### 统计代写|统计推断作业代写statistics interference代考|Interpretations of uncertainty

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|统计推断作业代写statistics interference代考|General remarks

We can now consider some issues involved in formulating and comparing the different approaches.

In some respects the Bayesian formulation is the simpler and in other respects the more difficult. Once a likelihood and a prior are specified to a reasonable approximation all problems are, in principle at least, straightforward. The resulting posterior distribution can be manipulated in accordance with the ordinary laws of probability. The difficulties centre on the concepts underlying the definition of the probabilities involved and then on the numerical specification of the prior to sufficient accuracy.

Sometimes, as in certain genetical problems, it is reasonable to think of $\theta$ as generated by a stochastic mechanism. There is no dispute that the Bayesian approach is at least part of a reasonable formulation and solution in such situations. In other cases to use the formulation in a literal way we have to regard probability as measuring uncertainty in a sense not necessarily directly linked to frequencies. We return to this issue later. Another possible justification of some Bayesian methods is that they provide an algorithm for extracting from the likelihood some procedures whose fundamental strength comes from frequentist considerations. This can be regarded, in particular, as supporting
$5.2$ Broad roles of probability
a broad class of procedures, known as shrinkage methods, including ridge regression.

The emphasis in this book is quite often on the close relation between answers possible from different approaches. This does not imply that the different views never conflict. Also the differences of interpretation between different numerically similar answers may be conceptually important.

## 统计代写|统计推断作业代写statistics interference代考|Broad roles of probability

A recurring theme in the discussion so far has concerned the broad distinction between the frequentist and the Bayesian formalization and meaning of probability. Kolmogorov’s axiomatic formulation of the theory of probability largely decoupled the issue of meaning from the mathematical aspects; his axioms were, however, firmly rooted in a frequentist view, although towards the end of his life he became concerned with a different interpretation based on complexity. But in the present context meaning is crucial.

There are two ways in which probability may be used in statistical discussions. The first is phenomenological, to describe in mathematical form the empirical regularities that characterize systems containing haphazard variation. This typically underlies the formulation of a probability model for the data, in particular leading to the unknown parameters which are regarded as a focus of interest. The probability of an event $\mathcal{E}$ is an idealized limiting proportion of times in which $\mathcal{E}$ occurs in a large number of repeat observations on the system under the same conditions. In some situations the notion of a large number of repetitions can be reasonably closely realized; in others, as for example with economic time series, the notion is a more abstract construction. In both cases the working assumption is that the parameters describe features of the underlying data-generating process divorced from special essentially accidental features of the data under analysis.

That first phenomenological notion is concerned with describing variability. The second role of probability is in connection with uncertainty and is thus epistemological. In the frequentist theory we adapt the frequency-based view of probability, using it only indirectly to calibrate the notions of confidence intervals and significance tests. In most applications of the Bayesian view we need an extended notion of probability as measuring the uncertainty of $\mathcal{E}$ given $\mathcal{F}$, where now $\mathcal{E}$, for example, is not necessarily the outcome of a random system, but may be a hypothesis or indeed any feature which is unknown to the investigator. In statistical applications $\mathcal{E}$ is typically some statement about the unknown parameter $\theta$ or more specifically about the parameter of interest $\psi$. The present

discussion is largely confined to such situations. The issue of whether a single number could usefully encapsulate uncertainty about the correctness of, say, the Fundamental Theory underlying particle physics is far outside the scope of the present discussion. It could, perhaps, be applied to a more specific question such as a prediction of the Fundamental Theory: will the Higgs boson have been discovered by 2010 ?

One extended notion of probability aims, in particular, to address the point that in interpretation of data there are often sources of uncertainty additional to those arising from narrow-sense statistical variability. In the frequentist approach these aspects, such as possible systematic errors of measurement, are addressed qualitatively, usually by formal or informal sensitivity analysis, rather than incorporated into a probability assessment.

## 统计代写|统计推断作业代写statistics interference代考|Frequentist interpretation of upper limits

First we consider the frequentist interpretation of upper limits obtained, for example, from a suitable pivot. We take the simplest example, Example 1.1, namely the normal mean when the variance is known, but the considerations are fairly general. The upper limit
$$\bar{y}+k_{c}^{} \sigma_{0} / \sqrt{n},$$ derived here from the probability statement $$P\left(\mu<\bar{Y}+k_{c}^{} \sigma_{0} / \sqrt{n}\right)=1-c,$$
is a particular instance of a hypothetical long run of statements a proportion $1-c$ of which will be true, always, of course, assuming our model is sound. We can, at least in principle, make such a statement for each $c$ and thereby generate a collection of statements, sometimes called a confidence distribution. There is no restriction to a single $c$, so long as some compatibility requirements hold.
Because this has the formal properties of a distribution for $\mu$ it was called by R. A. Fisher the fiducial distribution and sometimes the fiducial probability distribution. A crucial question is whether this distribution can be interpreted and manipulated like an ordinary probability distribution. The position is:

• a single set of limits for $\mu$ from some data can in some respects be considered just like a probability statement for $\mu$;
• such probability statements cannot in general be combined or manipulated by the laws of probability to evaluate, for example, the chance that $\mu$ exceeds some given constant, for example zero. This is clearly illegitimate in the present context.

That is, as a single statement a $1-c$ upper limit has the evidential force of a statement of a unique event within a probability system. But the rules for manipulating probabilities in general do not apply. The limits are, of course, directly based on probability calculations.

Nevertheless the treatment of the confidence interval statement about the parameter as if it is in some respects like a probability statement contains the important insights that, in inference for the normal mean, the unknown parameter is more likely to be near the centre of the interval than near the end-points and that, provided the model is reasonably appropriate, if the mean is outside the interval it is not likely to be far outside.

A more emphatic demonstration that the sets of upper limits defined in this way do not determine a probability distribution is to show that in general there is an inconsistency if such a formal distribution determined in one stage of analysis is used as input into a second stage of probability calculation. We shall not give details; see Note 5.2.

The following example illustrates in very simple form the care needed in passing from assumptions about the data, given the model, to inference about the model, given the data, and in particular the false conclusions that can follow from treating such statements as probability statements.

5.2概率

## 统计代写|统计推断作业代写statistics interference代考|Frequentist interpretation of upper limits

• 一组限制μ从某些数据中可以在某些方面被认为就像一个概率陈述μ;
• 这种概率陈述通常不能被概率法则组合或操纵来评估，例如，μ超过某个给定的常数，例如零。在目前的情况下，这显然是非法的。

## 广义线性模型代考

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