Probability of observables versus probability of true values

The criticism about the inconsistent interpretation of results may look like a philosophical quibble, but it is, in my opinion, a crucial point which needs to be clarified. Let us consider the example of $n$ independent measurements of the same quantity under identical conditions (with $n$ large enough to simplify the problem, and neglecting systematic effects). We can evaluate the arithmetic average $\overline{x}$ and the standard deviation $\sigma$. The result on the true value $\mu$ is

$$\mu = \overline{x} \pm \frac{\sigma}{\sqrt{n}}\,.$$ (1.2)

The reader will have no difficulty in admitting that the large majority of people interpret () as if it were^1.7

$$P(\overline{x} - \frac{\sigma}{\sqrt{n}} \le \mu \le \overline{x} + \frac{\sigma}{\sqrt{n}}) = 68\%\,.$$ (1.3)

However, conventional statistics says only that^1.8

$$P(\mu - \frac{\sigma}{\sqrt{n}} \le \overline{X} \le \mu + \frac{\sigma}{\sqrt{n}} ) = 68\%\,,$$ (1.4)

a probabilistic statement about $\overline{X}$, given $\mu$, $\sigma$ and $n$. Probabilistic statements concerning $\mu$ are not foreseen by the theory (``$\mu$ is a constant of unknown value''^1.9), although this is what we are, intuitively, looking for: Having observed the effect $\overline{x}$ we are interested in stating something about the possible true value responsible for it. In fact, when we do an experiment, we want to increase our knowledge about $\mu$ and, consciously or not, we want to know which values are more or less probable. A statement concerning the probability that an observed value falls within a certain interval around $\mu$ is meaningless if it cannot be turned into an expression which states the quality of the knowledge about $\mu$ itself. Since the usual probability theory does not help, the probability inversion is performed intuitively. In routine cases it usually works, but there are cases in which it fails (see Section ).