: 2588:) distribution, which is not proper when 2550: 2518: 2512: 2499: 2474: 2457: 2387: 2355: 2349: 2336: 2285: 2279: 2266: 2241: 2229: 2128: 2118: 2109: 2096: 2077: 2064: 2039: 2034: 1998: 1992: 1979: 1958: 1913: 1891: 1837: 1825: 1812: 1793: 1780: 1762: 1684: 1674: 1662: 1640: 1630: 1625: 1619: 1584: 1572: 1550: 1540: 1535: 1517: 1495: 1488: 1475: 1445: 1440: 1407: 1395: 1382: 1363: 1350: 1332: 1267: 1245: 1230: 1219: 1195: 1190: 1180: 1169: 1147: 1134: 1115: 1102: 1075: 1026: 997: 974: 961: 944: 771: 769: 559: 539: 534:If the number of observations increases, 510: 484: 476: 449: 431: 412: 387: 370: 291: 273: 254: 229: 217: 134:Learn how and when to remove this message 6585:Probability Theory: The Logic of Science 6512:Philosophy and Phenomenological Research 6435: 6433: 5401:Probability Theory: The Logic of Science 663:) more than blue points divided by two ( 6505:"On the Application of Inductive Logic" 6420: 2693:{\displaystyle \mathrm {Hyp} (s|N,n,S)} 2776:{\displaystyle N,S\rightarrow \infty } 857:be 1 if we observe a "success" on the 343: + 2 observations (known as 5441:denote the probability that category 3045:, and then dividing this estimate by 2744:{\displaystyle \mathrm {Bin} (r|n,p)} 1751:for more on integrals of this form). 7: 6374:{\displaystyle {\frac {s+0.5}{n+1}}} 6326:, which then gives a pseudocount of 6204:is different from that contained in 5454:denote the number of times category 2204:is conditional on the observed data 869:of success on each trial. Thus each 72:adding citations to reliable sources 6405:Krichevsky–Trofimov estimator 6022:Because the propositions or events 1064:under our observations, we use the 739:Given this circle, the estimate of 166:Statement of the rule of succession 5663: 5584: 3186: 3164: 2770: 2714: 2711: 2708: 2657: 2654: 2651: 2433:that expresses total ignorance of 2317: 1960: 711:moving clockwise will be equal to 25: 3743:So the posterior expectation for 1312:is the number of "successes" and 755:defines the first blue arc while 5319: = 10, and we observe 2940:{\displaystyle {1 \over S(N-S)}} 2896:{\displaystyle {1 \over p(1-p)}} 2816:remains fixed. One can think of 865:, otherwise 0, with probability 48: 6056:denote the sum of the relevant 5859: = 1, ..., 5743: 5462: = 1, ..., 2978:{\displaystyle 1\leq S\leq N-1} 2783:in such a way that their ratio 2188:is being treated as if it is a 795:is the number of blue arcs and 59:needs additional citations for 6330:to the denominator instead of 6300: 6285: 6276: 6131: 6085: 6076: 5956: 5910: 5896: 5685: 5666: 5632: 5613: 5565: 5495: 5187: 5184: 5178: 5162: 5144: 5141: 5135: 5119: 5111: 5105: 5080: 5077: 5071: 5062: 5044: 5038: 4953: 4900: 4894: 4730: 4717: 4665: 4652: 4587: 4574: 4534: 4516: 4384: 4371: 4343: 4330: 4303: 4285: 4193: 4180: 4174: 4156: 4090: 4072: 4010: 3992: 3919: 3888: 3881: 3805: 3718: 3700: 3635: 3617: 3581: 3556: 3549: 3514: 3496: 3454: 3436: 3425: 3407: 3398: 3373: 3366: 3353: = 0 case, we have: 3315: 3312: 3300: 3285: 3279: 3267: 3264: 3252: 3241: 3229: 3151: 3139: 3124: 3105: 3098: 2931: 2919: 2887: 2875: 2767: 2738: 2725: 2718: 2687: 2668: 2661: 2544: 2467: 2381: 2323: 2311: 2234: 2115: 2051: 2024: 1966: 1910: 1897: 1878: 1866: 1852: 1840: 1831: 1767: 1725: 1713: 1705: 1693: 1659: 1646: 1569: 1556: 1514: 1501: 1472: 1466: 1460: 1454: 1431: 1425: 1419: 1413: 1401: 1337: 1264: 1251: 1216: 1203: 1159: 1095: 1086: 1080: 1060:For the likelihood of a given 955: 949: 784:{\displaystyle {\frac {b}{t}}} 609:New riddle of induction#Carnap 443: 380: 285: 222: 158:in the course of treating the 1: 6052:categories as "failure". Let 5874: + ... + 5380:{\displaystyle p={s \over n}} 3769:{\displaystyle p={S \over N}} 2809:{\displaystyle p={S \over N}} 1303: + ... + 719:trials can be interpreted as 174:times independently, and get 27:Formula in probability theory 6597:Bretthost, G. Larry (1988). 6443:Cambridge University Press. 6334:, and adds a pseudocount of 6048:categories as "success" and 3337:And it can be seen that, if 3025:is equivalent to estimating 2641:hypergeometric distribution 6644: 5445:will be observed, and let 747:arcs corresponding to the 36: 29: 6606:(PhD thesis). p. 55. 6410:Principle of indifference 2985:. Working conditional to 2178:conditionally independent 887:conditionally independent 204:conditionally independent 5422:multinomial distribution 2859:The equivalent prior to 2194:law of total probability 736:are highlighted in red. 651:defining a blue arc and 358:Nevertheless, if we had 933:over the open interval 18:Laplace–Bayes estimator 6623:Probability assessment 6503:Rudolf Carnap (1947). 6460:Rudolf Carnap (1945). 6375: 6307: 6170: 6002: 5811: 5766: 5662: 5612: 5426:Dirichlet distribution 5381: 5306: 5197: 4911: 4515: 4485: 4436: 4284: 4257: 4212: 4155: 4106: 4071: 4041: 3991: 3964: 3874: 3770: 3734: 3699: 3669: 3616: 3527: 3495: 3328: 3079: 3059: 3039: 3019: 3005:means that estimating 2999: 2979: 2941: 2897: 2850: 2830: 2810: 2777: 2745: 2694: 2567: 2420: 2161: 1932: 1738: 1598: 1283: 1185: 1051: 879:Bernoulli distribution 785: 672: 639:(in blue) is equal to 573: 548: 525: 499: 466: 324: 6583:Jaynes, E.T. (2003), 6469:Philosophy of Science 6376: 6308: 6171: 6003: 5812: 5746: 5642: 5592: 5382: 5307: 5198: 4912: 4489: 4459: 4437: 4258: 4231: 4213: 4129: 4107: 4045: 4015: 3965: 3938: 3848: 3771: 3735: 3673: 3643: 3590: 3528: 3469: 3329: 3080: 3060: 3040: 3020: 3000: 2980: 2942: 2898: 2851: 2831: 2811: 2778: 2746: 2695: 2568: 2445:)) for 0 ≤ 2441:(1 − 2421: 2162: 1933: 1739: 1599: 1284: 1165: 1052: 786: 622: 574: 549: 526: 500: 467: 325: 6462:"On Inductive Logic" 6342: 6267: 6070: 5890: 5489: 5358: 5221: 4931: 4452: 4228: 4126: 3783: 3747: 3543: 3360: 3092: 3069: 3065:. The posterior for 3049: 3029: 3009: 2989: 2951: 2907: 2863: 2840: 2820: 2787: 2755: 2704: 2647: 2456: 2228: 1957: 1761: 1618: 1609:normalizing constant 1331: 1074: 943: 931:uniform distribution 825:Mathematical details 768: 558: 538: 509: 483: 477:Mathematical details 369: 216: 185:More abstractly: If 156:Pierre-Simon Laplace 83:"Rule of succession" 68:improve this article 6628:Inductive reasoning 5737: 5712: 4803: 4716: 4648: 4570: 4329: 2947:, with a domain of 2584: − 2044: 1635: 1545: 1450: 1066:likelihood function 627:is the zero point, 605:inductive reasoning 524:{\displaystyle s=n} 498:{\displaystyle s=0} 32:Order of succession 6400:Additive smoothing 6371: 6303: 6250:In principle (see 6166: 5998: 5807: 5802: 5738: 5716: 5691: 5377: 5302: 5206:where the base 10 5193: 4907: 4905: 4783: 4696: 4628: 4550: 4432: 4309: 4208: 4102: 3766: 3730: 3523: 3324: 3075: 3055: 3035: 3015: 2995: 2975: 2937: 2893: 2846: 2826: 2806: 2773: 2741: 2690: 2631: = 0 or 2616: = 0 or 2604: = 0 or 2563: 2416: 2157: 2030: 1928: 1734: 1621: 1594: 1531: 1436: 1279: 1047: 1042: 781: 673: 572:{\displaystyle P'} 569: 544: 521: 495: 462: 320: 152:rule of succession 148:probability theory 6552:Unexpected Values 6449:978-0-521-59271-0 6369: 6292: 6282: 6161: 6082: 5993: 5798: 5689: 5473:, ..., 5375: 5327: = 10, 5300: 5240: 5191: 5148: 5084: 5048: 5013: 4994: 4950: 4922:natural logarithm 4858: 4838: 4817: 4684: 4606: 4541: 4430: 4410: 4100: 4097: 3933: 3846: 3802: 3764: 3728: 3725: 3521: 3461: 3322: 3209: 3171: 3155: 3085:can be given as: 3078:{\displaystyle S} 3058:{\displaystyle N} 3038:{\displaystyle S} 3018:{\displaystyle p} 2998:{\displaystyle N} 2935: 2891: 2849:{\displaystyle N} 2829:{\displaystyle S} 2804: 2558: 2521: 2411: 2358: 2288: 2152: 2001: 1943:beta distribution 1885: 1732: 1592: 1483: 1029: 1000: 977: 845:it were random). 779: 547:{\displaystyle P} 457: 315: 144: 143: 136: 118: 16:(Redirected from 6635: 6608: 6607: 6605: 6594: 6588: 6581: 6575: 6570: 6564: 6563: 6561: 6559: 6543: 6537: 6535: 6509: 6500: 6494: 6493:; here: p.86, 97 6492: 6466: 6457: 6451: 6437: 6428: 6425: 6390:analysis used). 6380: 6378: 6377: 6372: 6370: 6368: 6357: 6346: 6312: 6310: 6309: 6304: 6293: 6290: 6288: 6283: 6280: 6215:Further analysis 6175: 6173: 6172: 6167: 6162: 6160: 6149: 6138: 6130: 6129: 6117: 6116: 6098: 6097: 6088: 6083: 6080: 6007: 6005: 6004: 5999: 5994: 5992: 5981: 5974: 5973: 5963: 5955: 5954: 5942: 5941: 5923: 5922: 5913: 5908: 5907: 5816: 5814: 5813: 5808: 5806: 5805: 5799: 5796: 5786: 5776: 5775: 5765: 5760: 5739: 5736: 5735: 5734: 5724: 5711: 5710: 5709: 5699: 5690: 5688: 5678: 5677: 5661: 5656: 5640: 5639: 5635: 5625: 5624: 5611: 5606: 5582: 5558: 5557: 5539: 5538: 5526: 5525: 5507: 5506: 5386: 5384: 5383: 5378: 5376: 5368: 5335: = 5, 5311: 5309: 5308: 5303: 5301: 5299: 5288: 5283: 5279: 5278: 5277: 5241: 5233: 5202: 5200: 5199: 5194: 5192: 5190: 5174: 5173: 5154: 5149: 5147: 5131: 5130: 5114: 5101: 5100: 5090: 5085: 5083: 5054: 5049: 5047: 5031: 5030: 5014: 5009: 5008: 4999: 4997: 4995: 4987: 4982: 4978: 4956: 4951: 4943: 4920:where ln is the 4916: 4914: 4913: 4908: 4906: 4887: 4886: 4868: 4864: 4863: 4859: 4851: 4839: 4837: 4823: 4818: 4813: 4805: 4802: 4791: 4777: 4776: 4755: 4744: 4743: 4715: 4704: 4685: 4680: 4679: 4678: 4650: 4647: 4636: 4618: 4607: 4602: 4601: 4600: 4572: 4569: 4558: 4542: 4537: 4514: 4503: 4487: 4484: 4473: 4441: 4439: 4438: 4433: 4431: 4426: 4425: 4416: 4411: 4406: 4405: 4404: 4392: 4391: 4369: 4357: 4356: 4328: 4317: 4283: 4272: 4256: 4245: 4217: 4215: 4214: 4209: 4207: 4206: 4154: 4143: 4111: 4109: 4108: 4103: 4101: 4099: 4098: 4093: 4070: 4059: 4043: 4040: 4029: 4013: 3990: 3979: 3963: 3952: 3936: 3934: 3926: 3891: 3873: 3862: 3847: 3839: 3834: 3830: 3808: 3803: 3795: 3775: 3773: 3772: 3767: 3765: 3757: 3739: 3737: 3736: 3731: 3729: 3727: 3726: 3721: 3698: 3687: 3671: 3668: 3657: 3638: 3615: 3604: 3588: 3559: 3532: 3530: 3529: 3524: 3522: 3517: 3494: 3483: 3467: 3462: 3460: 3431: 3405: 3376: 3333: 3331: 3330: 3325: 3323: 3321: 3247: 3221: 3216: 3215: 3214: 3208: 3197: 3185: 3178: 3177: 3176: 3163: 3156: 3154: 3131: 3108: 3084: 3082: 3081: 3076: 3064: 3062: 3061: 3056: 3044: 3042: 3041: 3036: 3024: 3022: 3021: 3016: 3004: 3002: 3001: 2996: 2984: 2982: 2981: 2976: 2946: 2944: 2943: 2938: 2936: 2934: 2911: 2902: 2900: 2899: 2894: 2892: 2890: 2867: 2855: 2853: 2852: 2847: 2835: 2833: 2832: 2827: 2815: 2813: 2812: 2807: 2805: 2797: 2782: 2780: 2779: 2774: 2750: 2748: 2747: 2742: 2728: 2717: 2699: 2697: 2696: 2691: 2671: 2660: 2572: 2570: 2569: 2564: 2559: 2551: 2522: 2519: 2517: 2516: 2504: 2503: 2485: 2484: 2466: 2431:(improper) prior 2425: 2423: 2422: 2417: 2412: 2410: 2399: 2388: 2359: 2356: 2354: 2353: 2341: 2340: 2289: 2286: 2284: 2283: 2271: 2270: 2252: 2251: 2166: 2164: 2163: 2158: 2153: 2151: 2140: 2129: 2114: 2113: 2101: 2100: 2082: 2081: 2069: 2068: 2043: 2038: 2002: 1999: 1997: 1996: 1984: 1983: 1937: 1935: 1934: 1929: 1924: 1923: 1896: 1895: 1886: 1884: 1858: 1838: 1830: 1829: 1817: 1816: 1798: 1797: 1785: 1784: 1743: 1741: 1740: 1735: 1733: 1731: 1711: 1685: 1673: 1672: 1645: 1644: 1634: 1629: 1603: 1601: 1600: 1595: 1593: 1591: 1583: 1582: 1555: 1554: 1544: 1539: 1529: 1528: 1527: 1500: 1499: 1489: 1484: 1482: 1449: 1444: 1434: 1408: 1400: 1399: 1387: 1386: 1368: 1367: 1355: 1354: 1288: 1286: 1285: 1280: 1278: 1277: 1250: 1249: 1237: 1236: 1235: 1234: 1202: 1201: 1200: 1199: 1184: 1179: 1152: 1151: 1139: 1138: 1120: 1119: 1107: 1106: 1056: 1054: 1053: 1048: 1046: 1045: 1030: 1027: 1001: 998: 978: 975: 881:. Suppose these 873:is 0 or 1; each 820:ratio of counts. 790: 788: 787: 782: 780: 772: 578: 576: 575: 570: 568: 553: 551: 550: 545: 530: 528: 527: 522: 504: 502: 501: 496: 471: 469: 468: 463: 458: 450: 436: 435: 417: 416: 398: 397: 379: 329: 327: 326: 321: 316: 314: 303: 292: 278: 277: 259: 258: 240: 239: 207:random variables 139: 132: 128: 125: 119: 117: 76: 52: 44: 21: 6643: 6642: 6638: 6637: 6636: 6634: 6633: 6632: 6613: 6612: 6611: 6603: 6596: 6595: 6591: 6582: 6578: 6571: 6567: 6557: 6555: 6545: 6544: 6540: 6524:10.2307/2102920 6507: 6502: 6501: 6497: 6464: 6459: 6458: 6454: 6438: 6431: 6426: 6422: 6418: 6396: 6358: 6347: 6340: 6339: 6265: 6264: 6252:Cromwell's rule 6241: 6225:sunrise problem 6217: 6210: 6203: 6194: 6185: 6150: 6139: 6121: 6108: 6089: 6068: 6067: 6062: 6043: 6030: 5982: 5965: 5964: 5946: 5933: 5914: 5899: 5888: 5887: 5882: 5873: 5850: 5841: 5832: 5801: 5800: 5793: 5787: 5784: 5783: 5767: 5744: 5726: 5701: 5669: 5641: 5616: 5591: 5587: 5583: 5572: 5549: 5530: 5517: 5498: 5487: 5486: 5481: 5472: 5453: 5440: 5430:conjugate prior 5415: 5397: 5356: 5355: 5292: 5269: 5231: 5227: 5219: 5218: 5165: 5158: 5122: 5115: 5092: 5091: 5058: 5016: 5015: 5000: 4941: 4937: 4929: 4928: 4904: 4903: 4872: 4846: 4827: 4806: 4782: 4778: 4762: 4753: 4752: 4729: 4664: 4651: 4616: 4615: 4586: 4573: 4543: 4488: 4450: 4449: 4417: 4396: 4383: 4370: 4342: 4226: 4225: 4192: 4124: 4123: 4044: 4014: 3937: 3793: 3789: 3781: 3780: 3745: 3744: 3672: 3639: 3589: 3541: 3540: 3468: 3432: 3406: 3358: 3357: 3248: 3222: 3198: 3187: 3180: 3158: 3135: 3090: 3089: 3067: 3066: 3047: 3046: 3027: 3026: 3007: 3006: 2987: 2986: 2949: 2948: 2915: 2905: 2904: 2871: 2861: 2860: 2838: 2837: 2818: 2817: 2785: 2784: 2753: 2752: 2702: 2701: 2645: 2644: 2520: for 2508: 2495: 2470: 2459: 2454: 2453: 2400: 2389: 2357: for 2345: 2332: 2287: for 2275: 2262: 2237: 2226: 2225: 2212: 2190:random variable 2141: 2130: 2105: 2092: 2073: 2060: 2000: for 1988: 1975: 1955: 1954: 1909: 1887: 1859: 1839: 1821: 1808: 1789: 1776: 1759: 1758: 1712: 1686: 1658: 1636: 1616: 1615: 1568: 1546: 1530: 1513: 1491: 1490: 1435: 1409: 1391: 1378: 1359: 1346: 1329: 1328: 1311: 1302: 1263: 1241: 1226: 1215: 1191: 1186: 1143: 1130: 1111: 1098: 1072: 1071: 1041: 1040: 1024: 1018: 1017: 995: 989: 988: 972: 962: 941: 940: 912: 904:given the data 856: 829:The proportion 827: 766: 765: 699:corresponds to 643:. The value of 617: 585: 561: 556: 555: 536: 535: 507: 506: 481: 480: 427: 408: 383: 372: 367: 366: 336: 304: 293: 269: 250: 225: 214: 213: 201: 191: 178:successes, and 168: 160:sunrise problem 140: 129: 123: 120: 77: 75: 65: 53: 42: 39:Bayes estimator 35: 28: 23: 22: 15: 12: 11: 5: 6641: 6639: 6631: 6630: 6625: 6615: 6614: 6610: 6609: 6589: 6576: 6565: 6546:Neyman, Eric. 6538: 6518:(1): 133–148. 6495: 6481:10.1086/286851 6452: 6429: 6419: 6417: 6414: 6413: 6412: 6407: 6402: 6395: 6392: 6367: 6364: 6361: 6356: 6353: 6350: 6302: 6299: 6296: 6291:something else 6287: 6278: 6275: 6272: 6239: 6216: 6213: 6208: 6199: 6190: 6183: 6177: 6176: 6165: 6159: 6156: 6153: 6148: 6145: 6142: 6136: 6133: 6128: 6124: 6120: 6115: 6111: 6107: 6104: 6101: 6096: 6092: 6087: 6078: 6075: 6060: 6039: 6026: 6009: 6008: 5997: 5991: 5988: 5985: 5980: 5977: 5972: 5968: 5961: 5958: 5953: 5949: 5945: 5940: 5936: 5932: 5929: 5926: 5921: 5917: 5912: 5906: 5902: 5898: 5895: 5878: 5871: 5846: 5837: 5828: 5818: 5817: 5804: 5794: 5792: 5789: 5788: 5785: 5782: 5779: 5774: 5770: 5764: 5759: 5756: 5753: 5749: 5745: 5742: 5733: 5729: 5723: 5719: 5715: 5708: 5704: 5698: 5694: 5687: 5684: 5681: 5676: 5672: 5668: 5665: 5660: 5655: 5652: 5649: 5645: 5638: 5634: 5631: 5628: 5623: 5619: 5615: 5610: 5605: 5602: 5599: 5595: 5590: 5586: 5578: 5577: 5575: 5570: 5567: 5564: 5561: 5556: 5552: 5548: 5545: 5542: 5537: 5533: 5529: 5524: 5520: 5516: 5513: 5510: 5505: 5501: 5497: 5494: 5477: 5470: 5449: 5436: 5411: 5396: 5393: 5374: 5371: 5366: 5363: 5313: 5312: 5298: 5295: 5291: 5286: 5282: 5276: 5272: 5268: 5265: 5262: 5259: 5256: 5253: 5250: 5247: 5244: 5239: 5236: 5230: 5226: 5204: 5203: 5189: 5186: 5183: 5180: 5177: 5172: 5168: 5164: 5161: 5157: 5152: 5146: 5143: 5140: 5137: 5134: 5129: 5125: 5121: 5118: 5113: 5110: 5107: 5104: 5099: 5095: 5088: 5082: 5079: 5076: 5073: 5070: 5067: 5064: 5061: 5057: 5052: 5046: 5043: 5040: 5037: 5034: 5029: 5026: 5023: 5019: 5012: 5007: 5003: 4993: 4990: 4985: 4981: 4977: 4974: 4971: 4968: 4965: 4962: 4959: 4955: 4949: 4946: 4940: 4936: 4918: 4917: 4902: 4899: 4896: 4893: 4890: 4885: 4882: 4879: 4875: 4871: 4867: 4862: 4857: 4854: 4849: 4845: 4842: 4836: 4833: 4830: 4826: 4821: 4816: 4812: 4809: 4801: 4798: 4795: 4790: 4786: 4781: 4775: 4772: 4769: 4765: 4761: 4758: 4756: 4754: 4751: 4748: 4742: 4739: 4736: 4732: 4728: 4725: 4722: 4719: 4714: 4711: 4708: 4703: 4699: 4695: 4692: 4689: 4683: 4677: 4674: 4671: 4667: 4663: 4660: 4657: 4654: 4646: 4643: 4640: 4635: 4631: 4627: 4624: 4621: 4619: 4617: 4614: 4611: 4605: 4599: 4596: 4593: 4589: 4585: 4582: 4579: 4576: 4568: 4565: 4562: 4557: 4553: 4549: 4546: 4544: 4540: 4536: 4533: 4530: 4527: 4524: 4521: 4518: 4513: 4510: 4507: 4502: 4499: 4496: 4492: 4483: 4480: 4477: 4472: 4469: 4466: 4462: 4458: 4457: 4443: 4442: 4429: 4424: 4420: 4414: 4409: 4403: 4399: 4395: 4390: 4386: 4382: 4379: 4376: 4373: 4367: 4364: 4361: 4355: 4352: 4349: 4345: 4341: 4338: 4335: 4332: 4327: 4324: 4321: 4316: 4312: 4308: 4305: 4302: 4299: 4296: 4293: 4290: 4287: 4282: 4279: 4276: 4271: 4268: 4265: 4261: 4255: 4252: 4249: 4244: 4241: 4238: 4234: 4219: 4218: 4205: 4202: 4199: 4195: 4191: 4188: 4185: 4182: 4179: 4176: 4173: 4170: 4167: 4164: 4161: 4158: 4153: 4150: 4147: 4142: 4139: 4136: 4132: 4113: 4112: 4096: 4092: 4089: 4086: 4083: 4080: 4077: 4074: 4069: 4066: 4063: 4058: 4055: 4052: 4048: 4039: 4036: 4033: 4028: 4025: 4022: 4018: 4012: 4009: 4006: 4003: 4000: 3997: 3994: 3989: 3986: 3983: 3978: 3975: 3972: 3968: 3962: 3959: 3956: 3951: 3948: 3945: 3941: 3932: 3929: 3924: 3921: 3918: 3915: 3912: 3909: 3906: 3903: 3900: 3897: 3894: 3890: 3886: 3883: 3880: 3877: 3872: 3869: 3866: 3861: 3858: 3855: 3851: 3845: 3842: 3837: 3833: 3829: 3826: 3823: 3820: 3817: 3814: 3811: 3807: 3801: 3798: 3792: 3788: 3763: 3760: 3755: 3752: 3741: 3740: 3724: 3720: 3717: 3714: 3711: 3708: 3705: 3702: 3697: 3694: 3691: 3686: 3683: 3680: 3676: 3667: 3664: 3661: 3656: 3653: 3650: 3646: 3642: 3637: 3634: 3631: 3628: 3625: 3622: 3619: 3614: 3611: 3608: 3603: 3600: 3597: 3593: 3586: 3583: 3580: 3577: 3574: 3571: 3568: 3565: 3562: 3558: 3554: 3551: 3548: 3534: 3533: 3520: 3516: 3513: 3510: 3507: 3504: 3501: 3498: 3493: 3490: 3487: 3482: 3479: 3476: 3472: 3465: 3459: 3456: 3453: 3450: 3447: 3444: 3441: 3438: 3435: 3430: 3427: 3424: 3421: 3418: 3415: 3412: 3409: 3403: 3400: 3397: 3394: 3391: 3388: 3385: 3382: 3379: 3375: 3371: 3368: 3365: 3335: 3334: 3320: 3317: 3314: 3311: 3308: 3305: 3302: 3299: 3296: 3293: 3290: 3287: 3284: 3281: 3278: 3275: 3272: 3269: 3266: 3263: 3260: 3257: 3254: 3251: 3246: 3243: 3240: 3237: 3234: 3231: 3228: 3225: 3219: 3213: 3207: 3204: 3201: 3196: 3193: 3190: 3184: 3175: 3170: 3167: 3162: 3153: 3150: 3147: 3144: 3141: 3138: 3134: 3129: 3126: 3123: 3120: 3117: 3114: 3111: 3107: 3103: 3100: 3097: 3074: 3054: 3034: 3014: 2994: 2974: 2971: 2968: 2965: 2962: 2959: 2956: 2933: 2930: 2927: 2924: 2921: 2918: 2914: 2889: 2886: 2883: 2880: 2877: 2874: 2870: 2845: 2825: 2803: 2800: 2795: 2792: 2772: 2769: 2766: 2763: 2760: 2740: 2737: 2734: 2731: 2727: 2723: 2720: 2716: 2713: 2710: 2689: 2686: 2683: 2680: 2677: 2674: 2670: 2666: 2663: 2659: 2656: 2653: 2574: 2573: 2562: 2557: 2554: 2549: 2546: 2543: 2540: 2537: 2534: 2531: 2528: 2525: 2515: 2511: 2507: 2502: 2498: 2494: 2491: 2488: 2483: 2480: 2477: 2473: 2469: 2465: 2462: 2427: 2426: 2415: 2409: 2406: 2403: 2398: 2395: 2392: 2386: 2383: 2380: 2377: 2374: 2371: 2368: 2365: 2362: 2352: 2348: 2344: 2339: 2335: 2331: 2328: 2325: 2322: 2319: 2316: 2313: 2310: 2307: 2304: 2301: 2298: 2295: 2292: 2282: 2278: 2274: 2269: 2265: 2261: 2258: 2255: 2250: 2247: 2244: 2240: 2236: 2233: 2208: 2170: 2169: 2168: 2167: 2156: 2150: 2147: 2144: 2139: 2136: 2133: 2127: 2124: 2121: 2117: 2112: 2108: 2104: 2099: 2095: 2091: 2088: 2085: 2080: 2076: 2072: 2067: 2063: 2059: 2056: 2053: 2050: 2047: 2042: 2037: 2033: 2029: 2026: 2023: 2020: 2017: 2014: 2011: 2008: 2005: 1995: 1991: 1987: 1982: 1978: 1974: 1971: 1968: 1965: 1962: 1947:expected value 1939: 1938: 1927: 1922: 1919: 1916: 1912: 1908: 1905: 1902: 1899: 1894: 1890: 1883: 1880: 1877: 1874: 1871: 1868: 1865: 1862: 1857: 1854: 1851: 1848: 1845: 1842: 1836: 1833: 1828: 1824: 1820: 1815: 1811: 1807: 1804: 1801: 1796: 1792: 1788: 1783: 1779: 1775: 1772: 1769: 1766: 1745: 1744: 1730: 1727: 1724: 1721: 1718: 1715: 1710: 1707: 1704: 1701: 1698: 1695: 1692: 1689: 1683: 1680: 1677: 1671: 1668: 1665: 1661: 1657: 1654: 1651: 1648: 1643: 1639: 1633: 1628: 1624: 1605: 1604: 1590: 1587: 1581: 1578: 1575: 1571: 1567: 1564: 1561: 1558: 1553: 1549: 1543: 1538: 1534: 1526: 1523: 1520: 1516: 1512: 1509: 1506: 1503: 1498: 1494: 1487: 1481: 1478: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1453: 1448: 1443: 1439: 1433: 1430: 1427: 1424: 1421: 1418: 1415: 1412: 1406: 1403: 1398: 1394: 1390: 1385: 1381: 1377: 1374: 1371: 1366: 1362: 1358: 1353: 1349: 1345: 1342: 1339: 1336: 1307: 1300: 1290: 1289: 1276: 1273: 1270: 1266: 1262: 1259: 1256: 1253: 1248: 1244: 1240: 1233: 1229: 1225: 1222: 1218: 1214: 1211: 1208: 1205: 1198: 1194: 1189: 1183: 1178: 1175: 1172: 1168: 1164: 1161: 1158: 1155: 1150: 1146: 1142: 1137: 1133: 1129: 1126: 1123: 1118: 1114: 1110: 1105: 1101: 1097: 1094: 1091: 1088: 1085: 1082: 1079: 1058: 1057: 1044: 1039: 1036: 1033: 1025: 1023: 1020: 1019: 1016: 1013: 1010: 1007: 1004: 996: 994: 991: 990: 987: 984: 981: 973: 971: 968: 967: 965: 960: 957: 954: 951: 948: 929:we assigned a 908: 898:Bayes' theorem 852: 826: 823: 778: 775: 616: 613: 599:In the 1940s, 584: 581: 567: 564: 543: 520: 517: 514: 494: 491: 488: 473: 472: 461: 456: 453: 448: 445: 442: 439: 434: 430: 426: 423: 420: 415: 411: 407: 404: 401: 396: 393: 390: 386: 382: 378: 375: 335: 334:Interpretation 332: 331: 330: 319: 313: 310: 307: 302: 299: 296: 290: 287: 284: 281: 276: 272: 268: 265: 262: 257: 253: 249: 246: 243: 238: 235: 232: 228: 224: 221: 196: 189: 167: 164: 142: 141: 56: 54: 47: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 6640: 6629: 6626: 6624: 6621: 6620: 6618: 6602: 6601: 6593: 6590: 6586: 6580: 6577: 6574: 6569: 6566: 6554:. Eric Neyman 6553: 6549: 6542: 6539: 6536:; here: p.145 6533: 6529: 6525: 6521: 6517: 6513: 6506: 6499: 6496: 6490: 6486: 6482: 6478: 6474: 6470: 6463: 6456: 6453: 6450: 6446: 6442: 6436: 6434: 6430: 6424: 6421: 6415: 6411: 6408: 6406: 6403: 6401: 6398: 6397: 6393: 6391: 6388: 6382: 6365: 6362: 6359: 6354: 6351: 6348: 6337: 6333: 6329: 6325: 6320: 6314: 6297: 6294: 6273: 6270: 6261: 6257: 6253: 6248: 6245: 6238: 6232: 6228: 6226: 6222: 6214: 6212: 6207: 6202: 6198: 6193: 6189: 6182: 6163: 6157: 6154: 6151: 6146: 6143: 6140: 6134: 6126: 6122: 6118: 6113: 6109: 6105: 6102: 6099: 6094: 6090: 6073: 6066: 6065: 6064: 6059: 6055: 6051: 6047: 6042: 6038: 6034: 6029: 6025: 6020: 6018: 6014: 5995: 5989: 5986: 5983: 5978: 5975: 5970: 5966: 5959: 5951: 5947: 5943: 5938: 5934: 5930: 5927: 5924: 5919: 5915: 5904: 5900: 5893: 5886: 5885: 5884: 5881: 5877: 5870: 5867: = 5866: 5862: 5858: 5854: 5849: 5845: 5840: 5836: 5831: 5827: 5823: 5790: 5780: 5777: 5772: 5768: 5762: 5757: 5754: 5751: 5747: 5740: 5731: 5727: 5721: 5717: 5713: 5706: 5702: 5696: 5692: 5682: 5679: 5674: 5670: 5658: 5653: 5650: 5647: 5643: 5636: 5629: 5626: 5621: 5617: 5608: 5603: 5600: 5597: 5593: 5588: 5573: 5568: 5562: 5559: 5554: 5550: 5546: 5543: 5540: 5535: 5531: 5527: 5522: 5518: 5514: 5511: 5508: 5503: 5499: 5492: 5485: 5484: 5483: 5482:is given by: 5480: 5476: 5469: 5465: 5461: 5457: 5452: 5448: 5444: 5439: 5435: 5431: 5427: 5423: 5417: 5414: 5410: 5404: 5402: 5394: 5392: 5390: 5372: 5369: 5364: 5361: 5353: 5349: 5345: 5340: 5338: 5334: 5330: 5326: 5322: 5318: 5296: 5293: 5289: 5284: 5280: 5274: 5270: 5266: 5263: 5260: 5257: 5254: 5251: 5248: 5245: 5242: 5237: 5234: 5228: 5224: 5217: 5216: 5215: 5213: 5209: 5181: 5175: 5170: 5166: 5159: 5155: 5150: 5138: 5132: 5127: 5123: 5116: 5108: 5102: 5097: 5093: 5086: 5074: 5068: 5065: 5059: 5055: 5050: 5041: 5035: 5032: 5027: 5024: 5021: 5017: 5010: 5005: 5001: 4991: 4988: 4983: 4979: 4975: 4972: 4969: 4966: 4963: 4960: 4957: 4947: 4944: 4938: 4934: 4927: 4926: 4925: 4923: 4897: 4891: 4888: 4883: 4880: 4877: 4873: 4869: 4865: 4860: 4855: 4852: 4847: 4843: 4840: 4834: 4831: 4828: 4824: 4819: 4814: 4810: 4807: 4799: 4796: 4793: 4788: 4784: 4779: 4773: 4770: 4767: 4763: 4759: 4757: 4749: 4746: 4740: 4737: 4734: 4726: 4723: 4720: 4712: 4709: 4706: 4701: 4697: 4693: 4690: 4687: 4681: 4675: 4672: 4669: 4661: 4658: 4655: 4644: 4641: 4638: 4633: 4629: 4625: 4622: 4620: 4612: 4609: 4603: 4597: 4594: 4591: 4583: 4580: 4577: 4566: 4563: 4560: 4555: 4551: 4547: 4545: 4538: 4531: 4528: 4525: 4522: 4519: 4511: 4508: 4505: 4500: 4497: 4494: 4490: 4481: 4478: 4475: 4470: 4467: 4464: 4460: 4448: 4447: 4446: 4427: 4422: 4418: 4412: 4407: 4401: 4397: 4393: 4388: 4380: 4377: 4374: 4365: 4362: 4359: 4353: 4350: 4347: 4339: 4336: 4333: 4325: 4322: 4319: 4314: 4310: 4306: 4300: 4297: 4294: 4291: 4288: 4280: 4277: 4274: 4269: 4266: 4263: 4259: 4253: 4250: 4247: 4242: 4239: 4236: 4232: 4224: 4223: 4222: 4203: 4200: 4197: 4189: 4186: 4183: 4177: 4171: 4168: 4165: 4162: 4159: 4151: 4148: 4145: 4140: 4137: 4134: 4130: 4122: 4121: 4120: 4118: 4094: 4087: 4084: 4081: 4078: 4075: 4067: 4064: 4061: 4056: 4053: 4050: 4046: 4037: 4034: 4031: 4026: 4023: 4020: 4016: 4007: 4004: 4001: 3998: 3995: 3987: 3984: 3981: 3976: 3973: 3970: 3966: 3960: 3957: 3954: 3949: 3946: 3943: 3939: 3930: 3927: 3922: 3916: 3913: 3910: 3907: 3904: 3901: 3898: 3895: 3892: 3884: 3878: 3875: 3870: 3867: 3864: 3859: 3856: 3853: 3849: 3843: 3840: 3835: 3831: 3827: 3824: 3821: 3818: 3815: 3812: 3809: 3799: 3796: 3790: 3786: 3779: 3778: 3777: 3761: 3758: 3753: 3750: 3722: 3715: 3712: 3709: 3706: 3703: 3695: 3692: 3689: 3684: 3681: 3678: 3674: 3665: 3662: 3659: 3654: 3651: 3648: 3644: 3640: 3632: 3629: 3626: 3623: 3620: 3612: 3609: 3606: 3601: 3598: 3595: 3591: 3584: 3578: 3575: 3572: 3569: 3566: 3563: 3560: 3552: 3546: 3539: 3538: 3537: 3518: 3511: 3508: 3505: 3502: 3499: 3491: 3488: 3485: 3480: 3477: 3474: 3470: 3463: 3457: 3451: 3448: 3445: 3442: 3439: 3433: 3428: 3422: 3419: 3416: 3413: 3410: 3401: 3395: 3392: 3389: 3386: 3383: 3380: 3377: 3369: 3363: 3356: 3355: 3354: 3352: 3348: 3344: 3341: = 3340: 3318: 3309: 3306: 3303: 3297: 3294: 3291: 3288: 3282: 3276: 3273: 3270: 3261: 3258: 3255: 3249: 3244: 3238: 3235: 3232: 3226: 3223: 3217: 3205: 3202: 3199: 3194: 3191: 3188: 3168: 3165: 3148: 3145: 3142: 3136: 3132: 3127: 3121: 3118: 3115: 3112: 3109: 3101: 3095: 3088: 3087: 3086: 3072: 3052: 3032: 3012: 2992: 2972: 2969: 2966: 2963: 2960: 2957: 2954: 2928: 2925: 2922: 2916: 2912: 2884: 2881: 2878: 2872: 2868: 2857: 2843: 2823: 2801: 2798: 2793: 2790: 2764: 2761: 2758: 2735: 2732: 2729: 2721: 2684: 2681: 2678: 2675: 2672: 2664: 2643:, denoted by 2642: 2638: 2635: = 2634: 2630: 2625: 2623: 2620: = 2619: 2615: 2611: 2608: = 2607: 2603: 2599: 2595: 2592: = 2591: 2587: 2583: 2579: 2560: 2555: 2552: 2547: 2541: 2538: 2535: 2532: 2529: 2526: 2523: 2513: 2509: 2505: 2500: 2496: 2492: 2489: 2486: 2481: 2478: 2475: 2471: 2463: 2460: 2452: 2451: 2450: 2448: 2444: 2440: 2436: 2432: 2413: 2407: 2404: 2401: 2396: 2393: 2390: 2384: 2378: 2375: 2372: 2369: 2366: 2363: 2360: 2350: 2346: 2342: 2337: 2333: 2329: 2326: 2320: 2314: 2308: 2305: 2302: 2299: 2296: 2293: 2290: 2280: 2276: 2272: 2267: 2263: 2259: 2256: 2253: 2248: 2245: 2242: 2238: 2231: 2224: 2223: 2222: 2220: 2216: 2211: 2207: 2203: 2199: 2195: 2191: 2187: 2183: 2179: 2175: 2154: 2148: 2145: 2142: 2137: 2134: 2131: 2125: 2122: 2119: 2110: 2106: 2102: 2097: 2093: 2089: 2086: 2083: 2078: 2074: 2070: 2065: 2061: 2057: 2054: 2048: 2045: 2040: 2035: 2031: 2027: 2021: 2018: 2015: 2012: 2009: 2006: 2003: 1993: 1989: 1985: 1980: 1976: 1972: 1969: 1963: 1953: 1952: 1951: 1950: 1949: 1948: 1944: 1925: 1920: 1917: 1914: 1906: 1903: 1900: 1892: 1888: 1881: 1875: 1872: 1869: 1863: 1860: 1855: 1849: 1846: 1843: 1834: 1826: 1822: 1818: 1813: 1809: 1805: 1802: 1799: 1794: 1790: 1786: 1781: 1777: 1773: 1770: 1764: 1757: 1756: 1755: 1752: 1750: 1749:beta function 1728: 1722: 1719: 1716: 1708: 1702: 1699: 1696: 1690: 1687: 1681: 1678: 1675: 1669: 1666: 1663: 1655: 1652: 1649: 1641: 1637: 1631: 1626: 1622: 1614: 1613: 1612: 1610: 1588: 1585: 1579: 1576: 1573: 1565: 1562: 1559: 1551: 1547: 1541: 1536: 1532: 1524: 1521: 1518: 1510: 1507: 1504: 1496: 1492: 1485: 1479: 1476: 1469: 1463: 1457: 1451: 1446: 1441: 1437: 1428: 1422: 1416: 1410: 1404: 1396: 1392: 1388: 1383: 1379: 1375: 1372: 1369: 1364: 1360: 1356: 1351: 1347: 1343: 1340: 1334: 1327: 1326: 1325: 1323: 1319: 1315: 1310: 1306: 1299: 1296: = 1295: 1274: 1271: 1268: 1260: 1257: 1254: 1246: 1242: 1238: 1231: 1227: 1223: 1220: 1212: 1209: 1206: 1196: 1192: 1187: 1181: 1176: 1173: 1170: 1166: 1162: 1156: 1153: 1148: 1144: 1140: 1135: 1131: 1127: 1124: 1121: 1116: 1112: 1108: 1103: 1099: 1092: 1089: 1083: 1077: 1070: 1069: 1068: 1067: 1063: 1037: 1034: 1031: 1021: 1014: 1011: 1008: 1005: 1002: 992: 985: 982: 979: 969: 963: 958: 952: 946: 939: 938: 937: 936: 932: 928: 924: 920: 916: 911: 907: 903: 899: 894: 892: 888: 884: 880: 876: 872: 868: 864: 860: 855: 851: 846: 844: 840: 836: 832: 824: 822: 821: 816: 814: 810: 806: 802: 798: 794: 776: 773: 763: 758: 754: 750: 746: 742: 737: 735: 731: 727: 722: 718: 714: 710: 706: 702: 698: 694: 689: 687: 683: 679: 670: 666: 662: 659:is then one ( 658: 654: 650: 646: 642: 638: 634: 630: 626: 621: 614: 612: 610: 606: 602: 601:Rudolf Carnap 597: 593: 589: 582: 580: 565: 562: 541: 532: 518: 515: 512: 492: 489: 486: 478: 459: 454: 451: 446: 440: 437: 432: 428: 424: 421: 418: 413: 409: 405: 402: 399: 394: 391: 388: 384: 376: 373: 365: 364: 363: 361: 356: 354: 350: 346: 342: 333: 317: 311: 308: 305: 300: 297: 294: 288: 282: 279: 274: 270: 266: 263: 260: 255: 251: 247: 244: 241: 236: 233: 230: 226: 219: 212: 211: 210: 208: 205: 199: 195: 188: 183: 181: 177: 173: 165: 163: 161: 157: 153: 149: 138: 135: 127: 124:February 2017 116: 113: 109: 106: 102: 99: 95: 92: 88: 85: – 84: 80: 79:Find sources: 73: 69: 63: 62: 57:This article 55: 51: 46: 45: 40: 33: 19: 6599: 6592: 6584: 6579: 6568: 6556:. Retrieved 6551: 6541: 6515: 6511: 6498: 6475:(2): 72–97. 6472: 6468: 6455: 6440: 6423: 6387:pseudocounts 6383: 6335: 6331: 6327: 6323: 6318: 6315: 6259: 6249: 6243: 6236: 6233: 6229: 6218: 6205: 6200: 6196: 6191: 6187: 6180: 6178: 6057: 6053: 6049: 6045: 6040: 6036: 6032: 6027: 6023: 6021: 6016: 6012: 6010: 5879: 5875: 5868: 5864: 5860: 5856: 5852: 5847: 5843: 5838: 5834: 5829: 5825: 5821: 5819: 5478: 5474: 5467: 5463: 5459: 5455: 5450: 5446: 5442: 5437: 5433: 5418: 5412: 5408: 5405: 5400: 5398: 5388: 5351: 5347: 5343: 5341: 5336: 5332: 5328: 5324: 5320: 5316: 5314: 5211: 5205: 4919: 4444: 4220: 4116: 4114: 3742: 3535: 3350: 3346: 3342: 3338: 3336: 2858: 2636: 2632: 2628: 2626: 2621: 2617: 2613: 2609: 2605: 2601: 2597: 2593: 2589: 2585: 2581: 2577: 2575: 2446: 2442: 2438: 2434: 2428: 2218: 2214: 2209: 2205: 2201: 2197: 2185: 2181: 2173: 2171: 1940: 1753: 1746: 1606: 1321: 1317: 1313: 1308: 1304: 1297: 1293: 1291: 1061: 1059: 934: 926: 918: 914: 909: 905: 901: 895: 890: 882: 874: 870: 866: 858: 853: 849: 847: 842: 838: 834: 830: 828: 818: 817: 812: 808: 804: 800: 796: 792: 761: 756: 752: 748: 744: 740: 738: 733: 729: 725: 720: 716: 712: 708: 704: 700: 696: 692: 690: 685: 681: 677: 674: 668: 664: 660: 656: 652: 648: 644: 640: 636: 632: 628: 624: 598: 594: 590: 586: 533: 474: 359: 357: 348: 345:pseudocounts 340: 337: 197: 193: 186: 184: 179: 175: 171: 169: 151: 145: 130: 121: 111: 104: 97: 90: 78: 66:Please help 61:verification 58: 5863:), and let 1607:To get the 896:We can use 6617:Categories 6416:References 6260:absolutely 6256:Bayes rule 5797:otherwise. 2221:, we have 2217:= 1, ..., 1941:This is a 1611:, we find 917:= 1, ..., 623:The point 588:tomorrow. 94:newspapers 6103:… 5928:… 5748:∑ 5714:⋯ 5664:Γ 5644:∏ 5594:∑ 5585:Γ 5544:… 5528:∣ 5512:… 5285:≈ 5243:∣ 5208:logarithm 5176:⁡ 5133:⁡ 5103:⁡ 5069:⁡ 5036:⁡ 5025:− 4984:≈ 4892:⁡ 4881:− 4870:≈ 4832:− 4820:− 4797:− 4785:∫ 4771:− 4738:− 4724:− 4710:− 4698:∫ 4694:− 4673:− 4659:− 4642:− 4630:∫ 4595:− 4581:− 4564:− 4552:∫ 4548:≈ 4529:− 4523:− 4509:− 4491:∏ 4479:− 4461:∑ 4413:≈ 4394:− 4378:− 4351:− 4337:− 4323:− 4311:∫ 4307:≈ 4298:− 4292:− 4278:− 4260:∏ 4251:− 4233:∑ 4201:− 4187:− 4178:≈ 4169:− 4163:− 4149:− 4131:∏ 4085:− 4079:− 4065:− 4047:∏ 4035:− 4017:∑ 4005:− 3999:− 3985:− 3967:∏ 3958:− 3940:∑ 3868:− 3850:∑ 3713:− 3707:− 3693:− 3675:∏ 3663:− 3645:∑ 3630:− 3624:− 3610:− 3592:∏ 3509:− 3503:− 3489:− 3471:∏ 3449:− 3443:− 3420:− 3414:− 3402:∝ 3307:− 3298:− 3292:− 3274:− 3259:− 3236:− 3218:∝ 3203:− 3192:− 3146:− 3128:∝ 2970:− 2964:≤ 2958:≤ 2926:− 2882:− 2771:∞ 2768:→ 2536:… 2493:∣ 2373:… 2330:∣ 2321:⁡ 2303:… 2260:∣ 2087:… 2058:∣ 2032:∫ 2016:… 1973:∣ 1964:⁡ 1918:− 1904:− 1873:− 1803:… 1774:∣ 1700:− 1667:− 1653:− 1623:∫ 1577:− 1563:− 1533:∫ 1522:− 1508:− 1438:∫ 1373:… 1344:∣ 1272:− 1258:− 1224:− 1210:− 1167:∏ 1154:∣ 1125:… 1035:≥ 1028:for 999:for 983:≤ 976:for 921:For the " 615:Intuition 422:⋯ 406:∣ 355:, below. 264:⋯ 248:∣ 6558:13 April 6489:14481246 6394:See also 5833:is just 5290:0.434294 5156:0.434294 2464:′ 2200:. Since 566:′ 475:But see 377:′ 6532:2102920 6223:on the 6221:Laplace 6081:success 843:p as if 811:) than 347:) with 192:, ..., 108:scholar 6530: 6487: 6447: 6258:takes 2192:, the 2172:Since 1292:where 889:given 885:s are 877:has a 791:where 715:. The 150:, the 110: 103: 96: 89: 81: 6604:(PDF) 6528:JSTOR 6508:(PDF) 6485:S2CID 6465:(PDF) 5346:, or 2184:. As 1945:with 1747:(see 935:(0,1) 923:prior 863:trial 505:, or 180:n − s 115:JSTOR 101:books 6560:2023 6445:ISBN 6281:data 3776:is: 2213:for 1012:< 1006:< 848:Let 807:and 732:and 667:and 554:and 202:are 87:news 6520:doi 6477:doi 6355:0.5 6050:m-c 5389:s=n 5167:log 5124:log 5094:log 3345:or 2903:is 2596:or 861:th 764:is 749:n+2 745:n+2 707:to 635:to 360:not 146:In 70:by 6619:: 6550:. 6526:. 6514:. 6510:. 6483:. 6473:12 6471:. 6467:. 6432:^ 6381:. 5403:. 5271:10 5212:10 5171:10 5128:10 5098:10 5066:ln 5033:ln 4889:ln 2856:. 919:n. 913:, 893:. 611:. 531:. 349:s 200:+1 6562:. 6534:. 6522:: 6516:8 6491:. 6479:: 6366:1 6363:+ 6360:n 6352:+ 6349:s 6336:m 6332:m 6328:1 6324:m 6319:m 6301:) 6298:I 6295:, 6286:| 6277:( 6274:r 6271:P 6244:m 6240:m 6237:I 6209:2 6206:I 6201:m 6197:I 6192:m 6188:I 6184:2 6181:I 6164:, 6158:m 6155:+ 6152:n 6147:c 6144:+ 6141:s 6135:= 6132:) 6127:m 6123:I 6119:, 6114:m 6110:n 6106:, 6100:, 6095:1 6091:n 6086:| 6077:( 6074:P 6061:i 6058:n 6054:s 6046:c 6041:i 6037:A 6033:m 6028:i 6024:A 6017:m 6013:n 5996:. 5990:m 5987:+ 5984:n 5979:1 5976:+ 5971:i 5967:n 5960:= 5957:) 5952:m 5948:I 5944:, 5939:m 5935:n 5931:, 5925:, 5920:1 5916:n 5911:| 5905:i 5901:A 5897:( 5894:P 5880:m 5876:n 5872:1 5869:n 5865:n 5861:m 5857:i 5855:( 5853:i 5848:i 5844:A 5839:i 5835:p 5830:i 5826:p 5822:i 5791:0 5781:1 5778:= 5773:i 5769:p 5763:m 5758:1 5755:= 5752:i 5741:, 5732:m 5728:n 5722:m 5718:p 5707:1 5703:n 5697:1 5693:p 5686:) 5683:1 5680:+ 5675:i 5671:n 5667:( 5659:m 5654:1 5651:= 5648:i 5637:) 5633:) 5630:1 5627:+ 5622:i 5618:n 5614:( 5609:m 5604:1 5601:= 5598:i 5589:( 5574:{ 5569:= 5566:) 5563:I 5560:, 5555:m 5551:n 5547:, 5541:, 5536:1 5532:n 5523:m 5519:p 5515:, 5509:, 5504:1 5500:p 5496:( 5493:f 5479:m 5475:p 5471:1 5468:p 5464:m 5460:i 5458:( 5456:i 5451:i 5447:n 5443:i 5438:i 5434:p 5413:m 5409:I 5373:n 5370:s 5365:= 5362:p 5352:N 5348:k 5344:N 5337:k 5333:n 5329:k 5325:n 5321:n 5317:k 5297:k 5294:n 5281:) 5275:k 5267:= 5264:N 5261:, 5258:0 5255:= 5252:s 5249:, 5246:n 5238:N 5235:S 5229:( 5225:E 5188:] 5185:) 5182:N 5179:( 5163:[ 5160:n 5151:= 5145:] 5142:) 5139:N 5136:( 5120:[ 5117:n 5112:) 5109:e 5106:( 5087:= 5081:] 5078:) 5075:N 5072:( 5063:[ 5060:n 5056:1 5051:= 5045:) 5042:N 5039:( 5028:1 5022:n 5018:N 5011:n 5006:n 5002:N 4992:N 4989:1 4980:) 4976:N 4973:, 4970:0 4967:= 4964:s 4961:, 4958:n 4954:| 4948:N 4945:S 4939:( 4935:E 4901:) 4898:N 4895:( 4884:1 4878:n 4874:N 4866:] 4861:) 4856:N 4853:1 4848:( 4844:O 4841:+ 4835:1 4829:n 4825:1 4815:R 4811:R 4808:d 4800:n 4794:N 4789:1 4780:[ 4774:1 4768:n 4764:N 4760:= 4750:R 4747:d 4741:2 4735:n 4731:) 4727:R 4721:N 4718:( 4713:n 4707:N 4702:1 4691:R 4688:d 4682:R 4676:2 4670:n 4666:) 4662:R 4656:N 4653:( 4645:n 4639:N 4634:1 4626:N 4623:= 4613:R 4610:d 4604:R 4598:1 4592:n 4588:) 4584:R 4578:N 4575:( 4567:n 4561:N 4556:1 4539:R 4535:) 4532:j 4526:R 4520:N 4517:( 4512:1 4506:n 4501:1 4498:= 4495:j 4482:n 4476:N 4471:1 4468:= 4465:R 4428:n 4423:n 4419:N 4408:n 4402:n 4398:n 4389:n 4385:) 4381:1 4375:N 4372:( 4366:= 4363:S 4360:d 4354:1 4348:n 4344:) 4340:S 4334:N 4331:( 4326:n 4320:N 4315:1 4304:) 4301:j 4295:S 4289:N 4286:( 4281:1 4275:n 4270:1 4267:= 4264:j 4254:n 4248:N 4243:1 4240:= 4237:S 4204:1 4198:n 4194:) 4190:R 4184:N 4181:( 4175:) 4172:j 4166:R 4160:N 4157:( 4152:1 4146:n 4141:1 4138:= 4135:j 4117:N 4095:R 4091:) 4088:j 4082:R 4076:N 4073:( 4068:1 4062:n 4057:1 4054:= 4051:j 4038:n 4032:N 4027:1 4024:= 4021:R 4011:) 4008:j 4002:S 3996:N 3993:( 3988:1 3982:n 3977:1 3974:= 3971:j 3961:n 3955:N 3950:1 3947:= 3944:S 3931:N 3928:1 3923:= 3920:) 3917:0 3914:= 3911:s 3908:, 3905:1 3902:= 3899:n 3896:, 3893:N 3889:| 3885:S 3882:( 3879:P 3876:S 3871:n 3865:N 3860:1 3857:= 3854:S 3844:N 3841:1 3836:= 3832:) 3828:N 3825:, 3822:0 3819:= 3816:s 3813:, 3810:n 3806:| 3800:N 3797:S 3791:( 3787:E 3762:N 3759:S 3754:= 3751:p 3723:R 3719:) 3716:j 3710:R 3704:N 3701:( 3696:1 3690:n 3685:1 3682:= 3679:j 3666:n 3660:N 3655:1 3652:= 3649:R 3641:S 3636:) 3633:j 3627:S 3621:N 3618:( 3613:1 3607:n 3602:1 3599:= 3596:j 3585:= 3582:) 3579:0 3576:= 3573:s 3570:, 3567:n 3564:, 3561:N 3557:| 3553:S 3550:( 3547:P 3519:S 3515:) 3512:j 3506:S 3500:N 3497:( 3492:1 3486:n 3481:1 3478:= 3475:j 3464:= 3458:! 3455:) 3452:n 3446:S 3440:N 3437:( 3434:S 3429:! 3426:) 3423:1 3417:S 3411:N 3408:( 3399:) 3396:0 3393:= 3390:s 3387:, 3384:n 3381:, 3378:N 3374:| 3370:S 3367:( 3364:P 3351:s 3347:s 3343:n 3339:s 3319:! 3316:) 3313:] 3310:s 3304:n 3301:[ 3295:S 3289:N 3286:( 3283:! 3280:) 3277:s 3271:S 3268:( 3265:) 3262:S 3256:N 3253:( 3250:S 3245:! 3242:) 3239:S 3233:N 3230:( 3227:! 3224:S 3212:) 3206:s 3200:n 3195:S 3189:N 3183:( 3174:) 3169:s 3166:S 3161:( 3152:) 3149:S 3143:N 3140:( 3137:S 3133:1 3125:) 3122:s 3119:, 3116:n 3113:, 3110:N 3106:| 3102:S 3099:( 3096:P 3073:S 3053:N 3033:S 3013:p 2993:N 2973:1 2967:N 2961:S 2955:1 2932:) 2929:S 2923:N 2920:( 2917:S 2913:1 2888:) 2885:p 2879:1 2876:( 2873:p 2869:1 2844:N 2824:S 2802:N 2799:S 2794:= 2791:p 2765:S 2762:, 2759:N 2739:) 2736:p 2733:, 2730:n 2726:| 2722:r 2719:( 2715:n 2712:i 2709:B 2688:) 2685:S 2682:, 2679:n 2676:, 2673:N 2669:| 2665:s 2662:( 2658:p 2655:y 2652:H 2637:n 2633:s 2629:s 2622:n 2618:s 2614:s 2610:n 2606:s 2602:s 2598:s 2594:n 2590:s 2586:s 2582:n 2580:, 2578:s 2561:. 2556:n 2553:s 2548:= 2545:) 2542:n 2539:, 2533:, 2530:1 2527:= 2524:i 2514:i 2510:x 2506:= 2501:i 2497:X 2490:1 2487:= 2482:1 2479:+ 2476:n 2472:X 2468:( 2461:P 2447:p 2443:p 2439:p 2435:p 2414:. 2408:2 2405:+ 2402:n 2397:1 2394:+ 2391:s 2385:= 2382:) 2379:n 2376:, 2370:, 2367:1 2364:= 2361:i 2351:i 2347:x 2343:= 2338:i 2334:X 2327:p 2324:( 2318:E 2315:= 2312:) 2309:n 2306:, 2300:, 2297:1 2294:= 2291:i 2281:i 2277:x 2273:= 2268:i 2264:X 2257:1 2254:= 2249:1 2246:+ 2243:n 2239:X 2235:( 2232:P 2219:n 2215:i 2210:i 2206:X 2202:p 2198:p 2186:p 2182:p 2174:p 2155:. 2149:2 2146:+ 2143:n 2138:1 2135:+ 2132:s 2126:= 2123:p 2120:d 2116:) 2111:n 2107:x 2103:= 2098:n 2094:X 2090:, 2084:, 2079:1 2075:x 2071:= 2066:1 2062:X 2055:p 2052:( 2049:f 2046:p 2041:1 2036:0 2028:= 2025:) 2022:n 2019:, 2013:, 2010:1 2007:= 2004:i 1994:i 1990:x 1986:= 1981:i 1977:X 1970:p 1967:( 1961:E 1926:. 1921:s 1915:n 1911:) 1907:p 1901:1 1898:( 1893:s 1889:p 1882:! 1879:) 1876:s 1870:n 1867:( 1864:! 1861:s 1856:! 1853:) 1850:1 1847:+ 1844:n 1841:( 1835:= 1832:) 1827:n 1823:x 1819:= 1814:n 1810:X 1806:, 1800:, 1795:1 1791:x 1787:= 1782:1 1778:X 1771:p 1768:( 1765:f 1729:! 1726:) 1723:1 1720:+ 1717:n 1714:( 1709:! 1706:) 1703:s 1697:n 1694:( 1691:! 1688:s 1682:= 1679:r 1676:d 1670:s 1664:n 1660:) 1656:r 1650:1 1647:( 1642:s 1638:r 1632:1 1627:0 1589:r 1586:d 1580:s 1574:n 1570:) 1566:r 1560:1 1557:( 1552:s 1548:r 1542:1 1537:0 1525:s 1519:n 1515:) 1511:p 1505:1 1502:( 1497:s 1493:p 1486:= 1480:r 1477:d 1473:) 1470:r 1467:( 1464:f 1461:) 1458:r 1455:( 1452:L 1447:1 1442:0 1432:) 1429:p 1426:( 1423:f 1420:) 1417:p 1414:( 1411:L 1405:= 1402:) 1397:n 1393:x 1389:= 1384:n 1380:X 1376:, 1370:, 1365:1 1361:x 1357:= 1352:1 1348:X 1341:p 1338:( 1335:f 1322:x 1318:X 1314:n 1309:n 1305:x 1301:1 1298:x 1294:s 1275:s 1269:n 1265:) 1261:p 1255:1 1252:( 1247:s 1243:p 1239:= 1232:i 1228:x 1221:1 1217:) 1213:p 1207:1 1204:( 1197:i 1193:x 1188:p 1182:n 1177:1 1174:= 1171:i 1163:= 1160:) 1157:p 1149:n 1145:x 1141:= 1136:n 1132:X 1128:, 1122:, 1117:1 1113:x 1109:= 1104:1 1100:X 1096:( 1093:P 1090:= 1087:) 1084:p 1081:( 1078:L 1062:p 1038:1 1032:p 1022:0 1015:1 1009:p 1003:0 993:1 986:0 980:p 970:0 964:{ 959:= 956:) 953:p 950:( 947:f 927:p 915:i 910:i 906:X 902:p 891:p 883:X 875:X 871:X 867:p 859:i 854:i 850:X 839:p 835:p 831:p 813:n 809:Z 805:P 801:Z 797:t 793:b 777:t 774:b 762:p 757:P 753:Z 741:p 734:Z 730:P 726:p 721:n 717:n 713:p 709:P 705:Z 701:p 697:P 693:Z 686:p 682:p 678:p 669:P 665:Z 661:Z 657:p 653:P 649:Z 645:p 641:p 637:P 633:Z 629:P 625:Z 563:P 542:P 519:n 516:= 513:s 493:0 490:= 487:s 460:. 455:n 452:s 447:= 444:) 441:s 438:= 433:n 429:X 425:+ 419:+ 414:1 410:X 403:1 400:= 395:1 392:+ 389:n 385:X 381:( 374:P 341:n 318:. 312:2 309:+ 306:n 301:1 298:+ 295:s 289:= 286:) 283:s 280:= 275:n 271:X 267:+ 261:+ 256:1 252:X 245:1 242:= 237:1 234:+ 231:n 227:X 223:( 220:P 198:n 194:X 190:1 187:X 176:s 172:n 137:) 131:( 126:) 122:( 112:· 105:· 98:· 91:· 64:. 41:. 34:. 20:)

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