{'fraction_non_alphanumeric': 0.04060944891564785, 'fraction_numerical': 0.011550039933648707, 'mean_word_length': 4.372277227722773, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Objective Bayesian Statistics

Bayesian inference --- although becoming popular in physics and chemistry --- is hampered up to now by the vagueness of its notion of prior probability. Some of its supporters argue that this vagueness is the unavoidable consequence of the subjectivity of judgements --- even scientific ones. We argue that priors can be defined uniquely if the statistical model at hand possesses a symmetry and if the ensuing confidence intervals are subjected to a frequentist criterion. Moreover, it is shown via an example taken from recent experimental nuclear physics, that this procedure can be extended to models with broken symmetry.


INTRODUCTION
Bayesian inference is becoming popular in the physical sciences. It finds eloquent and noteworthy defenders [1,2], it is treated in a growing number of textbooks [3][4][5][6][7], an annual series of conferences [8] as well as numerous articles report on its applications. Even the "Guide to the expression of uncertainty in measurement" [9] supported by the International Organization for Standardization (ISO), the Bureau International des Poids et Mésures (BIPM) and other international organizations implicitly favors Bayesian statistics -as D'Agostini points out (in Secs. 2.2 and 14.2 of [2]). Some authors that have collaborated in the formulation of the Guide [9] and of the German standard [10], adhere in their publications to "a Bayesian theory of measurement uncertainty" [11]. The change of paradigm from frequentist to Bayesian statistics (for this antagonism see e.g. the introduction of [11] and ch. 7 of [12]) is taking place despite the fact, that the disturbing vagueness of Bayesian prior probabilities persists up to now. D'Agostini [2] eloquently holds that the prior probabilities are the mathematical representation of an unavoidable subjectivity of judgments -even of scientific ones. In our opinion, however, a criterion exists that prior probabilities should meet -at least approximately -and which in many cases does not leave any freedom in their choice. This criterion is a consequence of the fact that Bayesian inference has a frequentist interpretation -as will be explained below. This fact has been pointed out earlier (by Welch, Peers, Stein and Villegas [13][14][15][16][17]) in the literature of mathematical statistics but it seems rarely known in practice. In the present note, the criterion is described as well as the circumstances under which it is met. They amount to the existence of a symmetry of the model that states the relation between event and hypothesis. The Bayesian prior is then a Haar measure of the symmetry group. By helps of a realistic example taken from current nuclear physics it is shown that this procedure can be extended to the usual case of models with broken symmetry. Finally the same example shows that the popular likelihood method -as described in [18]-yields confidence intervals of lower reliability than the Bayesian procedure.

BAYESIAN STATISTICS
Bayesian statistics is a very useful tool for statistical inference. Let x denote the continuous event and ξ the continuous hypotheses. Suppose that the conditional probability p(x|ξ)dx -the above mentioned model -as well as the event x are given. The problem of statistical inference is: What can we learn from x about ξ? The Bayesian answer [19] is a conditional probability P for ξ given x which can be expressed in terms of the given distribution p, Here, µ(ξ) is the prior distribution assigned to ξ in the absence of or "prior to" any experimental evidence. Hence, Bayesian statistics relies on the assumption that µ(ξ) is a meaningful object. The choice of the prior is the problem of Bayesian theory, because it is not clear what we can know "prior" about ξ.

SUBJECTIVE INTERPRETATION
The view held e.g. in the recent article by D'Agostini [2] or the recent textbook by Howson and Urbach [6] is that the choice of µ(ξ) should be left to the good taste and the experience of the scientist analyzing the event x. They argue that a such a subjective element should be in any honest theory of inference since it makes explicit the subjectivity of any judgement -including the scientific ones: nothing can be known about ξ in an objective way prior to the event. In subjective interpretation Bayesian probabilities reflect personal confidence in hypotheses. It can be expressed in bets (see [6]).
This view does in principle not preclude objectivity altogether: Objectivity exists as the limiting consequence of an infinite number of recorded events. Indeed one can show that in this limit Bayesian inference becomes independent of µ(ξ) in the sense that p(x|ξ) then tends towards a δ-distribution with respect to ξ -centered at the true valueξ of the hypothesis.

OBJECTIVE BAYESIAN STATISTICS
The subjective interpretation -taken literally and without appealing to some common sense -allows anything. This is obvious from eq. (1): if one is free to choose µ(ξ), one can generate any P (ξ|x) -given a finite number of events. To avoid this, we prefer to look for some criterion that would severely restrict the class of allowed priors. Fortunately there is a very natural one.
Consider a Bayesian confidence area A(x, C). It shall satisfy which is usually stated in the form: "given the event x, the hypothesis lies with confidence C in the area A". We want to reformulate this in a way which turns the vague notion of confidence into probability and by the same token defines the desired criterion. Imagine an ensemble X of events x with relative frequencies p(x|ξ)dx. Let x run over the ensemble and suppose that from every x the confidence area A(x, C) is derived in a unique way, e.g. by determing the smallest one. This yields an ensemble of confidence areas. The criterion then is: The prior must lead to an ensemble of confidence areas such that they cover the true valueξ with probability C.
Since this gedanken experiment -which can even be realized via Monte Carlo simulation -equates C with a well defined frequency (to coverξ), the criterion turns confidence into frequentist probability. In short: We require that Bayesian confidence areas are frequentist confidence areas. It is proven in the mathematical literature [14] that this criterion can be met exactly: Let the conditional probability p(x|ξ)dx be invariant under a Lie group G represented as transformations of x and ξ, i.e.
suppose furthermore that the definition of the confidence area A is invariant under G, i.e then the right Haar measure of the symmetry group is a suitable prior in the sense of the criterion. It is necessary for this, that the hypothesis can be identified with the symmetry group of the conditional probability, i.e. for every two hypotheses ξ 1 ,ξ 2 there must be exactly one transformation G ρ ∈ G such that ξ 1 = G ρ ξ 2 . Then the uniqueness of the right Haar measure implies the uniqueness of the prior and one has which is the inverse Jacobian of the transformation G ρ taken at the unit element of the group.
Note that the very interesting case of A to be the smallest area of confidence C, satisfies eq. (4), if the volume of A is determined by help of the measure that is invariant under G, i.e. the left Haar measure.
The prior distribution is not independent of the statistical model; rather it is defined by the structure of the model (see [16,20]). It is prior to the observations. Since the above symmetry uniquely defines the prior and since it ensures a frequentist interpretation of Bayesian confidence intervals and since frequentist probability is often termed "objective" probability, one may call the present procedure objective Bayesian statistics. However, we return to the word "Bayesian" implying for the rest of the paper the qualification "objective". By the following example, we show that this concept can be extended to models that lack the symmetry (3).

EXAMPLE
The measurement problems encountered in science, often do not have a symmetry (3). However, Bayesian inference based on a multi-dimensional event can usually be broken down into a succession of Bayesian arguments on more elementary events until a starting point that possesses a symmetry has been found. To be definite let us consider the parity violating matrix elements measured by the TRIPLE collaboration in resonant p-wave scattering of polarized neutrons on heavy nuclei [21][22][23][24]. For the present purpose it is not necessary to understand the details of those experiments. It suffices to know that the parity violating matrix elements -the events -have the Gaussian probability where ε is an experimental error -supposed to be given -and M is the root mean square parity violating matrix element. The latter one is the "hypothesis" to be determined. There is no symmetry (3) relating x and M. However, the probability of eq. (6) remains invariant under simultaneous change of the scale of x and ξ = √ M 2 + ε 2 . Hence, Bayesian inference should be done with respect to ξ rather than M. Afterwards -when the posterior distribution of ξ has been constructed -one can derive statements about M. E.g. one can decide whether with sufficiently high confidence ξ is larger than ε and thus M > 0. The symmetry considerations require as prior, which is the Haar measure of the Lie group of scale changes. The problem of [21][22][23][24] is complicated by the fact, that one usually does not know the total angular momentum of the p-wave resonance. Only p 1/2 resonances can show parity violation. If the event x is gathered in a p 3/2 resonance nothing can be learned about M and the distribution of the event is g(x|ε). One knows, however, the probability q p of the occurrence of p 1/2 resonances. Again it is not necessary to discuss here the contents of nuclear physics that create this complication. It suffices to state, that the event x follows with probability q p the distribution g(x|ξ) and with probability 1−q p the distribution g(x|ε), so that one has The presence of the second term on the right hand side precludes any symmetry (3) relating x and ξ -except in the limit of ε → 0 when g(x|ε) → δ(x) and p(x|ξ) recovers the symmetry under scale changes. The full experiment of [21][22][23][24] probes n resonances i = 1, . . . , n under varying experimental conditions, so that ε = ε i becomes a function of the resonance i. This alone precludes any symmetry of the multidimensional problem.
If, however, one knows for one of the resonances that it is a p 1/2 case, say for i = 1, then the event measured there has the distribution g(x 1 |ξ); the symmetry is scale-invariance and the prior is (7); one can use the event x 1 to construct the Bayesian inverse P 1 (ξ|x 1 ); it can be injected as prior distribution into the analysis of the results at i = 2, . . . , n. This procedure leads to a posterior distribution P (ξ|x 1 , . . . , x n ). In this way, one finds a starting point of the whole analysis which has a well defined symmetry and therefore a well defined prior. Let us call the whole procedure an approximate Bayesian (AB) one.
It is not our purpose to reanalyze the data of refs. [21][22][23][24]. We only want to demonstrate, that the AB analysis satisfies the criterion described above to a good approximation. For the purpose of this, the AB analysis has been subjected to a Monte Carlo test with parameters close to those of the experimental cases [21][22][23][24]. The number of resonances was chosen to be n = 15. The errors ε i , i = 1, . . . , 15 have been drawn from an exponential distribution with mean value rM. This allows one to study (on fig. 1) the result as a function of r =ε/Mthe mean error relative to the true valueM of the r.m.s. parity violating matrix element. In the experiments [21][22][23][24], r ranged from 0.23 up to the order of unity. The coordinates x i of the event x = (x 1 , . . . , x 15 ) were generated in two steps. First one decides with probability q p = 1 3 , whether the quantity x i should belong to the p 1/2 -wave resonances. If yes then x i was drawn from an ensemble with distribution g(x i | M2 + ε 2 i ), else the distribution g(x i |ε i ) was applied. The vector x -without the information from which of the two ensembles anyone of the x i comes -is equivalent to the experimental "event".
The event x was analyzed by assuming that the "resonance" i m , where the maximum of the ratio x i /ε i occurs, is a p 1/2 resonance and can serve as the starting point of the analysis in the above sense. By help of the posterior distribution P (ξ|x), the shortest confidence interval (ξ < , ξ > ) was found such that ξ> ξ< dηP (η|x 1 , . . . , x n ) = 0.68 and it was recorded whether the true valueM was inside (max(0, ξ 2 < − ε 2 1 ), max(0, ξ 2 > − ε 2 1 )). This procedure was applied to 10 4 vectors x. On fig. 1, the symbols labeled "Bayes" give the relative frequency of "success", i.e the probability to findM in the above mentioned range performed for different r which controls the experimental error. In this way, the curve on the figure was generated. (The full and dashed lines are 4th order polynomials fit to the points). Because of the lack of symmetry, the result is not identical but only close to 0.68 (the dotted line). However, in the limit of r → 0 which means ε i → 0 for all i, the criterion is obeyed exactly -as it should be, since scale invariance is recovered in this limit. In comparison, a "likelihood" analysis was performed. This type of analysis is very popular. It amounts to the Bayesian procedure with the prior distribution set constant. In the present case, we had to cut off the posterior distribution for M ≫M in order to normalize it very much as in [21][22][23]. The figure shows, that the likelihood method is inferior to even the approximate Bayesian method. For good data, i.e. r → 0, the likelihood method yields confidence intervals that are -in the light of the criterion -too wide. For data of marginal quality (r ≈ 1), it yields no reliable confidence intervals, because the frequency of successes is considerably lower than the prescribed confidence.

SUMMARY
We have defined objective Bayesian statistics by supplementing the Bayesian argument with the requirement that it should yield "objectively correct" confidence intervals. A theorem by Stein -which can be found in the published mathematical literature [14] but which seems to be unknown in practice -shows that this requirement can be met provided that the conditional probability p(x|ξ)dx posses the symmetry (3) defined by a Lie group. By way of an example we have extended objective Bayesian statistics to the common case in which p(x|ξ)dx does not have an exact symmetry. In the example, a complex event x = (x 1 , . . . , x n ) is broken down into elementary ones x i among which there is at least one -say x 1 -whose conditional probability p(x 1 |ξ)dx possesses a symmetry (3). It is used to define the prior µ(ξ). This has been termed approximate Bayesian procedure. We have shown numerically that the AB procedure is superior to the popular likelihood method as judged by the objectivity of the deduced confidence intervals.
Note that the arguments presented here amount to a reconciliation of the subjective and frequentist interpretations of probability. The Bayesian argument attributes a distribution to an object, i.e. the hypothesis, which is given by Nature once and for all. This is justified by interpreting probability distributions as a representation of subjective knowledge on that object. The frequentist interpretation insists that a probability distribution must be verifiable -at least in a gedanken experiment -as a frequency distribution that occurs in some stochastic process. We have described a gedanken experiment to generate a distribution of Bayesian confidence intervals from data that are conditioned by a fixed true value of the hypothesis. The rate of success, i.e. of the true value lying inside the confidence interval, turns out to be independent of the true value, moreover the rate of success is equal to the confidence prescribed in the Bayesian procedure -if the conditional distribution possesses the symmetry (3) and if the prior is chosen to be the right Haar measure. Then the Bayesian inference is found reasonable from a frequentist's point of view.