Prior probability - Knowledge (XXG)

2884:(e.g. the number of times a die is thrown) to the total number of events—and these considered purely deductively, i.e. without any experimenting. In the case of the die if we look at it on the table without throwing it, each elementary event is reasoned deductively to have the same probability—thus the probability of each outcome of an imaginary throwing of the (perfect) die or simply by counting the number of faces is 1/6. Each face of the die appears with equal probability—probability being a measure defined for each elementary event. The result is different if we throw the die twenty times and ask how many times (out of 20) the number 6 appears on the upper face. In this case time comes into play and we have a different type of probability depending on time or the number of times the die is thrown. On the other hand, the a priori probability is independent of time—you can look at the die on the table as long as you like without touching it and you deduce the probability for the number 6 to appear on the upper face is 1/6. 594:) = 1/3 seems intuitively like the only reasonable choice. More formally, we can see that the problem remains the same if we swap around the labels ("A", "B" and "C") of the cups. It would therefore be odd to choose a prior for which a permutation of the labels would cause a change in our predictions about which cup the ball will be found under; the uniform prior is the only one which preserves this invariance. If one accepts this invariance principle then one can see that the uniform prior is the logically correct prior to represent this state of knowledge. This prior is "objective" in the sense of being the correct choice to represent a particular state of knowledge, but it is not objective in the sense of being an observer-independent feature of the world: in reality the ball exists under a particular cup, and it only makes sense to speak of probabilities in this situation if there is an observer with limited knowledge about the system. 558:, i.e. probability distributions in some sense logically required by the nature of one's state of uncertainty; these are a subject of philosophical controversy, with Bayesians being roughly divided into two schools: "objective Bayesians", who believe such priors exist in many useful situations, and "subjective Bayesians" who believe that in practice priors usually represent subjective judgements of opinion that cannot be rigorously justified (Williamson 2010). Perhaps the strongest arguments for objective Bayesianism were given by 279: 2794: > 0) which would suggest that any value for the mean is "equally likely" and that a value for the positive variance becomes "less likely" in inverse proportion to its value. Many authors (Lindley, 1973; De Groot, 1937; Kass and Wasserman, 1996) warn against the danger of over-interpreting those priors since they are not probability densities. The only relevance they have is found in the corresponding posterior, as long as it is well-defined for all observations. (The 745:, one finds the distribution that is least informative in the sense that it contains the least amount of information consistent with the constraints that define the set. For example, the maximum entropy prior on a discrete space, given only that the probability is normalized to 1, is the prior that assigns equal probability to each state. And in the continuous case, the maximum entropy prior given that the density is normalized with mean zero and unit variance is the standard 3997:. It is for this system that one postulates in quantum statistics the "fundamental postulate of equal a priori probabilities of an isolated system." This says that the isolated system in equilibrium occupies each of its accessible states with the same probability. This fundamental postulate therefore allows us to equate the a priori probability to the degeneracy of a system, i.e. to the number of different states with the same energy. 2750:. If the summation in the denominator converges, the posterior probabilities will still sum (or integrate) to 1 even if the prior values do not, and so the priors may only need to be specified in the correct proportion. Taking this idea further, in many cases the sum or integral of the prior values may not even need to be finite to get sensible answers for the posterior probabilities. When this is the case, the prior is called an 2385:

occurs where the two distributions in the logarithm argument, improper or not, do not diverge. This in turn occurs when the prior distribution is proportional to the square root of the Fisher information of the likelihood function. Hence in the single parameter case, reference priors and Jeffreys priors are identical, even though Jeffreys has a very different rationale.

47: 551:, which assigns equal probabilities to all possibilities. In parameter estimation problems, the use of an uninformative prior typically yields results which are not too different from conventional statistical analysis, as the likelihood function often yields more information than the uninformative prior. 2473:

Practical problems associated with uninformative priors include the requirement that the posterior distribution be proper. The usual uninformative priors on continuous, unbounded variables are improper. This need not be a problem if the posterior distribution is proper. Another issue of importance is

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which leaves invariant our Bayesian state of knowledge. This can be seen as a generalisation of the invariance principle used to justify the uniform prior over the three cups in the example above. For example, in physics we might expect that an experiment will give the same results regardless of our

790:) is the "least informative" prior about X. The reference prior is defined in the asymptotic limit, i.e., one considers the limit of the priors so obtained as the number of data points goes to infinity. In the present case, the KL divergence between the prior and posterior distributions is given by 510:

expresses partial information about a variable, steering the analysis toward solutions that align with existing knowledge without overly constraining the results and preventing extreme estimates. An example is, when setting the prior distribution for the temperature at noon tomorrow in St. Louis, to

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do not need to be integrated, and a likelihood function that is uniformly 1 corresponds to the absence of data (all models are equally likely, given no data): Bayes' rule multiplies a prior by the likelihood, and an empty product is just the constant likelihood 1. However, without starting with a

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This is a quasi-KL divergence ("quasi" in the sense that the square root of the Fisher information may be the kernel of an improper distribution). Due to the minus sign, we need to minimise this in order to maximise the KL divergence with which we started. The minimum value of the last equation

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Philosophical problems associated with uninformative priors are associated with the choice of an appropriate metric, or measurement scale. Suppose we want a prior for the running speed of a runner who is unknown to us. We could specify, say, a normal distribution as the prior for his speed, but

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use a normal distribution with mean 50 degrees Fahrenheit and standard deviation 40 degrees, which very loosely constrains the temperature to the range (10 degrees, 90 degrees) with a small chance of being below -30 degrees or above 130 degrees. The purpose of a weakly informative prior is for

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is a preceding assumption, theory, concept or idea upon which, after taking account of new information, a current assumption, theory, concept or idea is founded. A strong prior is a type of informative prior in which the information contained in the prior distribution dominates the information

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accumulates, the posterior is determined largely by the evidence rather than any original assumption, provided that the original assumption admitted the possibility of what the evidence is suggesting. The terms "prior" and "posterior" are generally relative to a specific datum or observation.

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This example has a property in common with many priors, namely, that the posterior from one problem (today's temperature) becomes the prior for another problem (tomorrow's temperature); pre-existing evidence which has already been taken into account is part of the prior and, as more evidence

5692: 3228:. In order to understand this quantity as giving a number of states in quantum (i.e. wave) mechanics, recall that in quantum mechanics every particle is associated with a matter wave which is the solution of a Schrödinger equation. In the case of free particles (of energy 713:. Similarly, some measurements are naturally invariant to the choice of an arbitrary scale (e.g., whether centimeters or inches are used, the physical results should be equal). In such a case, the scale group is the natural group structure, and the corresponding prior on 3663:, i.e. the time independence of this phase space volume element and thus of the a priori probability. A time dependence of this quantity would imply known information about the dynamics of the system, and hence would not be an a priori probability. Thus the region 3766: 661:, indicating that the sample will either dissolve every time or never dissolve, with equal probability. However, if one has observed samples of the chemical to dissolve in one experiment and not to dissolve in another experiment then this prior is updated to the 5191: 740:

of a probability distribution measures the amount of information contained in the distribution. The larger the entropy, the less information is provided by the distribution. Thus, by maximizing the entropy over a suitable set of probability distributions on

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of the individual gas elements (atoms or molecules) are finite in the phase space spanned by these coordinates. In analogy to the case of the die, the a priori probability is here (in the case of a continuum) proportional to the phase space volume element

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Similarly, if asked to estimate an unknown proportion between 0 and 1, we might say that all proportions are equally likely, and use a uniform prior. Alternatively, we might say that all orders of magnitude for the proportion are equally likely, the

5871:. An important aspect in the derivation is the taking into account of the indistinguishability of particles and states in quantum statistics, i.e. there particles and states do not have labels. In the case of fermions, like electrons, obeying the 4190: 334:

before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a

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alternatively we could specify a normal prior for the time he takes to complete 100 metres, which is proportional to the reciprocal of the first prior. These are very different priors, but it is not clear which is to be preferred. Jaynes'

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As an example of an a priori prior, due to Jaynes (2003), consider a situation in which one knows a ball has been hidden under one of three cups, A, B, or C, but no other information is available about its location. In this case a

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etc. appear here suddenly? Above no mention was made of electric or other fields. Thus with no such fields present we have the Fermi-Dirac distribution as above. But with such fields present we have this additional dependence of

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As a more contentious example, Jaynes published an argument based on the invariance of the prior under a change of parameters that suggests that the prior representing complete uncertainty about a probability should be the

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Thus the a priori weighting in the classical context (a) corresponds to the a priori weighting here in the quantal context (b). In the case of the one-dimensional simple harmonic oscillator of natural frequency

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Uninformative priors can express "objective" information such as "the variable is positive" or "the variable is less than some limit". The simplest and oldest rule for determining a non-informative prior is the

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to the data set consisting of one observation of dissolving and one of not dissolving, using the above prior. The Haldane prior is an improper prior distribution (meaning that it has an infinite mass).

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under the adopted loss function. Unfortunately, admissibility is often difficult to check, although some results are known (e.g., Berger and Strawderman 1996). The issue is particularly acute with

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In the full quantum theory one has an analogous conservation law. In this case, the phase space region is replaced by a subspace of the space of states expressed in terms of a projection operator

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In the case of the hydrogen atom or Coulomb potential (where the evaluation of the phase space volume for constant energy is more complicated) one knows that the quantum mechanical degeneracy is

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is available. In these methods, either an information theory based criterion, such as KL divergence or log-likelihood function for binary supervised learning problems and mixture model problems.

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is constant under uniform conditions (as many particles as flow out of a volume element also flow in steadily, so that the situation in the element appears static), i.e. independent of time

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Esfahani, M. S.; Dougherty, E. R. (2014). "Incorporation of Biological Pathway Knowledge in the Construction of Priors for Optimal Bayesian Classification - IEEE Journals & Magazine".

3558: 2380: 3274: 2568: 3226: 2563: 1098: 609:). The example Jaynes gives is of finding a chemical in a lab and asking whether it will dissolve in water in repeated experiments. The Haldane prior gives by far the most weight to 5261: 5878: 5011: 4236: 3657: 2419:. Constructing objective priors have been recently introduced in bioinformatics, and specially inference in cancer systems biology, where sample size is limited and a vast amount of 5461: 5319: 3834: 1608: 3615: 1526:

Now we use the concept of entropy which, in the case of probability distributions, is the negative expected value of the logarithm of the probability mass or density function or

5735: 4844: 4790: 2190:. Indeed, the very idea goes against the philosophy of Bayesian inference in which 'true' values of parameters are replaced by prior and posterior distributions. So we remove 461:

expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a

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There are many ways to construct a prior distribution. In some cases, a prior may be determined from past information, such as previous experiments. A prior can also be

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attempts to solve this problem by computing a prior which expresses the same belief no matter which metric is used. The Jeffreys prior for an unknown proportion

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Jaynes (1968), pp. 17, see also Jaynes (2003), chapter 12. Note that chapter 12 is not available in the online preprint but can be previewed via Google Books.

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These functions are derived for (1) a system in dynamic equilibrium (i.e. under steady, uniform conditions) with (2) total (and huge) number of particles

721:. It sometimes matters whether we use the left-invariant or right-invariant Haar measure. For example, the left and right invariant Haar measures on the 4569:{\displaystyle \int _{0}^{\phi =2\pi }\int _{0}^{\theta =\pi }2I\pi E\sin \theta d\theta d\phi =8\pi ^{2}IE=\oint dp_{\theta }dp_{\phi }d\theta d\phi ,} 535:

expresses vague or general information about a variable. The term "uninformative prior" is somewhat of a misnomer. Such a prior might also be called a

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it can be taken out of the integral, and as this integral is over a probability space it equals one. Hence we can write the asymptotic form of KL as

5687:{\displaystyle f_{i}^{FD}={\frac {1}{e^{(\epsilon _{i}-\epsilon _{0})/kT}+1}},\quad f_{i}^{BE}={\frac {1}{e^{(\epsilon _{i}-\epsilon _{0})/kT}-1}}.} 182: 1893: 362:, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a 4849: 7353: 7193: 7166: 7085: 6889: 6762: 2880:

the a priori probability is used to describe the initial state of a system. The classical version is defined as the ratio of the number of

2754:. However, the posterior distribution need not be a proper distribution if the prior is improper. This is clear from the case where event 3660: 7364: 2498:; the usual priors (e.g., Jeffreys' prior) may give badly inadmissible decision rules if employed at the higher levels of the hierarchy. 4846:

for the rotating diatomic molecule. From wave mechanics it is known that the energy levels of a rotating diatomic molecule are given by

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Simpson, Daniel; et al. (2017). "Penalising Model Component Complexity: A Principled, Practical Approach to Constructing Priors".

3953:. In either case, the considerations assume a closed isolated system. This closed isolated system is a system with (1) a fixed energy 2428: 359: 755:

generalizes MAXENT to the case of "updating" an arbitrary prior distribution with suitable constraints in the maximum-entropy sense.

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The following example illustrates the a priori probability (or a priori weighting) in (a) classical and (b) quantal contexts.

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equal to the day-to-day variance of atmospheric temperature, or a distribution of the temperature for that day of the year.

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these states are indistinguishable (i.e. these states do not carry labels). An important consequence is a result known as

5501: 3761:{\displaystyle \Omega :={\frac {\Delta q\Delta p}{\int \Delta q\Delta p}},\;\;\;\int \Delta q\Delta p=\mathrm {const.} ,} 2876:

While in Bayesian statistics the prior probability is used to represent initial beliefs about an uncertain parameter, in

7427: 6345: 5186:{\displaystyle {\frac {dn}{dE_{n}}}={\frac {8\pi ^{2}I}{(2n+1)h^{2}}},\;\;\;(2n+1)dn={\frac {8\pi ^{2}I}{h^{2}}}dE_{n}.} 733: 512: 295: 187: 125: 6077: 3993:

in (c) a state of equilibrium. If one considers a huge number of replicas of this system, one obtains what is called a

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in "A note on inverse probability", Mathematical Proceedings of the Cambridge Philosophical Society 28, 55–61, 1932,

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would not be concerned with such issues, but it can be important in this situation. For example, one would want any

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Florens, Jean-Pierre; Mouchart, Michael; Rolin, Jean-Marie (1990). "Invariance Arguments in Bayesian Statistics".

2774:. For example, if they need a prior distribution for the mean and variance of a random variable, they may assume 400:

The prior distributions of model parameters will often depend on parameters of their own. Uncertainty about these

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Consider the rotational energy E of a diatomic molecule with moment of inertia I in spherical polar coordinates

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Dawid, A. P.; Stone, M.; Zidek, J. V. (1973). "Marginalization Paradoxes in Bayesian and Structural Inference".

6950:"Incorporating biological prior knowledge for Bayesian learning via maximal knowledge-driven information priors" 3231: 2412: 1036:. Splitting the logarithm into two parts, reversing the order of integrals in the second part and noting that 5351: 3174: 2509: 2495: 2491: 371: 331: 260: 172: 6854:. See also J. Haldane, "The precision of observed values of small frequencies", Biometrika, 35:297–300, 1948, 3949:

is the dimensionality of the subspace. The conservation law in this case is expressed by the unitarity of the

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are not equal. Berger (1985, p. 413) argues that the right-invariant Haar measure is the correct choice.

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of the posterior distribution relative to the prior. This maximizes the expected posterior information about

7289: 5427: 5287: 4365:{\displaystyle \oint dp_{\theta }dp_{\phi }=\pi {\sqrt {2IE}}{\sqrt {2IE}}\sin \theta =2\pi IE\sin \theta .} 2463: 2388:

Reference priors are often the objective prior of choice in multivariate problems, since other rules (e.g.,

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Bauwens, Luc; Lubrano, Michel; Richard, Jean-François (1999). "Prior Densities for the Regression Model".

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above, one finds that the approximate number of states in the range dE is given by the degeneracy, i.e.

4795: 4758: 4185:{\displaystyle E={\frac {1}{2I}}\left(p_{\theta }^{2}+{\frac {p_{\phi }^{2}}{\sin ^{2}\theta }}\right).} 2877: 2807: 54: 5324: 370:, for that it would result in a tractable posterior of the same family. The widespread availability of 278: 5386: 4729: 2965: 7311: 7217: 4677: 4605: 3618: 2483: 759: 697: 555: 381:

from the purely subjective assessment of an experienced expert. When no information is available, an

234: 115: 85: 5350:(no degeneracy). Thus in quantum mechanics the a priori probability is effectively a measure of the 6492:

Mikkola, Petrus; et al. (2023). "Prior Knowledge Elicitation: The Past, Present, and Future".

6003: 5847: 5740: 4063: 4017: 2865: 2811: 2802: 2771: 2713:{\displaystyle P(A_{i}\mid B)={\frac {P(B\mid A_{i})P(A_{i})}{\sum _{j}P(B\mid A_{j})P(A_{j})}}\,,} 2467: 1821:

is normal with a variance equal to the reciprocal of the Fisher information at the 'true' value of

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Boluki, Shahin; Esfahani, Mohammad Shahrokh; Qian, Xiaoning; Dougherty, Edward R (December 2017).

1274:{\displaystyle KL=\int p(t)\int p(x\mid t)\log\,dx\,dt\,-\,\int \log\,\int p(t)p(x\mid t)\,dt\,dx} 7327: 7301: 7265: 7025: 6930: 6863: 6735: 6690: 6672: 6633: 6515: 3120: 2396: 494:

combines the information contained in the prior with that extracted from the data to produce the

283: 208: 80: 5968: 5958:{\displaystyle 0\leq f_{i}^{FD}\leq 1,\quad {\text{whereas}}\quad 0\leq f_{i}^{BE}\leq \infty .} 955: 445:

In principle, priors can be decomposed into many conditional levels of distributions, so-called

389:. In modern applications, priors are also often chosen for their mechanical properties, such as 110: 6351: 7387: 7349: 7189: 7162: 7139: 7081: 7048: 6989: 6971: 6922: 6885: 6783: 6758: 6573: 6472: 6444: 6407: 3922:{\displaystyle \Sigma :={\frac {P}{{\text{Tr}}(P)}},\;\;\;N={\text{Tr}}(P)=\mathrm {const.} ,} 3057: 3014: 2834: 2795: 702: 666: 491: 409: 394: 213: 90: 5196: 4375: 4241: 3457: 3279: 1844: 498:

which, in the case of a "strong prior", would be little changed from the prior distribution.

7379: 7319: 7244: 7226: 7073: 6979: 6961: 6914: 6855: 6847: 6825: 6725: 6717: 6682: 6643: 6602: 6542: 6505: 6497: 6464: 6436: 2881: 1306: 105: 7277: 7240: 7203: 7176: 6177: 6130: 6050: 5820: 5359: 4395: 3425: 2937: 2910: 7339: 7273: 7248: 7236: 7199: 7172: 6806: 6467:(1996). "Elicitation of Prior Distributions". In Berry, Donald A.; Stangl, Dalene (eds.). 5872: 5267: 4628: 2487: 2236: 1341: 990: 737: 729: 671: 562:, based mainly on the consequences of symmetries and on the principle of maximum entropy. 559: 363: 340: 141: 6533:

Icazatti, Alejandro; Abril-Pla, Oriol; Klami, Arto; Martin, Osvaldo A. (September 2023).

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Iba, Y. (1989). "Bayesian Statistics and Statistical Mechanics". In Takayama, H. (ed.).

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Bayesian Ideas and Data Analysis : An Introduction for Scientists and Statisticians

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Price, Harold J.; Manson, Allison R. (2001). "Uninformative priors for Bayes' theorem".

6337:{\displaystyle {\frac {df_{i}}{dt}}=0,\quad f_{i}=f_{i}(t,{\bf {v}}_{i},{\bf {r}}_{i}).} 4579: 2193: 2170: 2167:

is proportional to the (asymptotically large) sample size. We do not know the value of

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then it is clear that the same result would be obtained if all the prior probabilities

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and taking the expected value of the normal entropy, which we obtain by multiplying by

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is the arbitrarily large sample size (to which Fisher information is proportional) and

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Bernardo, Jose M. (1979). "Reference Posterior Distributions for Bayesian Inference".

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is the same as at time zero. One describes this also as conservation of information.

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These functions, interpreted as uniform distributions, can also be interpreted as the

7421: 7154: 6739: 6622:"Sparsity information and regularization in the horseshoe and other shrinkage priors" 6519: 2400: 701:

choice of the origin of a coordinate system. This induces the group structure of the

599: 17: 7072:. Springer Series in Synergetics. Vol. 43. Berlin: Springer. pp. 235–236. 6934: 6694: 7331: 6417: 722: 689: 255: 903:{\displaystyle KL=\int p(t)\int p(x\mid t)\log {\frac {p(x\mid t)}{p(x)}}\,dx\,dt} 7077: 7043:

Christensen, Ronald; Johnson, Wesley; Branscum, Adam; Hanson, Timothy E. (2010).

3831:, and instead of the probability in phase space, one has the probability density 2395:

Objective prior distributions may also be derived from other principles, such as

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is a measure of the fraction of states actually occupied by electrons at energy

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distribution, and thus cannot integrate or compute expected values or loss. See

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Expressing this equation in terms of its partial derivatives, one obtains the

2140:{\displaystyle KL=-\log \left(1{\sqrt {kI(x^{*})}}\right)-\,\int p(x)\log\,dx} 1841:. The entropy of a normal density function is equal to half the logarithm of 405: 7383: 7231: 7212: 6975: 6829: 6753:

Congdon, Peter D. (2020). "Regression Techniques using Hierarchical Priors".

1517:{\displaystyle KL=\int p(t)\int p(x\mid t)\log\,dx\,dt\,-\,\int p(x)\log\,dx} 7213:"Choice of hierarchical priors: admissibility in estimation of normal means" 6402: 2857: 336: 6993: 6926: 3617:

by considering the area covered by these points. Moreover, in view of the

6918: 3097:(here for simplicity considered in one dimension). In 1 dimension (length 2887:

In statistical mechanics, e.g. that of a gas contained in a finite volume

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yields zero (with the help of Hamilton's equations): The volume at time

3415:{\displaystyle \psi \propto \sin(l\pi x/L)\sin(m\pi y/L)\sin(n\pi z/L),} 7269: 7029: 6867: 6721: 6708:

Fortuin, Vincent (2022). "Priors in Bayesian Deep Learning: A Review".

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is a measure of the number of wave mechanical states available. Hence

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be mutually exclusive and exhaustive. If Bayes' theorem is written as

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prescribes how to update the prior with new information to obtain the

1777:. In the limiting case where the sample size tends to infinity, the 1016:

distributions and the result is the weighted mean over all values of

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devised a systematic way for designing uninformative priors as e.g.,

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is a parameter of the underlying system (Bernoulli distribution), and

7411: 6859: 4792:, the number of states in the energy range dE is, as seen under (a) 2830:

on an infinite interval (i.e., a half-line or the entire real line).

2746:) were multiplied by a given constant; the same would be true for a 437:

are parameters of the prior distribution (beta distribution); hence

7365:"review of Bruno di Finetti. Philosophical Lectures on Probability" 7292:; Dongchu Sun (2009). "The formal definition of reference priors". 6638: 4412:

the total volume of phase space covered for constant energy E is

3011:, and is the number of standing waves (i.e. states) therein, where 7306: 6677: 5473:

In statistical mechanics (see any book) one derives the so-called

6907:

IEEE/ACM Transactions on Computational Biology and Bioinformatics

5875:(only one particle per state or none allowed), one has therefore 2443:, which is the uniform prior on the logarithm of proportion. The 952:. The inner integral is the KL divergence between the posterior 4755:

for each direction of motion is given, per element, by a factor

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and hence the classical a priori weighting in the energy range

6780:

Economic Decision-Making: Games, Econometrics and Optimisation

3117:) this number or statistical weight or a priori weighting is 2806:

prior probability distribution, one does not end up getting a

6568:(1971). "Prior Distributions to Represent 'Knowing Little'". 1890:

is the variance of the distribution. In this case therefore

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each such level being (2n+1)-fold degenerate. By evaluating

2478:, i.e., with many different data sets, it should have good 1283:

The inner integral in the second part is the integral over

1953:{\displaystyle H=\log {\sqrt {\frac {2\pi e}{NI(x^{*})}}}} 6757:(2nd ed.). Boca Raton: CRC Press. pp. 253–315. 6439:(1994). "From Prior Information to Prior Distributions". 4924:{\displaystyle E_{n}={\frac {n(n+1)h^{2}}{8\pi ^{2}I}},} 688:

Priors can be constructed which are proportional to the

7105:(2nd ed.). Singapore: World Scientific. Chapter 6. 709:, which determines the prior probability as a constant 4726:

Assuming that the number of quantum states in a range

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probability distributions. For example, if one uses a

6882:

Probability Matching Priors: Higher Order Asymptotics

6570:

An Introduction to Bayesian Inference in Econometrics

6379: 6354: 6227: 6207: 6180: 6160: 6133: 6080: 6053: 6033: 6006: 5971: 5881: 5850: 5823: 5770: 5743: 5700: 5510: 5482: 5430: 5389: 5362: 5354:, i.e. the number of states having the same energy. 5327: 5290: 5270: 5219: 5199: 5019: 4937: 4852: 4798: 4761: 4732: 4680: 4660: 4631: 4608: 4582: 4418: 4398: 4378: 4264: 4244: 4198: 4092: 4066: 4046: 4020: 3979: 3959: 3935: 3837: 3817: 3794: 3774: 3669: 3627: 3566: 3498: 3460: 3428: 3315: 3282: 3234: 3177: 3157: 3123: 3103: 3083: 3060: 3040: 3017: 2997: 2968: 2940: 2913: 2893: 2571: 2512: 2288: 2282:. This allows us to combine the logarithms yielding 2268: 2239: 2219: 2196: 2173: 2153: 2029: 2009: 1986: 1966: 1896: 1876: 1847: 1827: 1807: 1787: 1763: 1743: 1723: 1703: 1616: 1373: 1344: 1309: 1289: 1101: 1081: 1042: 1022: 993: 958: 938: 918: 796: 641: 615: 515:, that is, to keep inferences in a reasonable range. 7161:(2nd ed.). Boca Raton: Chapman & Hall/CRC. 6818:

IEEE Transactions on Systems Science and Cybernetics

2003:

is the 'true' value. Since this does not depend on

7414:

a collaborative database of models and their priors

6572:. New York: John Wiley & Sons. pp. 41–53. 3171:) the corresponding number can be calculated to be 2868:in the absence of data, but are not proper priors. 2470:as a basis for induction in very general settings. 374:methods, however, has made this less of a concern. 7258:Journal of the Royal Statistical Society, Series B 7211:Berger, James O.; Strawderman, William E. (1996). 6385: 6364: 6336: 6213: 6193: 6166: 6146: 6119: 6066: 6039: 6019: 5992: 5957: 5863: 5836: 5809: 5756: 5729: 5686: 5488: 5455: 5416: 5375: 5342: 5313: 5276: 5255: 5205: 5185: 5005: 4923: 4838: 4784: 4747: 4710: 4666: 4646: 4617: 4591: 4568: 4404: 4384: 4364: 4250: 4230: 4184: 4078: 4052: 4032: 3985: 3965: 3941: 3921: 3823: 3800: 3780: 3760: 3651: 3609: 3552: 3484: 3446: 3414: 3301: 3268: 3220: 3163: 3143: 3109: 3089: 3069: 3046: 3026: 3003: 2983: 2953: 2926: 2899: 2712: 2557: 2392:) may result in priors with problematic behavior. 2374: 2274: 2254: 2225: 2205: 2182: 2159: 2139: 2015: 1995: 1972: 1952: 1882: 1862: 1833: 1813: 1793: 1769: 1757:plus the marginal (i.e. unconditional) entropy of 1749: 1729: 1709: 1688:{\displaystyle KL=-\int p(t)H(x\mid t)\,dt+\,H(x)} 1687: 1602: 1516: 1359: 1330: 1295: 1273: 1087: 1067: 1028: 1008: 979: 944: 924: 902: 653: 627: 326:of an uncertain quantity, often simply called the 7186:Statistical decision theory and Bayesian analysis 1697:In words, KL is the negative expected value over 7136:Bayesian Inference in Dynamic Econometric Models 7070:Cooperative Dynamics in Complex Physical Systems 665:on the interval . This is obtained by applying 5810:{\displaystyle E=\Sigma _{i}n_{i}\epsilon _{i}} 2770:Statisticians sometimes use improper priors as 6047:. On the other hand, the a priori probability 3492:values and hence states in the region between 2459:), which differs from Jaynes' recommendation. 543:, i.e. one that is not subjectively elicited. 6880:Datta, Gauri Sankar; Mukerjee, Rahul (2004). 6801: 6799: 6560: 6558: 6471:. New York: Marcel Dekker. pp. 141–156. 5469:Priori probability and distribution functions 2474:that if an uninformative prior is to be used 2431:can answer this question in some situations. 932:is a sufficient statistic for some parameter 766:. Here, the idea is to maximize the expected 303: 8: 7138:. Oxford University Press. pp. 94–128. 2812:Likelihood function § Non-integrability 469:equal to today's noontime temperature, with 3553:{\displaystyle p,p+dp,p^{2}={\bf {p}}^{2},} 2375:{\displaystyle KL=-\int p(x)\log \left\,dx} 412:to model the distribution of the parameter 6535:"PreliZ: A tool-box for prior elicitation" 5107: 5106: 5105: 3872: 3871: 3870: 3716: 3715: 3714: 3269:{\displaystyle \epsilon ={\bf {p}}^{2}/2m} 2872:Prior probability in statistical mechanics 2490:based on the posterior distribution to be 2417:Solomonoff's theory of inductive inference 490:contained in the data being analyzed. The 310: 296: 29: 7305: 7230: 6983: 6965: 6729: 6676: 6647: 6637: 6546: 6509: 6378: 6356: 6355: 6353: 6322: 6316: 6315: 6305: 6299: 6298: 6282: 6269: 6238: 6228: 6226: 6206: 6185: 6179: 6159: 6138: 6132: 6108: 6098: 6085: 6079: 6058: 6052: 6032: 6011: 6005: 5981: 5976: 5970: 5937: 5932: 5916: 5897: 5892: 5880: 5855: 5849: 5828: 5822: 5801: 5791: 5781: 5769: 5748: 5742: 5721: 5711: 5699: 5659: 5650: 5637: 5629: 5619: 5607: 5602: 5572: 5563: 5550: 5542: 5532: 5520: 5515: 5509: 5481: 5441: 5429: 5408: 5399: 5388: 5367: 5361: 5326: 5303: 5289: 5269: 5218: 5198: 5174: 5159: 5145: 5135: 5093: 5060: 5050: 5038: 5020: 5018: 4989: 4983: 4968: 4956: 4944: 4936: 4906: 4891: 4866: 4857: 4851: 4830: 4821: 4806: 4797: 4774: 4760: 4731: 4690: 4685: 4679: 4659: 4630: 4607: 4581: 4545: 4532: 4507: 4452: 4447: 4428: 4423: 4417: 4397: 4377: 4313: 4300: 4288: 4275: 4263: 4243: 4219: 4206: 4197: 4159: 4148: 4143: 4137: 4128: 4123: 4099: 4091: 4065: 4045: 4019: 3978: 3958: 3934: 3896: 3879: 3850: 3844: 3836: 3816: 3793: 3773: 3768:when differentiated with respect to time 3735: 3676: 3668: 3626: 3601: 3592: 3580: 3565: 3560:is then found to be the above expression 3541: 3535: 3534: 3524: 3497: 3459: 3427: 3398: 3369: 3340: 3314: 3293: 3281: 3276:) like those of a gas in a box of volume 3255: 3249: 3243: 3242: 3233: 3221:{\displaystyle V4\pi p^{2}\Delta p/h^{3}} 3212: 3203: 3191: 3176: 3156: 3133: 3122: 3102: 3082: 3059: 3039: 3016: 2996: 2967: 2945: 2939: 2918: 2912: 2892: 2706: 2694: 2675: 2653: 2638: 2619: 2600: 2582: 2570: 2558:{\displaystyle A_{1},A_{2},\ldots ,A_{n}} 2549: 2530: 2517: 2511: 2365: 2326: 2287: 2267: 2238: 2218: 2195: 2172: 2152: 2130: 2090: 2071: 2056: 2028: 2008: 1985: 1965: 1937: 1909: 1895: 1875: 1846: 1826: 1806: 1801:conditional on a given observed value of 1786: 1762: 1742: 1722: 1702: 1672: 1662: 1615: 1590: 1531: 1507: 1467: 1463: 1456: 1449: 1372: 1343: 1308: 1288: 1264: 1257: 1223: 1195: 1191: 1184: 1177: 1100: 1080: 1046: 1041: 1021: 992: 957: 937: 917: 893: 886: 848: 795: 640: 614: 7345:Probability Theory: The Logic of Science 7018:Journal of the Royal Statistical Society 5737:(this condition determines the constant 5256:{\displaystyle \Sigma \propto (2n+1)dn.} 554:Some attempts have been made at finding 183:Integrated nested Laplace approximations 6443:. New York: Springer. pp. 89–136. 6428: 5496:for various statistics. In the case of 5006:{\displaystyle dn/dE_{n}=1/(dE_{n}/dn)} 4231:{\displaystyle (p_{\theta },p_{\phi })} 3652:{\displaystyle \Delta q\Delta p\geq h,} 3454:are integers. The number of different 1610:Using this in the last equation yields 247: 226: 200: 164: 133: 72: 37: 5456:{\displaystyle \Sigma \propto n^{2}dn} 5314:{\displaystyle \Omega \propto dE/\nu } 7157:; John B. Carlin; Stern, Hal (2003). 6620:Piironen, Juho; Vehtari, Aki (2017). 2822:Examples of improper priors include: 1603:{\textstyle H(x)=-\int p(x)\log\,dx.} 1338:. This is the marginal distribution 736:(MAXENT). The motivation is that the 685:) for the Bernoulli random variable. 27:Distribution of an uncertain quantity 7: 3973:and (2) a fixed number of particles 3151:. In customary 3 dimensions (volume 6782:. North-Holland. pp. 351–367. 3610:{\displaystyle V4\pi p^{2}dp/h^{3}} 385:may be adopted as justified by the 7047:. Hoboken: CRC Press. p. 69. 5949: 5778: 5730:{\displaystyle N=\Sigma _{i}n_{i}} 5708: 5504:these functions are respectively 5431: 5328: 5291: 5220: 5200: 4839:{\displaystyle 8\pi ^{2}IdE/h^{2}} 4785:{\displaystyle \Delta q\Delta p/h} 4768: 4762: 4739: 4733: 4609: 3909: 3906: 3903: 3900: 3897: 3838: 3748: 3745: 3742: 3739: 3736: 3726: 3720: 3702: 3696: 3685: 3679: 3670: 3634: 3628: 3621:, which in 1 spatial dimension is 3197: 3127: 3061: 3018: 2975: 2969: 360:posterior probability distribution 25: 7101:Müller-Kirsten, H. J. W. (2013). 6120:{\displaystyle n_{i}=f_{i}g_{i}.} 5343:{\displaystyle \Sigma \propto dn} 3309:such a matter wave is explicitly 6710:International Statistical Review 6626:Electronic Journal of Statistics 6357: 6317: 6300: 5417:{\displaystyle E\propto 1/n^{2}} 4748:{\displaystyle \Delta q\Delta p} 4654:) minus (phase space volume at 3536: 3244: 2984:{\displaystyle \Delta q\Delta p} 1781:states that the distribution of 277: 193:Approximate Bayesian computation 45: 6539:Journal of Open Source Software 6264: 5921: 5915: 5597: 5284:one finds correspondingly: (a) 4711:{\displaystyle 8{\pi }^{2}IdE.} 4618:{\displaystyle \Omega \propto } 2907:, both the spatial coordinates 2429:method of transformation groups 219:Maximum a posteriori estimation 7348:. Cambridge University Press. 7120:. Singapore: World Scientific. 6328: 6288: 5656: 5630: 5569: 5543: 5241: 5226: 5123: 5108: 5086: 5071: 5000: 4973: 4884: 4872: 4225: 4199: 4011:Classical a priori probability 3890: 3884: 3861: 3855: 3479: 3461: 3406: 3386: 3377: 3357: 3348: 3328: 2798:is a typical counterexample.) 2700: 2687: 2681: 2662: 2644: 2631: 2625: 2606: 2594: 2575: 2355: 2349: 2338: 2332: 2313: 2307: 2249: 2243: 2127: 2124: 2118: 2112: 2103: 2097: 2077: 2064: 1943: 1930: 1682: 1676: 1659: 1647: 1641: 1635: 1587: 1584: 1578: 1572: 1563: 1557: 1542: 1536: 1504: 1501: 1495: 1489: 1480: 1474: 1446: 1443: 1431: 1425: 1416: 1404: 1395: 1389: 1354: 1348: 1325: 1313: 1254: 1242: 1236: 1230: 1220: 1217: 1211: 1205: 1174: 1171: 1159: 1153: 1144: 1132: 1123: 1117: 1062: 1059: 1053: 1047: 1003: 997: 974: 962: 880: 874: 866: 854: 839: 827: 818: 812: 404:can, in turn, be expressed as 324:prior probability distribution 1: 7103:Basics of Statistical Physics 7020:. Series B (Methodological). 6020:{\displaystyle \epsilon _{i}} 5864:{\displaystyle \epsilon _{i}} 5757:{\displaystyle \epsilon _{0}} 4079:{\displaystyle \theta ,\phi } 4033:{\displaystyle \theta ,\phi } 3077:is the range of the variable 3034:is the range of the variable 2934:and the momentum coordinates 2852:The logarithmic prior on the 2758:is independent of all of the 7078:10.1007/978-3-642-74554-6_60 6755:Bayesian Hierarchical Models 6346:Boltzmann transport equation 6221:as shown earlier, we obtain 6201:is also independent of time 5844:particles having the energy 4723:Quantum a priori probability 2845:=0 (uniform distribution on 2415:). Such methods are used in 734:principle of maximum entropy 728:Another idea, championed by 126:Principle of maximum entropy 7188:. Berlin: Springer-Verlag. 6842:This prior was proposed by 3144:{\displaystyle L\Delta p/h} 2462:Priors based on notions of 2413:probability matching priors 1779:Bernstein-von Mises theorem 768:Kullback–Leibler divergence 96:Bernstein–von Mises theorem 7449: 5993:{\displaystyle f_{i}^{FD}} 4238:-curve for constant E and 2748:continuous random variable 2405:minimum description length 980:{\displaystyle p(x\mid t)} 774:when the prior density is 537:not very informative prior 7184:Berger, James O. (1985). 6967:10.1186/s12859-017-1893-4 6852:10.1017/S0305004100010495 6365:{\displaystyle {\bf {r}}} 2856:(uniform distribution on 2496:hierarchical Bayes models 549:principle of indifference 502:Weakly informative priors 387:principle of indifference 121:Principle of indifference 7363:Williamson, Jon (2010). 6830:10.1109/TSSC.1968.300117 5817:, i.e. with each of the 5764:), and (3) total energy 5502:Bose–Einstein statistics 5193:Thus by comparison with 3070:{\displaystyle \Delta p} 3027:{\displaystyle \Delta q} 782:); thus, in some sense, 508:weakly informative prior 372:Markov chain Monte Carlo 332:probability distribution 173:Markov chain Monte Carlo 7372:Philosophia Mathematica 5206:{\displaystyle \Omega } 4625:(phase space volume at 4385:{\displaystyle \theta } 4258:is an ellipse of area 4251:{\displaystyle \theta } 3995:microcanonical ensemble 3485:{\displaystyle (l,m,n)} 3302:{\displaystyle V=L^{3}} 2482:properties. Normally a 2464:algorithmic probability 1863:{\displaystyle 2\pi ev} 1068:{\displaystyle \log \,} 698:natural group structure 692:if the parameter space 178:Laplace's approximation 165:Posterior approximation 7433:Probability assessment 7384:10.1093/philmat/nkp019 7232:10.1214/aos/1032526950 7159:Bayesian Data Analysis 6469:Bayesian Biostatistics 6387: 6366: 6338: 6215: 6195: 6168: 6148: 6121: 6068: 6041: 6021: 5994: 5959: 5865: 5838: 5811: 5758: 5731: 5688: 5498:Fermi–Dirac statistics 5490: 5475:distribution functions 5457: 5418: 5377: 5344: 5315: 5278: 5257: 5207: 5187: 5007: 4925: 4840: 4786: 4749: 4712: 4668: 4648: 4619: 4593: 4570: 4406: 4386: 4366: 4252: 4232: 4186: 4080: 4054: 4034: 3987: 3967: 3943: 3923: 3825: 3802: 3782: 3762: 3653: 3611: 3554: 3486: 3448: 3416: 3303: 3270: 3222: 3165: 3145: 3111: 3091: 3071: 3048: 3028: 3005: 2985: 2955: 2928: 2901: 2714: 2559: 2409:frequentist statistics 2376: 2276: 2256: 2227: 2207: 2184: 2161: 2141: 2017: 1997: 1974: 1954: 1884: 1864: 1835: 1815: 1795: 1771: 1751: 1731: 1711: 1689: 1604: 1518: 1361: 1332: 1331:{\displaystyle p(x,t)} 1297: 1275: 1089: 1069: 1030: 1010: 981: 946: 926: 904: 655: 629: 556:a priori probabilities 496:posterior distribution 418:Bernoulli distribution 284:Mathematics portal 227:Evidence approximation 7116:Ben-Naim, A. (2007). 6919:10.1109/TCBB.2013.143 6811:"Prior Probabilities" 6413:Bayesian epistemology 6388: 6367: 6348:. How do coordinates 6339: 6216: 6196: 6194:{\displaystyle g_{i}} 6169: 6149: 6147:{\displaystyle n_{i}} 6122: 6069: 6067:{\displaystyle g_{i}} 6042: 6022: 5995: 5960: 5866: 5839: 5837:{\displaystyle n_{i}} 5812: 5759: 5732: 5689: 5491: 5458: 5419: 5378: 5376:{\displaystyle n^{2}} 5345: 5316: 5279: 5258: 5208: 5188: 5008: 4926: 4841: 4787: 4750: 4713: 4669: 4649: 4620: 4594: 4571: 4407: 4405:{\displaystyle \phi } 4387: 4367: 4253: 4233: 4187: 4081: 4055: 4035: 3988: 3968: 3944: 3924: 3826: 3803: 3783: 3763: 3654: 3612: 3555: 3487: 3449: 3447:{\displaystyle l,m,n} 3417: 3304: 3271: 3223: 3166: 3146: 3112: 3092: 3072: 3049: 3029: 3006: 2986: 2956: 2954:{\displaystyle p_{i}} 2929: 2927:{\displaystyle q_{i}} 2902: 2878:statistical mechanics 2808:posterior probability 2715: 2560: 2455:(1 − 2377: 2277: 2262:and integrating over 2257: 2228: 2213:by replacing it with 2208: 2185: 2162: 2142: 2018: 1998: 1975: 1955: 1885: 1865: 1836: 1816: 1796: 1772: 1752: 1732: 1712: 1690: 1605: 1519: 1362: 1333: 1303:of the joint density 1298: 1276: 1090: 1070: 1031: 1011: 982: 947: 927: 905: 752:minimum cross-entropy 717:is proportional to 1/ 681:(1 − 656: 630: 605:(1 − 188:Variational inference 18:Non-informative prior 7294:Annals of Statistics 7218:Annals of Statistics 6649:10.1214/17-EJS1337SI 6377: 6352: 6225: 6205: 6178: 6158: 6131: 6078: 6051: 6031: 6004: 5969: 5879: 5848: 5821: 5768: 5741: 5698: 5508: 5480: 5428: 5424:. Thus in this case 5387: 5360: 5325: 5288: 5277:{\displaystyle \nu } 5268: 5217: 5197: 5017: 4935: 4850: 4796: 4759: 4730: 4678: 4658: 4647:{\displaystyle E+dE} 4629: 4606: 4580: 4416: 4396: 4376: 4372:By integrating over 4262: 4242: 4196: 4090: 4064: 4044: 4018: 3977: 3957: 3933: 3835: 3815: 3792: 3772: 3667: 3625: 3619:uncertainty relation 3564: 3496: 3458: 3426: 3313: 3280: 3232: 3175: 3155: 3121: 3101: 3081: 3058: 3038: 3015: 2995: 2966: 2938: 2911: 2891: 2828:uniform distribution 2803:likelihood functions 2772:uninformative priors 2569: 2510: 2286: 2266: 2255:{\displaystyle p(x)} 2237: 2217: 2194: 2171: 2151: 2027: 2007: 1984: 1964: 1894: 1874: 1845: 1825: 1805: 1785: 1761: 1741: 1721: 1701: 1614: 1530: 1371: 1360:{\displaystyle p(x)} 1342: 1307: 1287: 1099: 1079: 1040: 1020: 1009:{\displaystyle p(x)} 991: 956: 936: 916: 794: 764:José-Miguel Bernardo 762:, was introduced by 749:. The principle of 663:uniform distribution 639: 613: 519:Uninformative priors 266:Posterior predictive 235:Evidence lower bound 116:Likelihood principle 86:Bayesian probability 7428:Bayesian statistics 7316:2009arXiv0904.0156B 7118:Entropy Demystified 6731:20.500.11850/547969 6665:Statistical Science 6548:10.21105/joss.05499 6441:The Bayesian Choice 5989: 5945: 5905: 5615: 5528: 4463: 4442: 4153: 4133: 3661:Liouville's theorem 2866:likelihood function 2468:inductive inference 1075:does not depend on 747:normal distribution 654:{\displaystyle p=1} 628:{\displaystyle p=0} 463:normal distribution 447:hierarchical priors 383:uninformative prior 368:likelihood function 352:Bayesian statistics 345:observable variable 39:Bayesian statistics 33:Part of a series on 7153:Rubin, Donald B.; 6954:BMC Bioinformatics 6722:10.1111/insr.12502 6383: 6362: 6334: 6211: 6191: 6164: 6144: 6117: 6064: 6037: 6017: 5990: 5972: 5955: 5928: 5888: 5861: 5834: 5807: 5754: 5727: 5684: 5598: 5511: 5486: 5453: 5414: 5373: 5340: 5311: 5274: 5253: 5203: 5183: 5003: 4921: 4836: 4782: 4745: 4708: 4664: 4644: 4615: 4592:{\displaystyle dE} 4589: 4566: 4443: 4419: 4402: 4382: 4362: 4248: 4228: 4182: 4139: 4119: 4076: 4050: 4030: 3983: 3963: 3939: 3919: 3821: 3798: 3778: 3758: 3649: 3607: 3550: 3482: 3444: 3412: 3299: 3266: 3218: 3161: 3141: 3107: 3087: 3067: 3044: 3024: 3001: 2981: 2951: 2924: 2897: 2710: 2658: 2555: 2372: 2272: 2252: 2223: 2206:{\displaystyle x*} 2203: 2183:{\displaystyle x*} 2180: 2157: 2137: 2013: 1996:{\displaystyle x*} 1993: 1970: 1950: 1880: 1860: 1831: 1811: 1791: 1767: 1747: 1727: 1717:of the entropy of 1707: 1685: 1600: 1514: 1357: 1328: 1293: 1271: 1085: 1065: 1026: 1006: 977: 942: 922: 900: 651: 625: 453:Informative priors 339:of the model or a 209:Bayesian estimator 157:Hierarchical model 81:Bayesian inference 7355:978-0-521-59271-0 7324:10.1214/07-AOS587 7195:978-0-387-96098-2 7168:978-1-58488-388-3 7087:978-3-642-74556-0 6891:978-0-387-20329-4 6764:978-1-03-217715-1 6687:10.1214/16-STS576 6607:10.1063/1.1477060 6502:10.1214/23-BA1381 6494:Bayesian Analysis 6465:Chaloner, Kathryn 6437:Robert, Christian 6408:Base rate fallacy 6386:{\displaystyle f} 6253: 6214:{\displaystyle t} 6167:{\displaystyle t} 6040:{\displaystyle T} 5919: 5679: 5592: 5489:{\displaystyle f} 5165: 5100: 5045: 4916: 4667:{\displaystyle E} 4324: 4311: 4172: 4112: 4053:{\displaystyle q} 3986:{\displaystyle N} 3966:{\displaystyle E} 3942:{\displaystyle N} 3882: 3865: 3853: 3824:{\displaystyle P} 3801:{\displaystyle t} 3781:{\displaystyle t} 3709: 3164:{\displaystyle V} 3110:{\displaystyle L} 3090:{\displaystyle p} 3047:{\displaystyle q} 3004:{\displaystyle h} 2900:{\displaystyle V} 2882:elementary events 2835:beta distribution 2704: 2649: 2439:logarithmic prior 2359: 2358: 2275:{\displaystyle x} 2226:{\displaystyle x} 2160:{\displaystyle k} 2080: 2016:{\displaystyle t} 1973:{\displaystyle N} 1948: 1947: 1883:{\displaystyle v} 1834:{\displaystyle x} 1814:{\displaystyle t} 1794:{\displaystyle x} 1770:{\displaystyle x} 1750:{\displaystyle t} 1730:{\displaystyle x} 1710:{\displaystyle t} 1296:{\displaystyle t} 1088:{\displaystyle t} 1029:{\displaystyle t} 945:{\displaystyle x} 925:{\displaystyle t} 884: 703:translation group 492:Bayesian analysis 459:informative prior 410:beta distribution 395:feature selection 330:, is its assumed 320: 319: 214:Credible interval 147:Linear regression 16:(Redirected from 7440: 7401: 7399: 7398: 7392: 7386:. Archived from 7369: 7359: 7340:Jaynes, Edwin T. 7335: 7309: 7290:José M. Bernardo 7281: 7252: 7234: 7207: 7180: 7149: 7122: 7121: 7113: 7107: 7106: 7098: 7092: 7091: 7065: 7059: 7058: 7040: 7034: 7033: 7013: 7007: 7004: 6998: 6997: 6987: 6969: 6945: 6939: 6938: 6902: 6896: 6895: 6877: 6871: 6840: 6834: 6833: 6815: 6807:Jaynes, Edwin T. 6803: 6794: 6793: 6775: 6769: 6768: 6750: 6744: 6743: 6733: 6705: 6699: 6698: 6680: 6660: 6654: 6653: 6651: 6641: 6632:(2): 5018–5051. 6617: 6611: 6610: 6590: 6584: 6583: 6562: 6553: 6552: 6550: 6530: 6524: 6523: 6513: 6489: 6483: 6482: 6461: 6455: 6454: 6433: 6392: 6390: 6389: 6384: 6371: 6369: 6368: 6363: 6361: 6360: 6343: 6341: 6340: 6335: 6327: 6326: 6321: 6320: 6310: 6309: 6304: 6303: 6287: 6286: 6274: 6273: 6254: 6252: 6244: 6243: 6242: 6229: 6220: 6218: 6217: 6212: 6200: 6198: 6197: 6192: 6190: 6189: 6173: 6171: 6170: 6165: 6153: 6151: 6150: 6145: 6143: 6142: 6126: 6124: 6123: 6118: 6113: 6112: 6103: 6102: 6090: 6089: 6073: 6071: 6070: 6065: 6063: 6062: 6046: 6044: 6043: 6038: 6027:and temperature 6026: 6024: 6023: 6018: 6016: 6015: 5999: 5997: 5996: 5991: 5988: 5980: 5964: 5962: 5961: 5956: 5944: 5936: 5920: 5917: 5904: 5896: 5870: 5868: 5867: 5862: 5860: 5859: 5843: 5841: 5840: 5835: 5833: 5832: 5816: 5814: 5813: 5808: 5806: 5805: 5796: 5795: 5786: 5785: 5763: 5761: 5760: 5755: 5753: 5752: 5736: 5734: 5733: 5728: 5726: 5725: 5716: 5715: 5693: 5691: 5690: 5685: 5680: 5678: 5671: 5670: 5663: 5655: 5654: 5642: 5641: 5620: 5614: 5606: 5593: 5591: 5584: 5583: 5576: 5568: 5567: 5555: 5554: 5533: 5527: 5519: 5495: 5493: 5492: 5487: 5462: 5460: 5459: 5454: 5446: 5445: 5423: 5421: 5420: 5415: 5413: 5412: 5403: 5382: 5380: 5379: 5374: 5372: 5371: 5349: 5347: 5346: 5341: 5320: 5318: 5317: 5312: 5307: 5283: 5281: 5280: 5275: 5262: 5260: 5259: 5254: 5212: 5210: 5209: 5204: 5192: 5190: 5189: 5184: 5179: 5178: 5166: 5164: 5163: 5154: 5150: 5149: 5136: 5101: 5099: 5098: 5097: 5069: 5065: 5064: 5051: 5046: 5044: 5043: 5042: 5029: 5021: 5012: 5010: 5009: 5004: 4993: 4988: 4987: 4972: 4961: 4960: 4948: 4930: 4928: 4927: 4922: 4917: 4915: 4911: 4910: 4897: 4896: 4895: 4867: 4862: 4861: 4845: 4843: 4842: 4837: 4835: 4834: 4825: 4811: 4810: 4791: 4789: 4788: 4783: 4778: 4754: 4752: 4751: 4746: 4717: 4715: 4714: 4709: 4695: 4694: 4689: 4673: 4671: 4670: 4665: 4653: 4651: 4650: 4645: 4624: 4622: 4621: 4616: 4598: 4596: 4595: 4590: 4575: 4573: 4572: 4567: 4550: 4549: 4537: 4536: 4512: 4511: 4462: 4451: 4441: 4427: 4411: 4409: 4408: 4403: 4391: 4389: 4388: 4383: 4371: 4369: 4368: 4363: 4325: 4314: 4312: 4301: 4293: 4292: 4280: 4279: 4257: 4255: 4254: 4249: 4237: 4235: 4234: 4229: 4224: 4223: 4211: 4210: 4191: 4189: 4188: 4183: 4178: 4174: 4173: 4171: 4164: 4163: 4152: 4147: 4138: 4132: 4127: 4113: 4111: 4100: 4085: 4083: 4082: 4077: 4059: 4057: 4056: 4051: 4039: 4037: 4036: 4031: 3992: 3990: 3989: 3984: 3972: 3970: 3969: 3964: 3948: 3946: 3945: 3940: 3928: 3926: 3925: 3920: 3915: 3883: 3880: 3866: 3864: 3854: 3851: 3845: 3830: 3828: 3827: 3822: 3807: 3805: 3804: 3799: 3787: 3785: 3784: 3779: 3767: 3765: 3764: 3759: 3754: 3710: 3708: 3691: 3677: 3658: 3656: 3655: 3650: 3616: 3614: 3613: 3608: 3606: 3605: 3596: 3585: 3584: 3559: 3557: 3556: 3551: 3546: 3545: 3540: 3539: 3529: 3528: 3491: 3489: 3488: 3483: 3453: 3451: 3450: 3445: 3421: 3419: 3418: 3413: 3402: 3373: 3344: 3308: 3306: 3305: 3300: 3298: 3297: 3275: 3273: 3272: 3267: 3259: 3254: 3253: 3248: 3247: 3227: 3225: 3224: 3219: 3217: 3216: 3207: 3196: 3195: 3170: 3168: 3167: 3162: 3150: 3148: 3147: 3142: 3137: 3116: 3114: 3113: 3108: 3096: 3094: 3093: 3088: 3076: 3074: 3073: 3068: 3053: 3051: 3050: 3045: 3033: 3031: 3030: 3025: 3010: 3008: 3007: 3002: 2990: 2988: 2987: 2982: 2960: 2958: 2957: 2952: 2950: 2949: 2933: 2931: 2930: 2925: 2923: 2922: 2906: 2904: 2903: 2898: 2786:) ~ 1/ 2719: 2717: 2716: 2711: 2705: 2703: 2699: 2698: 2680: 2679: 2657: 2647: 2643: 2642: 2624: 2623: 2601: 2587: 2586: 2564: 2562: 2561: 2556: 2554: 2553: 2535: 2534: 2522: 2521: 2441: 2440: 2381: 2379: 2378: 2373: 2364: 2360: 2342: 2341: 2327: 2281: 2279: 2278: 2273: 2261: 2259: 2258: 2253: 2232: 2230: 2229: 2224: 2212: 2210: 2209: 2204: 2189: 2187: 2186: 2181: 2166: 2164: 2163: 2158: 2146: 2144: 2143: 2138: 2086: 2082: 2081: 2076: 2075: 2057: 2022: 2020: 2019: 2014: 2002: 2000: 1999: 1994: 1979: 1977: 1976: 1971: 1959: 1957: 1956: 1951: 1949: 1946: 1942: 1941: 1922: 1911: 1910: 1889: 1887: 1886: 1881: 1869: 1867: 1866: 1861: 1840: 1838: 1837: 1832: 1820: 1818: 1817: 1812: 1800: 1798: 1797: 1792: 1776: 1774: 1773: 1768: 1756: 1754: 1753: 1748: 1736: 1734: 1733: 1728: 1716: 1714: 1713: 1708: 1694: 1692: 1691: 1686: 1609: 1607: 1606: 1601: 1523: 1521: 1520: 1515: 1366: 1364: 1363: 1358: 1337: 1335: 1334: 1329: 1302: 1300: 1299: 1294: 1280: 1278: 1277: 1272: 1094: 1092: 1091: 1086: 1074: 1072: 1071: 1066: 1035: 1033: 1032: 1027: 1015: 1013: 1012: 1007: 986: 984: 983: 978: 951: 949: 948: 943: 931: 929: 928: 923: 909: 907: 906: 901: 885: 883: 869: 849: 760:reference priors 758:A related idea, 732:, is to use the 660: 658: 657: 652: 634: 632: 631: 626: 364:conjugate family 312: 305: 298: 282: 281: 248:Model evaluation 49: 30: 21: 7448: 7447: 7443: 7442: 7441: 7439: 7438: 7437: 7418: 7417: 7408: 7396: 7394: 7390: 7367: 7362: 7356: 7338: 7286:James O. Berger 7284: 7255: 7210: 7196: 7183: 7169: 7152: 7146: 7133: 7130: 7125: 7115: 7114: 7110: 7100: 7099: 7095: 7088: 7067: 7066: 7062: 7055: 7042: 7041: 7037: 7015: 7014: 7010: 7005: 7001: 6947: 6946: 6942: 6904: 6903: 6899: 6892: 6879: 6878: 6874: 6860:10.2307/2332350 6841: 6837: 6813: 6805: 6804: 6797: 6790: 6777: 6776: 6772: 6765: 6752: 6751: 6747: 6707: 6706: 6702: 6662: 6661: 6657: 6619: 6618: 6614: 6592: 6591: 6587: 6580: 6566:Zellner, Arnold 6564: 6563: 6556: 6532: 6531: 6527: 6496:. Forthcoming. 6491: 6490: 6486: 6479: 6463: 6462: 6458: 6451: 6435: 6434: 6430: 6426: 6399: 6375: 6374: 6350: 6349: 6314: 6297: 6278: 6265: 6245: 6234: 6230: 6223: 6222: 6203: 6202: 6181: 6176: 6175: 6156: 6155: 6134: 6129: 6128: 6104: 6094: 6081: 6076: 6075: 6054: 6049: 6048: 6029: 6028: 6007: 6002: 6001: 5967: 5966: 5877: 5876: 5873:Pauli principle 5851: 5846: 5845: 5824: 5819: 5818: 5797: 5787: 5777: 5766: 5765: 5744: 5739: 5738: 5717: 5707: 5696: 5695: 5646: 5633: 5625: 5624: 5559: 5546: 5538: 5537: 5506: 5505: 5478: 5477: 5471: 5466: 5437: 5426: 5425: 5404: 5385: 5384: 5363: 5358: 5357: 5323: 5322: 5286: 5285: 5266: 5265: 5215: 5214: 5195: 5194: 5170: 5155: 5141: 5137: 5089: 5070: 5056: 5052: 5034: 5030: 5022: 5015: 5014: 4979: 4952: 4933: 4932: 4902: 4898: 4887: 4868: 4853: 4848: 4847: 4826: 4802: 4794: 4793: 4757: 4756: 4728: 4727: 4684: 4676: 4675: 4656: 4655: 4627: 4626: 4604: 4603: 4578: 4577: 4541: 4528: 4503: 4414: 4413: 4394: 4393: 4374: 4373: 4284: 4271: 4260: 4259: 4240: 4239: 4215: 4202: 4194: 4193: 4155: 4154: 4118: 4114: 4104: 4088: 4087: 4062: 4061: 4042: 4041: 4016: 4015: 4003: 3975: 3974: 3955: 3954: 3931: 3930: 3849: 3833: 3832: 3813: 3812: 3790: 3789: 3770: 3769: 3692: 3678: 3665: 3664: 3623: 3622: 3597: 3576: 3562: 3561: 3533: 3520: 3494: 3493: 3456: 3455: 3424: 3423: 3311: 3310: 3289: 3278: 3277: 3241: 3230: 3229: 3208: 3187: 3173: 3172: 3153: 3152: 3119: 3118: 3099: 3098: 3079: 3078: 3056: 3055: 3036: 3035: 3013: 3012: 2993: 2992: 2964: 2963: 2941: 2936: 2935: 2914: 2909: 2908: 2889: 2888: 2874: 2833:Beta(0,0), the 2820: 2766: 2745: 2732: 2690: 2671: 2648: 2634: 2615: 2602: 2578: 2567: 2566: 2545: 2526: 2513: 2508: 2507: 2504: 2502:Improper priors 2438: 2437: 2421:prior knowledge 2328: 2322: 2284: 2283: 2264: 2263: 2235: 2234: 2215: 2214: 2192: 2191: 2169: 2168: 2149: 2148: 2067: 2052: 2048: 2025: 2024: 2005: 2004: 1982: 1981: 1962: 1961: 1933: 1923: 1912: 1892: 1891: 1872: 1871: 1843: 1842: 1823: 1822: 1803: 1802: 1783: 1782: 1759: 1758: 1739: 1738: 1737:conditional on 1719: 1718: 1699: 1698: 1612: 1611: 1528: 1527: 1369: 1368: 1340: 1339: 1305: 1304: 1285: 1284: 1097: 1096: 1077: 1076: 1038: 1037: 1018: 1017: 989: 988: 954: 953: 934: 933: 914: 913: 870: 850: 792: 791: 738:Shannon entropy 730:Edwin T. Jaynes 672:Harold Jeffreys 637: 636: 611: 610: 560:Edwin T. Jaynes 541:objective prior 521: 504: 483: 455: 402:hyperparameters 343:rather than an 341:latent variable 316: 276: 261:Model averaging 240:Nested sampling 152:Empirical Bayes 142:Conjugate prior 111:Cromwell's rule 28: 23: 22: 15: 12: 11: 5: 7446: 7444: 7436: 7435: 7430: 7420: 7419: 7416: 7415: 7407: 7406:External links 7404: 7403: 7402: 7378:(1): 130–135. 7360: 7354: 7336: 7300:(2): 905–938. 7282: 7264:(2): 113–147. 7253: 7225:(3): 931–951. 7208: 7194: 7181: 7167: 7155:Gelman, Andrew 7150: 7144: 7129: 7126: 7124: 7123: 7108: 7093: 7086: 7060: 7053: 7035: 7024:(2): 189–233. 7008: 6999: 6940: 6897: 6890: 6872: 6844:J.B.S. Haldane 6835: 6824:(3): 227–241. 6795: 6788: 6770: 6763: 6745: 6716:(3): 563–591. 6700: 6655: 6612: 6595:AIP Conf. Proc 6585: 6578: 6554: 6525: 6484: 6477: 6456: 6449: 6427: 6425: 6422: 6421: 6420: 6415: 6410: 6405: 6398: 6395: 6382: 6359: 6333: 6330: 6325: 6319: 6313: 6308: 6302: 6296: 6293: 6290: 6285: 6281: 6277: 6272: 6268: 6263: 6260: 6257: 6251: 6248: 6241: 6237: 6233: 6210: 6188: 6184: 6163: 6141: 6137: 6116: 6111: 6107: 6101: 6097: 6093: 6088: 6084: 6061: 6057: 6036: 6014: 6010: 5987: 5984: 5979: 5975: 5954: 5951: 5948: 5943: 5940: 5935: 5931: 5927: 5924: 5914: 5911: 5908: 5903: 5900: 5895: 5891: 5887: 5884: 5858: 5854: 5831: 5827: 5804: 5800: 5794: 5790: 5784: 5780: 5776: 5773: 5751: 5747: 5724: 5720: 5714: 5710: 5706: 5703: 5683: 5677: 5674: 5669: 5666: 5662: 5658: 5653: 5649: 5645: 5640: 5636: 5632: 5628: 5623: 5618: 5613: 5610: 5605: 5601: 5596: 5590: 5587: 5582: 5579: 5575: 5571: 5566: 5562: 5558: 5553: 5549: 5545: 5541: 5536: 5531: 5526: 5523: 5518: 5514: 5485: 5470: 5467: 5465: 5464: 5452: 5449: 5444: 5440: 5436: 5433: 5411: 5407: 5402: 5398: 5395: 5392: 5370: 5366: 5339: 5336: 5333: 5330: 5310: 5306: 5302: 5299: 5296: 5293: 5273: 5252: 5249: 5246: 5243: 5240: 5237: 5234: 5231: 5228: 5225: 5222: 5202: 5182: 5177: 5173: 5169: 5162: 5158: 5153: 5148: 5144: 5140: 5134: 5131: 5128: 5125: 5122: 5119: 5116: 5113: 5110: 5104: 5096: 5092: 5088: 5085: 5082: 5079: 5076: 5073: 5068: 5063: 5059: 5055: 5049: 5041: 5037: 5033: 5028: 5025: 5002: 4999: 4996: 4992: 4986: 4982: 4978: 4975: 4971: 4967: 4964: 4959: 4955: 4951: 4947: 4943: 4940: 4920: 4914: 4909: 4905: 4901: 4894: 4890: 4886: 4883: 4880: 4877: 4874: 4871: 4865: 4860: 4856: 4833: 4829: 4824: 4820: 4817: 4814: 4809: 4805: 4801: 4781: 4777: 4773: 4770: 4767: 4764: 4744: 4741: 4738: 4735: 4720: 4719: 4718: 4707: 4704: 4701: 4698: 4693: 4688: 4683: 4674:) is given by 4663: 4643: 4640: 4637: 4634: 4614: 4611: 4588: 4585: 4565: 4562: 4559: 4556: 4553: 4548: 4544: 4540: 4535: 4531: 4527: 4524: 4521: 4518: 4515: 4510: 4506: 4502: 4499: 4496: 4493: 4490: 4487: 4484: 4481: 4478: 4475: 4472: 4469: 4466: 4461: 4458: 4455: 4450: 4446: 4440: 4437: 4434: 4431: 4426: 4422: 4401: 4381: 4361: 4358: 4355: 4352: 4349: 4346: 4343: 4340: 4337: 4334: 4331: 4328: 4323: 4320: 4317: 4310: 4307: 4304: 4299: 4296: 4291: 4287: 4283: 4278: 4274: 4270: 4267: 4247: 4227: 4222: 4218: 4214: 4209: 4205: 4201: 4181: 4177: 4170: 4167: 4162: 4158: 4151: 4146: 4142: 4136: 4131: 4126: 4122: 4117: 4110: 4107: 4103: 4098: 4095: 4075: 4072: 4069: 4060:above is here 4049: 4029: 4026: 4023: 4007: 4002: 3999: 3982: 3962: 3938: 3918: 3914: 3911: 3908: 3905: 3902: 3899: 3895: 3892: 3889: 3886: 3878: 3875: 3869: 3863: 3860: 3857: 3848: 3843: 3840: 3820: 3797: 3777: 3757: 3753: 3750: 3747: 3744: 3741: 3738: 3734: 3731: 3728: 3725: 3722: 3719: 3713: 3707: 3704: 3701: 3698: 3695: 3690: 3687: 3684: 3681: 3675: 3672: 3648: 3645: 3642: 3639: 3636: 3633: 3630: 3604: 3600: 3595: 3591: 3588: 3583: 3579: 3575: 3572: 3569: 3549: 3544: 3538: 3532: 3527: 3523: 3519: 3516: 3513: 3510: 3507: 3504: 3501: 3481: 3478: 3475: 3472: 3469: 3466: 3463: 3443: 3440: 3437: 3434: 3431: 3411: 3408: 3405: 3401: 3397: 3394: 3391: 3388: 3385: 3382: 3379: 3376: 3372: 3368: 3365: 3362: 3359: 3356: 3353: 3350: 3347: 3343: 3339: 3336: 3333: 3330: 3327: 3324: 3321: 3318: 3296: 3292: 3288: 3285: 3265: 3262: 3258: 3252: 3246: 3240: 3237: 3215: 3211: 3206: 3202: 3199: 3194: 3190: 3186: 3183: 3180: 3160: 3140: 3136: 3132: 3129: 3126: 3106: 3086: 3066: 3063: 3043: 3023: 3020: 3000: 2980: 2977: 2974: 2971: 2948: 2944: 2921: 2917: 2896: 2873: 2870: 2862: 2861: 2854:positive reals 2850: 2831: 2819: 2816: 2762: 2752:improper prior 2741: 2728: 2709: 2702: 2697: 2693: 2689: 2686: 2683: 2678: 2674: 2670: 2667: 2664: 2661: 2656: 2652: 2646: 2641: 2637: 2633: 2630: 2627: 2622: 2618: 2614: 2611: 2608: 2605: 2599: 2596: 2593: 2590: 2585: 2581: 2577: 2574: 2552: 2548: 2544: 2541: 2538: 2533: 2529: 2525: 2520: 2516: 2503: 2500: 2445:Jeffreys prior 2390:Jeffreys' rule 2371: 2368: 2363: 2357: 2354: 2351: 2348: 2345: 2340: 2337: 2334: 2331: 2325: 2321: 2318: 2315: 2312: 2309: 2306: 2303: 2300: 2297: 2294: 2291: 2271: 2251: 2248: 2245: 2242: 2222: 2202: 2199: 2179: 2176: 2156: 2136: 2133: 2129: 2126: 2123: 2120: 2117: 2114: 2111: 2108: 2105: 2102: 2099: 2096: 2093: 2089: 2085: 2079: 2074: 2070: 2066: 2063: 2060: 2055: 2051: 2047: 2044: 2041: 2038: 2035: 2032: 2012: 1992: 1989: 1969: 1945: 1940: 1936: 1932: 1929: 1926: 1921: 1918: 1915: 1908: 1905: 1902: 1899: 1879: 1859: 1856: 1853: 1850: 1830: 1810: 1790: 1766: 1746: 1726: 1706: 1684: 1681: 1678: 1675: 1671: 1668: 1665: 1661: 1658: 1655: 1652: 1649: 1646: 1643: 1640: 1637: 1634: 1631: 1628: 1625: 1622: 1619: 1599: 1596: 1593: 1589: 1586: 1583: 1580: 1577: 1574: 1571: 1568: 1565: 1562: 1559: 1556: 1553: 1550: 1547: 1544: 1541: 1538: 1535: 1513: 1510: 1506: 1503: 1500: 1497: 1494: 1491: 1488: 1485: 1482: 1479: 1476: 1473: 1470: 1466: 1462: 1459: 1455: 1452: 1448: 1445: 1442: 1439: 1436: 1433: 1430: 1427: 1424: 1421: 1418: 1415: 1412: 1409: 1406: 1403: 1400: 1397: 1394: 1391: 1388: 1385: 1382: 1379: 1376: 1356: 1353: 1350: 1347: 1327: 1324: 1321: 1318: 1315: 1312: 1292: 1270: 1267: 1263: 1260: 1256: 1253: 1250: 1247: 1244: 1241: 1238: 1235: 1232: 1229: 1226: 1222: 1219: 1216: 1213: 1210: 1207: 1204: 1201: 1198: 1194: 1190: 1187: 1183: 1180: 1176: 1173: 1170: 1167: 1164: 1161: 1158: 1155: 1152: 1149: 1146: 1143: 1140: 1137: 1134: 1131: 1128: 1125: 1122: 1119: 1116: 1113: 1110: 1107: 1104: 1084: 1064: 1061: 1058: 1055: 1052: 1049: 1045: 1025: 1005: 1002: 999: 996: 976: 973: 970: 967: 964: 961: 941: 921: 899: 896: 892: 889: 882: 879: 876: 873: 868: 865: 862: 859: 856: 853: 847: 844: 841: 838: 835: 832: 829: 826: 823: 820: 817: 814: 811: 808: 805: 802: 799: 711:improper prior 676:Jeffreys prior 667:Bayes' theorem 650: 647: 644: 624: 621: 618: 520: 517: 513:regularization 503: 500: 482: 479: 467:expected value 454: 451: 443: 442: 428: 391:regularization 318: 317: 315: 314: 307: 300: 292: 289: 288: 287: 286: 271: 270: 269: 268: 263: 258: 250: 249: 245: 244: 243: 242: 237: 229: 228: 224: 223: 222: 221: 216: 211: 203: 202: 198: 197: 196: 195: 190: 185: 180: 175: 167: 166: 162: 161: 160: 159: 154: 149: 144: 136: 135: 134:Model building 131: 130: 129: 128: 123: 118: 113: 108: 103: 98: 93: 91:Bayes' theorem 88: 83: 75: 74: 70: 69: 51: 50: 42: 41: 35: 34: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 7445: 7434: 7431: 7429: 7426: 7425: 7423: 7413: 7410: 7409: 7405: 7393:on 2011-06-09 7389: 7385: 7381: 7377: 7373: 7366: 7361: 7357: 7351: 7347: 7346: 7341: 7337: 7333: 7329: 7325: 7321: 7317: 7313: 7308: 7303: 7299: 7295: 7291: 7287: 7283: 7279: 7275: 7271: 7267: 7263: 7259: 7254: 7250: 7246: 7242: 7238: 7233: 7228: 7224: 7220: 7219: 7214: 7209: 7205: 7201: 7197: 7191: 7187: 7182: 7178: 7174: 7170: 7164: 7160: 7156: 7151: 7147: 7145:0-19-877313-7 7141: 7137: 7132: 7131: 7127: 7119: 7112: 7109: 7104: 7097: 7094: 7089: 7083: 7079: 7075: 7071: 7064: 7061: 7056: 7054:9781439894798 7050: 7046: 7039: 7036: 7031: 7027: 7023: 7019: 7012: 7009: 7003: 7000: 6995: 6991: 6986: 6981: 6977: 6973: 6968: 6963: 6959: 6955: 6951: 6944: 6941: 6936: 6932: 6928: 6924: 6920: 6916: 6913:(1): 202–18. 6912: 6908: 6901: 6898: 6893: 6887: 6883: 6876: 6873: 6869: 6865: 6861: 6857: 6853: 6849: 6845: 6839: 6836: 6831: 6827: 6823: 6819: 6812: 6808: 6802: 6800: 6796: 6791: 6789:0-444-88422-X 6785: 6781: 6774: 6771: 6766: 6760: 6756: 6749: 6746: 6741: 6737: 6732: 6727: 6723: 6719: 6715: 6711: 6704: 6701: 6696: 6692: 6688: 6684: 6679: 6674: 6670: 6666: 6659: 6656: 6650: 6645: 6640: 6635: 6631: 6627: 6623: 6616: 6613: 6608: 6604: 6600: 6596: 6589: 6586: 6581: 6579:0-471-98165-6 6575: 6571: 6567: 6561: 6559: 6555: 6549: 6544: 6540: 6536: 6529: 6526: 6521: 6517: 6512: 6507: 6503: 6499: 6495: 6488: 6485: 6480: 6478:0-8247-9334-X 6474: 6470: 6466: 6460: 6457: 6452: 6450:0-387-94296-3 6446: 6442: 6438: 6432: 6429: 6423: 6419: 6416: 6414: 6411: 6409: 6406: 6404: 6401: 6400: 6396: 6394: 6380: 6347: 6331: 6323: 6311: 6306: 6294: 6291: 6283: 6279: 6275: 6270: 6266: 6261: 6258: 6255: 6249: 6246: 6239: 6235: 6231: 6208: 6186: 6182: 6161: 6139: 6135: 6114: 6109: 6105: 6099: 6095: 6091: 6086: 6082: 6059: 6055: 6034: 6012: 6008: 5985: 5982: 5977: 5973: 5952: 5946: 5941: 5938: 5933: 5929: 5925: 5922: 5912: 5909: 5906: 5901: 5898: 5893: 5889: 5885: 5882: 5874: 5856: 5852: 5829: 5825: 5802: 5798: 5792: 5788: 5782: 5774: 5771: 5749: 5745: 5722: 5718: 5712: 5704: 5701: 5681: 5675: 5672: 5667: 5664: 5660: 5651: 5647: 5643: 5638: 5634: 5626: 5621: 5616: 5611: 5608: 5603: 5599: 5594: 5588: 5585: 5580: 5577: 5573: 5564: 5560: 5556: 5551: 5547: 5539: 5534: 5529: 5524: 5521: 5516: 5512: 5503: 5499: 5483: 5476: 5468: 5450: 5447: 5442: 5438: 5434: 5409: 5405: 5400: 5396: 5393: 5390: 5368: 5364: 5355: 5353: 5337: 5334: 5331: 5308: 5304: 5300: 5297: 5294: 5271: 5250: 5247: 5244: 5238: 5235: 5232: 5229: 5223: 5180: 5175: 5171: 5167: 5160: 5156: 5151: 5146: 5142: 5138: 5132: 5129: 5126: 5120: 5117: 5114: 5111: 5102: 5094: 5090: 5083: 5080: 5077: 5074: 5066: 5061: 5057: 5053: 5047: 5039: 5035: 5031: 5026: 5023: 4997: 4994: 4990: 4984: 4980: 4976: 4969: 4965: 4962: 4957: 4953: 4949: 4945: 4941: 4938: 4918: 4912: 4907: 4903: 4899: 4892: 4888: 4881: 4878: 4875: 4869: 4863: 4858: 4854: 4831: 4827: 4822: 4818: 4815: 4812: 4807: 4803: 4799: 4779: 4775: 4771: 4765: 4742: 4736: 4724: 4721: 4705: 4702: 4699: 4696: 4691: 4686: 4681: 4661: 4641: 4638: 4635: 4632: 4612: 4602: 4601: 4600: 4586: 4583: 4563: 4560: 4557: 4554: 4551: 4546: 4542: 4538: 4533: 4529: 4525: 4522: 4519: 4516: 4513: 4508: 4504: 4500: 4497: 4494: 4491: 4488: 4485: 4482: 4479: 4476: 4473: 4470: 4467: 4464: 4459: 4456: 4453: 4448: 4444: 4438: 4435: 4432: 4429: 4424: 4420: 4399: 4379: 4359: 4356: 4353: 4350: 4347: 4344: 4341: 4338: 4335: 4332: 4329: 4326: 4321: 4318: 4315: 4308: 4305: 4302: 4297: 4294: 4289: 4285: 4281: 4276: 4272: 4268: 4265: 4245: 4220: 4216: 4212: 4207: 4203: 4179: 4175: 4168: 4165: 4160: 4156: 4149: 4144: 4140: 4134: 4129: 4124: 4120: 4115: 4108: 4105: 4101: 4096: 4093: 4073: 4070: 4067: 4047: 4027: 4024: 4021: 4012: 4009: 4008: 4006: 4000: 3998: 3996: 3980: 3960: 3952: 3936: 3916: 3912: 3893: 3887: 3876: 3873: 3867: 3858: 3846: 3841: 3818: 3809: 3795: 3775: 3755: 3751: 3732: 3729: 3723: 3717: 3711: 3705: 3699: 3693: 3688: 3682: 3673: 3662: 3646: 3643: 3640: 3637: 3631: 3620: 3602: 3598: 3593: 3589: 3586: 3581: 3577: 3573: 3570: 3567: 3547: 3542: 3530: 3525: 3521: 3517: 3514: 3511: 3508: 3505: 3502: 3499: 3476: 3473: 3470: 3467: 3464: 3441: 3438: 3435: 3432: 3429: 3409: 3403: 3399: 3395: 3392: 3389: 3383: 3380: 3374: 3370: 3366: 3363: 3360: 3354: 3351: 3345: 3341: 3337: 3334: 3331: 3325: 3322: 3319: 3316: 3294: 3290: 3286: 3283: 3263: 3260: 3256: 3250: 3238: 3235: 3213: 3209: 3204: 3200: 3192: 3188: 3184: 3181: 3178: 3158: 3138: 3134: 3130: 3124: 3104: 3084: 3064: 3041: 3021: 2998: 2978: 2972: 2946: 2942: 2919: 2915: 2894: 2885: 2883: 2879: 2871: 2869: 2867: 2859: 2855: 2851: 2848: 2844: 2840: 2836: 2832: 2829: 2825: 2824: 2823: 2817: 2815: 2814:for details. 2813: 2809: 2804: 2801:By contrast, 2799: 2797: 2796:Haldane prior 2793: 2789: 2785: 2781: 2777: 2773: 2768: 2765: 2761: 2757: 2753: 2749: 2744: 2740: 2736: 2731: 2727: 2723: 2707: 2695: 2691: 2684: 2676: 2672: 2668: 2665: 2659: 2654: 2650: 2639: 2635: 2628: 2620: 2616: 2612: 2609: 2603: 2597: 2591: 2588: 2583: 2579: 2572: 2550: 2546: 2542: 2539: 2536: 2531: 2527: 2523: 2518: 2514: 2501: 2499: 2497: 2493: 2489: 2488:decision rule 2485: 2481: 2477: 2471: 2469: 2465: 2460: 2458: 2454: 2450: 2446: 2442: 2432: 2430: 2424: 2422: 2418: 2414: 2410: 2406: 2402: 2401:coding theory 2398: 2393: 2391: 2386: 2382: 2369: 2366: 2361: 2352: 2346: 2343: 2335: 2329: 2323: 2319: 2316: 2310: 2304: 2301: 2298: 2295: 2292: 2289: 2269: 2246: 2240: 2220: 2200: 2197: 2177: 2174: 2154: 2134: 2131: 2121: 2115: 2109: 2106: 2100: 2094: 2091: 2087: 2083: 2072: 2068: 2061: 2058: 2053: 2049: 2045: 2042: 2039: 2036: 2033: 2030: 2010: 1990: 1987: 1967: 1938: 1934: 1927: 1924: 1919: 1916: 1913: 1906: 1903: 1900: 1897: 1877: 1857: 1854: 1851: 1848: 1828: 1808: 1788: 1780: 1764: 1744: 1724: 1704: 1695: 1679: 1673: 1669: 1666: 1663: 1656: 1653: 1650: 1644: 1638: 1632: 1629: 1626: 1623: 1620: 1617: 1597: 1594: 1591: 1581: 1575: 1569: 1566: 1560: 1554: 1551: 1548: 1545: 1539: 1533: 1524: 1511: 1508: 1498: 1492: 1486: 1483: 1477: 1471: 1468: 1464: 1460: 1457: 1453: 1450: 1440: 1437: 1434: 1428: 1422: 1419: 1413: 1410: 1407: 1401: 1398: 1392: 1386: 1383: 1380: 1377: 1374: 1367:, so we have 1351: 1345: 1322: 1319: 1316: 1310: 1290: 1281: 1268: 1265: 1261: 1258: 1251: 1248: 1245: 1239: 1233: 1227: 1224: 1214: 1208: 1202: 1199: 1196: 1192: 1188: 1185: 1181: 1178: 1168: 1165: 1162: 1156: 1150: 1147: 1141: 1138: 1135: 1129: 1126: 1120: 1114: 1111: 1108: 1105: 1102: 1082: 1056: 1050: 1043: 1023: 1000: 994: 971: 968: 965: 959: 939: 919: 910: 897: 894: 890: 887: 877: 871: 863: 860: 857: 851: 845: 842: 836: 833: 830: 824: 821: 815: 809: 806: 803: 800: 797: 789: 785: 781: 777: 773: 769: 765: 761: 756: 754: 753: 748: 744: 739: 735: 731: 726: 724: 720: 716: 712: 708: 704: 699: 695: 691: 686: 684: 680: 677: 673: 668: 664: 648: 645: 642: 622: 619: 616: 608: 604: 601: 600:Haldane prior 595: 593: 589: 585: 581: 577: 573: 569: 568:uniform prior 563: 561: 557: 552: 550: 544: 542: 538: 534: 533:diffuse prior 530: 526: 525:uninformative 518: 516: 514: 509: 501: 499: 497: 493: 488: 480: 478: 474: 472: 468: 464: 460: 452: 450: 448: 440: 436: 432: 429: 426: 423: 422: 421: 419: 415: 411: 407: 403: 398: 396: 392: 388: 384: 380: 375: 373: 369: 365: 361: 357: 353: 348: 346: 342: 338: 333: 329: 325: 313: 308: 306: 301: 299: 294: 293: 291: 290: 285: 280: 275: 274: 273: 272: 267: 264: 262: 259: 257: 254: 253: 252: 251: 246: 241: 238: 236: 233: 232: 231: 230: 225: 220: 217: 215: 212: 210: 207: 206: 205: 204: 199: 194: 191: 189: 186: 184: 181: 179: 176: 174: 171: 170: 169: 168: 163: 158: 155: 153: 150: 148: 145: 143: 140: 139: 138: 137: 132: 127: 124: 122: 119: 117: 114: 112: 109: 107: 106:Cox's theorem 104: 102: 99: 97: 94: 92: 89: 87: 84: 82: 79: 78: 77: 76: 71: 68: 64: 60: 56: 53: 52: 48: 44: 43: 40: 36: 32: 31: 19: 7395:. Retrieved 7388:the original 7375: 7371: 7344: 7297: 7293: 7261: 7257: 7222: 7216: 7185: 7158: 7135: 7117: 7111: 7102: 7096: 7069: 7063: 7044: 7038: 7021: 7017: 7011: 7002: 6960:(S14): 552. 6957: 6953: 6943: 6910: 6906: 6900: 6884:. Springer. 6881: 6875: 6838: 6821: 6817: 6809:(Sep 1968). 6779: 6773: 6754: 6748: 6713: 6709: 6703: 6668: 6664: 6658: 6629: 6625: 6615: 6598: 6594: 6588: 6569: 6538: 6528: 6511:11336/183197 6493: 6487: 6468: 6459: 6440: 6431: 6418:Strong prior 5472: 5013:one obtains 4725: 4722: 4040:(this means 4013: 4010: 4004: 3994: 3810: 2886: 2875: 2863: 2842: 2838: 2821: 2800: 2791: 2787: 2783: 2779: 2775: 2769: 2763: 2759: 2755: 2751: 2742: 2738: 2734: 2729: 2725: 2721: 2505: 2475: 2472: 2466:are used in 2461: 2456: 2452: 2448: 2436: 2433: 2425: 2420: 2394: 2387: 2383: 1696: 1525: 1282: 911: 787: 783: 779: 775: 771: 757: 750: 742: 727: 723:affine group 718: 714: 706: 693: 690:Haar measure 687: 682: 678: 606: 602: 596: 591: 587: 583: 579: 575: 571: 567: 564: 553: 545: 540: 536: 532: 528: 524: 522: 507: 505: 487:strong prior 486: 484: 481:Strong prior 475: 458: 456: 446: 444: 438: 434: 430: 424: 413: 399: 382: 378: 376: 349: 327: 323: 321: 256:Bayes factor 62: 6671:(1): 1–28. 6601:: 379–391. 2991:divided by 2506:Let events 2480:frequentist 2411:(so-called 2397:information 441:parameters. 366:of a given 356:Bayes' rule 7422:Categories 7397:2010-07-02 7249:0865.62004 7128:References 6639:1707.01694 5352:degeneracy 5321:, and (b) 2492:admissible 2403:(see e.g. 987:and prior 696:carries a 406:hyperprior 201:Estimators 73:Background 59:Likelihood 7307:0904.0156 6976:1471-2105 6740:234681651 6678:1403.4630 6520:244798734 6403:Base rate 6009:ϵ 5950:∞ 5947:≤ 5926:≤ 5907:≤ 5886:≤ 5853:ϵ 5799:ϵ 5779:Σ 5746:ϵ 5709:Σ 5673:− 5648:ϵ 5644:− 5635:ϵ 5561:ϵ 5557:− 5548:ϵ 5435:∝ 5432:Σ 5394:∝ 5332:∝ 5329:Σ 5309:ν 5295:∝ 5292:Ω 5272:ν 5224:∝ 5221:Σ 5201:Ω 5143:π 5058:π 4904:π 4804:π 4769:Δ 4763:Δ 4740:Δ 4734:Δ 4687:π 4613:∝ 4610:Ω 4561:ϕ 4555:θ 4547:ϕ 4534:θ 4523:∮ 4505:π 4495:ϕ 4489:θ 4483:θ 4480:⁡ 4471:π 4460:π 4454:θ 4445:∫ 4439:π 4430:ϕ 4421:∫ 4400:ϕ 4380:θ 4357:θ 4354:⁡ 4342:π 4333:θ 4330:⁡ 4298:π 4290:ϕ 4277:θ 4266:∮ 4246:θ 4221:ϕ 4208:θ 4169:θ 4166:⁡ 4145:ϕ 4125:θ 4074:ϕ 4068:θ 4028:ϕ 4022:θ 3839:Σ 3727:Δ 3721:Δ 3718:∫ 3703:Δ 3697:Δ 3694:∫ 3686:Δ 3680:Δ 3671:Ω 3641:≥ 3635:Δ 3629:Δ 3574:π 3393:π 3384:⁡ 3364:π 3355:⁡ 3335:π 3326:⁡ 3320:∝ 3317:ψ 3236:ϵ 3198:Δ 3185:π 3128:Δ 3062:Δ 3019:Δ 2976:Δ 2970:Δ 2858:log scale 2669:∣ 2651:∑ 2613:∣ 2589:∣ 2540:… 2476:routinely 2320:⁡ 2302:∫ 2299:− 2201:∗ 2178:∗ 2110:⁡ 2092:∫ 2088:− 2073:∗ 2046:⁡ 2040:− 1991:∗ 1939:∗ 1917:π 1907:⁡ 1852:π 1654:∣ 1630:∫ 1627:− 1570:⁡ 1552:∫ 1549:− 1487:⁡ 1469:∫ 1465:− 1438:∣ 1423:⁡ 1411:∣ 1399:∫ 1384:∫ 1249:∣ 1225:∫ 1203:⁡ 1197:∫ 1193:− 1166:∣ 1151:⁡ 1139:∣ 1127:∫ 1112:∫ 969:∣ 861:∣ 846:⁡ 834:∣ 822:∫ 807:∫ 586:) = 578:) = 337:parameter 101:Coherence 55:Posterior 7342:(2003). 6994:29297278 6935:10096507 6927:26355519 6695:88513041 6397:See also 4086:), i.e. 3951:S-matrix 2847:log-odds 2818:Examples 2484:Bayesian 539:, or an 471:variance 420:, then: 379:elicited 67:Evidence 7412:PriorDB 7332:3221355 7312:Bibcode 7278:0547240 7270:2985028 7241:1401831 7204:0804611 7177:2027492 7030:2984907 6985:5751802 6868:2332350 5918:whereas 4001:Example 2849:scale). 2782:, 1095:yields 7352: 7330: 7276: 7268: 7247: 7239: 7202: 7192: 7175: 7165: 7142: 7084: 7051: 7028: 6992: 6982: 6974: 6933: 6925: 6888: 6866: 6786: 6761: 6738: 6693: 6576: 6518: 6475: 6447: 6174:, and 6127:Since 3929:where 3422:where 2733:) and 2147:where 1960:where 1870:where 912:Here, 7391:(PDF) 7368:(PDF) 7328:S2CID 7302:arXiv 7266:JSTOR 7026:JSTOR 6931:S2CID 6864:JSTOR 6814:(PDF) 6736:S2CID 6691:S2CID 6673:arXiv 6634:arXiv 6516:S2CID 6424:Notes 5965:Thus 5383:with 2790:(for 2407:) or 531:, or 465:with 439:hyper 416:of a 328:prior 63:Prior 7350:ISBN 7190:ISBN 7163:ISBN 7140:ISBN 7082:ISBN 7049:ISBN 6990:PMID 6972:ISSN 6923:PMID 6886:ISBN 6784:ISBN 6759:ISBN 6574:ISBN 6473:ISBN 6445:ISBN 5500:and 4392:and 4192:The 3054:and 2841:=0, 2837:for 2826:The 635:and 529:flat 433:and 393:and 7380:doi 7320:doi 7245:Zbl 7227:doi 7074:doi 6980:PMC 6962:doi 6915:doi 6856:doi 6848:doi 6826:doi 6726:hdl 6718:doi 6683:doi 6644:doi 6603:doi 6599:617 6543:doi 6506:hdl 6498:doi 4599:is 4477:sin 4351:sin 4327:sin 4157:sin 3381:sin 3352:sin 3323:sin 2451:is 2399:or 2317:log 2107:log 2043:log 1904:log 1567:log 1484:log 1420:log 1200:log 1148:log 1044:log 843:log 705:on 570:of 523:An 457:An 350:In 7424:: 7376:18 7374:. 7370:. 7326:. 7318:. 7310:. 7298:37 7296:. 7288:; 7274:MR 7272:. 7262:41 7260:. 7243:. 7237:MR 7235:. 7223:24 7221:. 7215:. 7200:MR 7198:. 7173:MR 7171:. 7080:. 7022:35 6988:. 6978:. 6970:. 6958:18 6956:. 6952:. 6929:. 6921:. 6911:11 6909:. 6862:, 6820:. 6816:. 6798:^ 6734:. 6724:. 6714:90 6712:. 6689:. 6681:. 6669:32 6667:. 6642:. 6630:11 6628:. 6624:. 6597:. 6557:^ 6541:. 6537:. 6514:. 6504:. 6393:. 3881:Tr 3852:Tr 3842::= 3674::= 2860:). 2767:. 527:, 506:A 485:A 449:. 397:. 354:, 347:. 322:A 65:÷ 61:× 57:= 7400:. 7382:: 7358:. 7334:. 7322:: 7314:: 7304:: 7280:. 7251:. 7229:: 7206:. 7179:. 7148:. 7090:. 7076:: 7057:. 7032:. 6996:. 6964:: 6937:. 6917:: 6894:. 6870:. 6858:: 6850:: 6832:. 6828:: 6822:4 6792:. 6767:. 6742:. 6728:: 6720:: 6697:. 6685:: 6675:: 6652:. 6646:: 6636:: 6609:. 6605:: 6582:. 6551:. 6545:: 6522:. 6508:: 6500:: 6481:. 6453:. 6381:f 6358:r 6332:. 6329:) 6324:i 6318:r 6312:, 6307:i 6301:v 6295:, 6292:t 6289:( 6284:i 6280:f 6276:= 6271:i 6267:f 6262:, 6259:0 6256:= 6250:t 6247:d 6240:i 6236:f 6232:d 6209:t 6187:i 6183:g 6162:t 6140:i 6136:n 6115:. 6110:i 6106:g 6100:i 6096:f 6092:= 6087:i 6083:n 6060:i 6056:g 6035:T 6013:i 5986:D 5983:F 5978:i 5974:f 5953:. 5942:E 5939:B 5934:i 5930:f 5923:0 5913:, 5910:1 5902:D 5899:F 5894:i 5890:f 5883:0 5857:i 5830:i 5826:n 5803:i 5793:i 5789:n 5783:i 5775:= 5772:E 5750:0 5723:i 5719:n 5713:i 5705:= 5702:N 5682:. 5676:1 5668:T 5665:k 5661:/ 5657:) 5652:0 5639:i 5631:( 5627:e 5622:1 5617:= 5612:E 5609:B 5604:i 5600:f 5595:, 5589:1 5586:+ 5581:T 5578:k 5574:/ 5570:) 5565:0 5552:i 5544:( 5540:e 5535:1 5530:= 5525:D 5522:F 5517:i 5513:f 5484:f 5463:. 5451:n 5448:d 5443:2 5439:n 5410:2 5406:n 5401:/ 5397:1 5391:E 5369:2 5365:n 5338:n 5335:d 5305:/ 5301:E 5298:d 5251:. 5248:n 5245:d 5242:) 5239:1 5236:+ 5233:n 5230:2 5227:( 5181:. 5176:n 5172:E 5168:d 5161:2 5157:h 5152:I 5147:2 5139:8 5133:= 5130:n 5127:d 5124:) 5121:1 5118:+ 5115:n 5112:2 5109:( 5103:, 5095:2 5091:h 5087:) 5084:1 5081:+ 5078:n 5075:2 5072:( 5067:I 5062:2 5054:8 5048:= 5040:n 5036:E 5032:d 5027:n 5024:d 5001:) 4998:n 4995:d 4991:/ 4985:n 4981:E 4977:d 4974:( 4970:/ 4966:1 4963:= 4958:n 4954:E 4950:d 4946:/ 4942:n 4939:d 4919:, 4913:I 4908:2 4900:8 4893:2 4889:h 4885:) 4882:1 4879:+ 4876:n 4873:( 4870:n 4864:= 4859:n 4855:E 4832:2 4828:h 4823:/ 4819:E 4816:d 4813:I 4808:2 4800:8 4780:h 4776:/ 4772:p 4766:q 4743:p 4737:q 4706:. 4703:E 4700:d 4697:I 4692:2 4682:8 4662:E 4642:E 4639:d 4636:+ 4633:E 4587:E 4584:d 4564:, 4558:d 4552:d 4543:p 4539:d 4530:p 4526:d 4520:= 4517:E 4514:I 4509:2 4501:8 4498:= 4492:d 4486:d 4474:E 4468:I 4465:2 4457:= 4449:0 4436:2 4433:= 4425:0 4360:. 4348:E 4345:I 4339:2 4336:= 4322:E 4319:I 4316:2 4309:E 4306:I 4303:2 4295:= 4286:p 4282:d 4273:p 4269:d 4226:) 4217:p 4213:, 4204:p 4200:( 4180:. 4176:) 4161:2 4150:2 4141:p 4135:+ 4130:2 4121:p 4116:( 4109:I 4106:2 4102:1 4097:= 4094:E 4071:, 4048:q 4025:, 3981:N 3961:E 3937:N 3917:, 3913:. 3910:t 3907:s 3904:n 3901:o 3898:c 3894:= 3891:) 3888:P 3885:( 3877:= 3874:N 3868:, 3862:) 3859:P 3856:( 3847:P 3819:P 3796:t 3776:t 3756:, 3752:. 3749:t 3746:s 3743:n 3740:o 3737:c 3733:= 3730:p 3724:q 3712:, 3706:p 3700:q 3689:p 3683:q 3647:, 3644:h 3638:p 3632:q 3603:3 3599:h 3594:/ 3590:p 3587:d 3582:2 3578:p 3571:4 3568:V 3548:, 3543:2 3537:p 3531:= 3526:2 3522:p 3518:, 3515:p 3512:d 3509:+ 3506:p 3503:, 3500:p 3480:) 3477:n 3474:, 3471:m 3468:, 3465:l 3462:( 3442:n 3439:, 3436:m 3433:, 3430:l 3410:, 3407:) 3404:L 3400:/ 3396:z 3390:n 3387:( 3378:) 3375:L 3371:/ 3367:y 3361:m 3358:( 3349:) 3346:L 3342:/ 3338:x 3332:l 3329:( 3295:3 3291:L 3287:= 3284:V 3264:m 3261:2 3257:/ 3251:2 3245:p 3239:= 3214:3 3210:h 3205:/ 3201:p 3193:2 3189:p 3182:4 3179:V 3159:V 3139:h 3135:/ 3131:p 3125:L 3105:L 3085:p 3065:p 3042:q 3022:q 2999:h 2979:p 2973:q 2947:i 2943:p 2920:i 2916:q 2895:V 2843:β 2839:α 2792:v 2788:v 2784:v 2780:m 2778:( 2776:p 2764:j 2760:A 2756:B 2743:j 2739:A 2737:( 2735:P 2730:i 2726:A 2724:( 2722:P 2708:, 2701:) 2696:j 2692:A 2688:( 2685:P 2682:) 2677:j 2673:A 2666:B 2663:( 2660:P 2655:j 2645:) 2640:i 2636:A 2632:( 2629:P 2626:) 2621:i 2617:A 2610:B 2607:( 2604:P 2598:= 2595:) 2592:B 2584:i 2580:A 2576:( 2573:P 2551:n 2547:A 2543:, 2537:, 2532:2 2528:A 2524:, 2519:1 2515:A 2457:p 2453:p 2449:p 2370:x 2367:d 2362:] 2356:) 2353:x 2350:( 2347:I 2344:k 2339:) 2336:x 2333:( 2330:p 2324:[ 2314:) 2311:x 2308:( 2305:p 2296:= 2293:L 2290:K 2270:x 2250:) 2247:x 2244:( 2241:p 2221:x 2198:x 2175:x 2155:k 2135:x 2132:d 2128:] 2125:) 2122:x 2119:( 2116:p 2113:[ 2104:) 2101:x 2098:( 2095:p 2084:) 2078:) 2069:x 2065:( 2062:I 2059:k 2054:1 2050:( 2037:= 2034:L 2031:K 2011:t 1988:x 1968:N 1944:) 1935:x 1931:( 1928:I 1925:N 1920:e 1914:2 1901:= 1898:H 1878:v 1858:v 1855:e 1849:2 1829:x 1809:t 1789:x 1765:x 1745:t 1725:x 1705:t 1683:) 1680:x 1677:( 1674:H 1670:+ 1667:t 1664:d 1660:) 1657:t 1651:x 1648:( 1645:H 1642:) 1639:t 1636:( 1633:p 1624:= 1621:L 1618:K 1598:. 1595:x 1592:d 1588:] 1585:) 1582:x 1579:( 1576:p 1573:[ 1564:) 1561:x 1558:( 1555:p 1546:= 1543:) 1540:x 1537:( 1534:H 1512:x 1509:d 1505:] 1502:) 1499:x 1496:( 1493:p 1490:[ 1481:) 1478:x 1475:( 1472:p 1461:t 1458:d 1454:x 1451:d 1447:] 1444:) 1441:t 1435:x 1432:( 1429:p 1426:[ 1417:) 1414:t 1408:x 1405:( 1402:p 1396:) 1393:t 1390:( 1387:p 1381:= 1378:L 1375:K 1355:) 1352:x 1349:( 1346:p 1326:) 1323:t 1320:, 1317:x 1314:( 1311:p 1291:t 1269:x 1266:d 1262:t 1259:d 1255:) 1252:t 1246:x 1243:( 1240:p 1237:) 1234:t 1231:( 1228:p 1221:] 1218:) 1215:x 1212:( 1209:p 1206:[ 1189:t 1186:d 1182:x 1179:d 1175:] 1172:) 1169:t 1163:x 1160:( 1157:p 1154:[ 1145:) 1142:t 1136:x 1133:( 1130:p 1124:) 1121:t 1118:( 1115:p 1109:= 1106:L 1103:K 1083:t 1063:] 1060:) 1057:x 1054:( 1051:p 1048:[ 1024:t 1004:) 1001:x 998:( 995:p 975:) 972:t 966:x 963:( 960:p 940:x 920:t 898:t 895:d 891:x 888:d 881:) 878:x 875:( 872:p 867:) 864:t 858:x 855:( 852:p 840:) 837:t 831:x 828:( 825:p 819:) 816:t 813:( 810:p 804:= 801:L 798:K 788:x 786:( 784:p 780:x 778:( 776:p 772:X 743:X 719:x 715:X 707:X 694:X 683:p 679:p 649:1 646:= 643:p 623:0 620:= 617:p 607:p 603:p 592:C 590:( 588:p 584:B 582:( 580:p 576:A 574:( 572:p 435:β 431:α 425:p 414:p 311:e 304:t 297:v 20:)

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