scholarly journals On the relation between direct and inverse methods in statistics

1937 ◽  
Vol 160 (902) ◽  
pp. 325-348 ◽  
Keyword(s):  
Standard Error ◽  
Inverse Method ◽  
Posterior Probability ◽  
Direct Methods ◽  
Previous Knowledge ◽  
Inverse Probability ◽  
True Value ◽  
Scientific Inference ◽  
Normal Law ◽  
True Values

1—In my “Scientific Inference” (1931) I gave a discussion of the posterior probability of a parameter based on a series of observations derived from the normal law of errors, when the parameter and its standard error are originally unknown. I afterwards (1932) generalized the result to the case of several unknowns. Dr J. Wishart pointed out to me that the form that I obtained by using the principle of inverse probability is identical with one obtained by direct methods by “Student” (1908). A formula identical with my general one was also given by T. E. Sterne (1934). The direct methods, however, deal with a different problem from the inverse one. They give results about the probability of the observations, or certain functions of them, the true values and the standard error being taken as known; the practical problem is usually to estimate the true value, the observations being known, and this is the problem treated by the inverse method. As the data and the propositions whose probabilities on the data are to be assessed are interchanged in the two cases it appeared to me that the identity of the results in form must be accidental. It turns out, however, that there is a definite reason why they should be identical, and that this throws a light on the use of direct methods for estimation and on their relation to the theory of probability. Suppose that the true value and the standard error are x and σ; the observed values are n in number with mean x̅ and standard deviation σ'. Then my result (1931, p. 69) was, for previous knowledge k expressing the truth of the normal law but nothing about x and σ, P ( dx | x̅ , σ', k ) ∝ {1 + ( x - x̅ ) 2 /σ' 2 } -½ n dx ; (1)

scholarly journals On the theory of errors and least squares

1932 ◽  
Vol 138 (834) ◽  
pp. 48-55 ◽  
Keyword(s):  
Least Squares ◽  
Prior Probability ◽  
Alternative Form ◽  
Probability Of Error ◽  
The Third ◽  
True Value ◽  
Scientific Inference ◽  
Two Measures ◽  
Normal Law

1. In my “Scientific Inference,” chapter V, I found that the usual presentation of the theory of errors of observation needed some modification, even where the probability of error is distributed according to the normal law. One change made was in the distribution of the prior probability of the precision constant h . Whereas this is usually taken as uniform (or ignored), I considered it better to assume that the prior probability that the constant lies in a range dh is proportional to dh/h . This is equivalent to assuming that if h 1 / h 2 = h 3 / h 4 , h is as likely to lie between h 1 and h 2 as between h 3 and h 4 ; this was thought to be the best way of expressing the condition that there is no previous know­ ledge of the magnitude of the errors. The relation must break down for very small h , comparable with the reciprocal of the whole length of the scale used, and for large h comparable with the reciprocal of the step of the scale; but for the range of practically admissible values it appeared to be the most plausible distribution. The argument for this law can now be expressed in an alternative form. The normal law of error is supposed to hold, but the true value x and the pre­cision constant h are unknown. Two measures are made: what is the pro ability that the third observation will lie between them ? The answer is easily seen to be one-third.

scholarly journals An alternative to the rejection of observations

1932 ◽  
Vol 137 (831) ◽  
pp. 78-87 ◽  
Keyword(s):  
Standard Deviation ◽  
Continuous Function ◽  
Large Deviations ◽  
Opposite Direction ◽  
Standard Error ◽  
The Other ◽  
Cause Of Error ◽  
The Mean ◽  
Normal Law ◽  
Usual Rule

1. It is widely felt that any method of rejecting observations with large deviations from the mean is open to some suspicion. Suppose that by some criterion, such as Peirce’s and Chauvenet’s, we decide to reject observations with deviations greater than 4 σ, where σ is the standard error, computed from the standard deviation by the usual rule; then we reject an observation deviating by 4·5 σ, and thereby alter the mean by about 4·5 σ/ n , where n is the number of observations, and at the same time we reduce the computed standard error. This may lead to the rejection of another observation deviating from the original mean by less than 4 σ, and if the process is repeated the mean may be shifted so much as to lead to doubt as to whether it is really sufficiently representative of the observations. In many cases, where we suspect that some abnormal cause has affected a fraction of the observations, there is a legitimate doubt as to whether it has affected a particular observation. Suppose that we have 50 observations. Then there is an even chance, according to the normal law, of a deviation exceeding 2·33 σ. But a deviation of 3 σ or more is not impossible, and if we make a mistake in rejecting it the mean of the remainder is not the most probable value. On the other hand, an observation deviating by only 2 σ may be affected by an abnormal cause of error, and then we should err in retaining it, even though no existing rule will instruct us to reject such an observation. It seems clear that the probability that a given observation has been affected by an abnormal cause of error is a continuous function of the deviation; it is never certain or impossible that it has been so affected, and a process that completely rejects certain observations, while retaining with full weight others with comparable deviations, possibly in the opposite direction, is unsatisfactory in principle.

On the Smoothing of Observed Data

1937 ◽  
Vol 33 (4) ◽  
pp. 444-450 ◽  
Author(s):  
Harold Jeffreys
Keyword(s):  
Standard Error ◽  
Random Error ◽  
Simple Function ◽  
Standard Errors ◽  
Random Errors ◽  
Previous Theory ◽  
Know How ◽  
True Value ◽  
Actual Error ◽  
Range Of Variation

1. It often happens that we have a series of observed data for different values of the argument and with known standard errors, and wish to remove the random errors as far as possible before interpolation. In many cases previous considerations suggest a form for the true value of the function; then the best method is to determine the adjustable parameters in this function by least squares. If the number required is not initially known, as for a polynomial where we do not know how many terms to retain, the number can be determined by finding out at what stage the introduction of a new parameter is not supported by the observations*. In many other cases, again, existing theory does not suggest a form for the solution, but the observations themselves suggest one when the departures from some simple function are found to be much less than the whole range of variation and to be consistent with the standard errors. The same method can then be used. There are, however, further cases where no simple function is suggested either by previous theory or by the data themselves. Even in these the presence of errors in the data is expected. If ε is the actual error of any observed value and σ the standard error, the expectation of Σε2/σ2 is equal to the number of observed values. Part, at least, of any irregularity in the data, such as is revealed by the divided differences, can therefore be attributed to random error, and we are entitled to try to reduce it.

Optimizing the liquidity parameter of logarithmic market scoring rules prediction markets

2018 ◽  
Vol 13 (3) ◽  
pp. 736-754
Author(s):  
Suparerk Lekwijit ◽  
Daricha Sutivong
Keyword(s):  
Standard Error ◽  
Market Price ◽  
Simulation Models ◽  
Scoring Rules ◽  
Prediction Markets ◽  
Market Mechanisms ◽  
Content Type ◽  
Market Makers ◽  
True Value ◽  
Price Functions

Purpose Prediction markets are techniques to aggregate dispersed public opinions via market mechanisms to predict uncertain future events’ outcome. Many experiments have shown that prediction markets outperform other traditional forecasting methods in terms of accuracy. Logarithmic market scoring rules (LMSR) is one of the most simple and widely used market mechanisms; however, market makers have to confront crucial design decisions including the setting of the parameter “b” or the “liquidity parameter” in the price functions. As the liquidity parameter has significant effects on the market performance, this paper aims to provide a comprehensive basis for the setting of the parameter. Design/methodology/approach The analyses include the effects of the liquidity parameter on the forecast standard error and the amount of time for the market price to converge to the true value. These experiments use artificial prediction markets, the proposed simulation models that mimic real prediction markets. Findings The simulation results indicate that prediction market’s forecast standard error decreases as the value of the liquidity parameter increases. Moreover, for any given number of traders in the market, there exists an optimal liquidity parameter value that yields appropriate price adaptability and leads to the fastest price convergence. Originality/value Understanding these tradeoffs, the market makers can effectively determine the liquidity parameter value under various objectives on the standard error, the time to convergence and cost.

On Gauss's proof of the normal law of errors

1933 ◽  
Vol 29 (2) ◽  
pp. 231-234 ◽  
Author(s):  
Harold Jeffreys
Keyword(s):  
Arithmetic Mean ◽  
Practical Case ◽  
Apparent Discrepancy ◽  
True Value ◽  
The Mean ◽  
Normal Law

Gauss gave a well-known proof that under certain conditions the postulate that the arithmetic mean of a number of measures is the most probable estimate of the true value, given the observations, implies the normal law of error. I found recently that in an important practical case the mean is the most probable value, although the normal law does not hold. I suggested an explanation of the apparent discrepancy, but it does not seem to be the true one in the case under consideration.

scholarly journals Estimating Mean Fruit Weight and Mean Fruit Value for Apple Trees: Comparison of Two Sampling Methods with the True Mean

2001 ◽  
Vol 126 (4) ◽  
pp. 503-510 ◽  
Author(s):  
Richard P. Marini
Keyword(s):  
Sampling Methods ◽  
Fruit Weight ◽  
Estimation Methods ◽  
Statistical Techniques ◽  
Apple Trees ◽  
Data Set ◽  
True Value ◽  
Malus Sylvestris ◽  
Fruit Sample ◽  
True Values

Data obtained over two years from chemical thinning experiments with `Redchief Delicious' apple [Malus sylvestris (L.) Mill. var. domestica (Borkh.) Manst.] on Malling 26 (M.26) rootstock were used to estimate mean fruit weight (MFW) and mean fruit value (MFV) using two sampling methods. The estimated values were compared with the true MFW and the true MFV calculated from the entire crop from a tree. Statistical techniques were used to assess agreement between the values obtained with estimation methods and the true values. Estimates of MFW obtained from a 20-fruit sample per tree may differ from the true value by ≈13% and estimates obtained from weighing all fruit on three limbs per tree may range from 11% to 19% of the true mean. Estimates of MFV obtained from packouts of a 20-fruit sample may differ from the true value by about $0.04 (U.S. dollars)/fruit and estimates from packing out all fruit on three limbs per tree may differ from the true mean by about $0.07/fruit. Analysis of variance was performed on each data set. The resulting P values differed for the three methods of calculating MFW and MFV. Therefore, erroneous conclusions may result from experiments where MFW and MFV are estimated from subsamples. Error associated with estimating fruit weight and fruit value from the sampling methods employed in this study may be larger than many pomologists can accept. Until protocols for sampling apple trees, which account for the important sources of within-tree variation, are developed, researchers should consider harvesting the entire crop to calculate MFW and MFV.

Some Tests of Significance, Treated by the Theory of Probability

1935 ◽  
Vol 31 (2) ◽  
pp. 203-222 ◽  
Author(s):  
Harold Jeffreys
Keyword(s):  
Standard Error ◽  
Tests Of Significance ◽  
Usual Procedure ◽  
True Value ◽  
The Difference

It often happens that when two sets of data obtained by observation give slightly different estimates of the true value we wish to know whether the difference is significant. The usual procedure is to say that it is significant if it exceeds a certain rather arbitrary multiple of the standard error; but this is not very satisfactory, and it seems worth while to see whether any precise criterion can be obtained by a thorough application of the theory of probability.

scholarly journals An Information-Efficient Bayesian Model for AMS Data Analysis

Radiocarbon ◽  
2007 ◽  
Vol 49 (2) ◽  
pp. 369-377 ◽  
Author(s):  
V Palonen ◽  
P Tikkanen
Keyword(s):  
Data Analysis ◽  
Gaussian Distribution ◽  
Bayesian Model ◽  
Simulated Data ◽  
True Value ◽  
Standard Level ◽  
Changing Trend ◽  
Non Gaussian ◽  
Instrumental Drift ◽  
True Values

A Bayesian model for accelerator mass spectrometry (AMS) data analysis is presented. Instrumental drift is modeled with a continuous autoregressive (CAR) process, and measurement uncertainties are taken to be Gaussian. All samples have a parameter describing their true value. The model adapts itself to different instrumental parameters based on the data, and yields the most probable true values for the unknown samples. The model is able to use the information in the measurements more efficiently. First, all measurements tell something about the overall instrument performance and possible drift. The overall machine uncertainty can be used to obtain realistic uncertainties even when the number of measurements per sample is small. Second, even the measurements of the unknown samples can be used to estimate the variations in the standard level, provided that the samples have been measured more than once. Third, the uncertainty of the standard level is known to be smaller nearer a standard. Fourth, even though individual measurements follow a Gaussian distribution, the end result may not.For simulated data, the new Bayesian method gives more accurate results and more realistic uncertainties than the conventional mean-based (MB) method. In some cases, the latter gives unrealistically small uncertainties. This can be due to the non-Gaussian nature of the final result, which results from combining few samples from a Gaussian distribution without knowing the underlying variance and from the normalization with an uncertain standard level. In addition, in some cases the standard error of the mean does not represent well the true error due to correlations within the measurements resulting from, for example, a changing trend. While the conventional method fails in these cases, the CAR model gives representative uncertainties.

scholarly journals Least Squares Estimation Without Priors or Supervision

2011 ◽  
Vol 23 (2) ◽  
pp. 374-420 ◽  
Author(s):  
Martin Raphan ◽  
Eero P. Simoncelli
Keyword(s):  
Least Squares ◽  
Empirical Bayes ◽  
Estimation Error ◽  
Kernel Functions ◽  
Least Squares Estimator ◽  
Supervised Training ◽  
True Value ◽  
Optimal Estimator ◽  
Unbiased Risk ◽  
True Values

Selection of an optimal estimator typically relies on either supervised training samples (pairs of measurements and their associated true values) or a prior probability model for the true values. Here, we consider the problem of obtaining a least squares estimator given a measurement process with known statistics (i.e., a likelihood function) and a set of unsupervised measurements, each arising from a corresponding true value drawn randomly from an unknown distribution. We develop a general expression for a nonparametric empirical Bayes least squares (NEBLS) estimator, which expresses the optimal least squares estimator in terms of the measurement density, with no explicit reference to the unknown (prior) density. We study the conditions under which such estimators exist and derive specific forms for a variety of different measurement processes. We further show that each of these NEBLS estimators may be used to express the mean squared estimation error as an expectation over the measurement density alone, thus generalizing Stein's unbiased risk estimator (SURE), which provides such an expression for the additive gaussian noise case. This error expression may then be optimized over noisy measurement samples, in the absence of supervised training data, yielding a generalized SURE-optimized parametric least squares (SURE2PLS) estimator. In the special case of a linear parameterization (i.e., a sum of nonlinear kernel functions), the objective function is quadratic, and we derive an incremental form for learning this estimator from data. We also show that combining the NEBLS form with its corresponding generalized SURE expression produces a generalization of the score-matching procedure for parametric density estimation. Finally, we have implemented several examples of such estimators, and we show that their performance is comparable to their optimal Bayesian or supervised regression counterparts for moderate to large amounts of data.

Mass and composition of the fat-free tissues of patients with weight-loss

1985 ◽  
Vol 68 (4) ◽  
pp. 455-462 ◽  
Author(s):  
L. Burkinshaw ◽  
D. B. Morgan
Keyword(s):  
Factor Analysis ◽  
Standard Error ◽  
Healthy Subjects ◽  
The Body ◽  
Fat Free Mass ◽  
Random Component ◽  
Total Body Potassium ◽  
Random Components ◽  
Total Body ◽  
True Values

1. An estimate of the mass of fat-free tissue in the body can be calculated from body weight and skinfold thickness; this estimate is called the ‘fat-free mass'. Total body potassium and nitrogen are alternative estimates. Factor analysis of data for healthy subjects has defined relationships between the true values of these three quantities and estimated the random component of the variance of each, i.e. the component independent of variations in the mass of fat-free tissue. The results indicated that all three were reliable measures of the mass of fat-free tissue. However, it is not known whether these findings are valid for patients who have lost weight. 2. We have measured the same three quantities in 104 wasted patients with heart disease or disorders of the gastrointestinal tract. The patients’ mean values were significantly less than corresponding values for healthy volunteers. The patients had a mean ratio of total body nitrogen to fat-free mass similar to that of healthy subjects, but lower mean ratios of potassium to fat-free mass and nitrogen. These findings suggest that the potassium content of the patients’ fat-free tissues was abnormally low. 3. Factor analysis of the patients’ data gave relationships between the true values of the three quantities similar to those for healthy subjects; however, total body potassium was 100-300 mmol lower in patients than in healthy subjects with the same fat-free mass or total body nitrogen. 4. Factor analysis also showed that the random components of variance of fat-free mass and total body nitrogen were similar to those in healthy subjects. Therefore, in the patients as in healthy subjects, fat-free mass was as valid a measure of fat-free tissue as the more complex measurement of total body nitrogen. The random component of total body potassium was twice as big as in healthy subjects; however, it formed no greater a proportion of total variance than did the random components of the other two quantities. 5. Total body nitrogen, and hence body protein, could be estimated from measured fat-free mass with a standard error of approximately 136 g (compared with 139 g for healthy individuals), and from total body potassium with a standard error of 129 g (compared with 91 g in healthy subjects).

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