Bridging Probability Theory and Statistical Estimation

Bridging Probability Theory and Statistical Estimation is summary on the exercise about precision of estimating the mean value of a quantity from independent measurements.

The result of every measurement is assent to be a real number \(X\) with the same normally-distributed deviation from some "true value" \(X_0\).

This note presents a mathematically grounded interpretation of the uncertainty in an estimate \(\bar{X}\) of an unknown true value \(X_0\), using the probability density function.

The approach builds a bridge between Kolmogorov-style probability theory and practical statistical inference — without requiring commitment to Bayesian or frequentist ideology.

The goal of this article is to suggest the efficient terminology and eliminate the ambiguous terms that cause confusion.

Introduction

Editors and practitioners often face vague questions like:

“How precise is your estimate?”
“What’s the accuracy of your estimate?”
“What’s the error of your estimate?”
“How confident are you in your estimate?”
“What’s the possible deviation of your estimate from the true value?”

These questions need reformulation into precise statistical quantities.

Model Assumptions

There is an unknown true value \(X_0 \in \mathbb{R}\) to be estimated.

The Expert performs \(N\) independent measurements \(X_1, X_2, \dots, X_N\), modeled as \[ X_i \sim \mathcal{N}(X_0, S^2) \] with both \(X_0\) and \(S\) unknown.

The Expert computes the following quantities:

Sample mean (point estimate): \[ \bar{X} = \frac{1}{N} \sum_{i=1}^N X_i \]

Sample standard deviation: \[ s = \sqrt{ \frac{1}{N - 1} \sum_{i=1}^N (X_i - \bar{X})^2 } \]

Naive standard error: \[ c_N = \frac{s}{\sqrt{N}} \]

Likelihood-Like Density

Let

\[ f_N(x) = \frac{1}{c_N} \cdot \mathrm{Student}_{N-1}\!\left( \frac{x - \bar{X}}{c_N} \right) \]

This is the probability density function of a Student’s t-distribution with \(N\!-\!1\) degrees of freedom, centered at the sample mean \(\bar{X}\) and scaled by the standard error \(c_N\).

Function \(f_N\) is confidence density or predictive belief distribution over the possible values of \(X_0\), conditional on the data.

It answers questions like:

"How close is \(\bar{X}\) likely to be to the true value \(X_0\)?"

"What’s the uncertainty of the estimate?"

One may interpret

\[ \int_A^B f_N(x) \, \mathrm dx \]

as probability that \(X_0\in(A,B)\), given the data and modeling assumptions.

Mean Square Width (Expected Squared Error)

The expected squared deviation is: \[ \sigma_N^2 = \int_{-\infty}^{\infty} (x - \bar{X})^2 \ f_N(x) \ \mathrm d x \]

which gives \[ \sigma_N=\sqrt{\frac{N-1}{N-3}}\;c_N \]

This is the corrected standard error. It has the property: \[ \mathbb{E}[(\bar{X} - X_0)^2]=\mathbb{E}[\sigma_N^2] \]

In such a way, \(\sigma_N\) properly accounts for small‑sample variability.

The correnction factor \( \sqrt{(N{-}1)/(N{-}3)} \) arises from computing the second moment of the Student Distribution and accounts for extra variability in \(s\) due to estimating \(S\).

However, \(\sigma_N\) has sense only for \(N>3\) while the probability density function \(f_N\) has sense for any integer \(N>1\)

Future Measurement Expectation

In this section, the additional question is considered:

What may Expert expect from the \((N{+}1)\)th similar measurement?

Assuming the same model, the Expert may want to predict the result of an additional, independent measurement \(X_{N+1}\).

Given the sample data \(X_1, \dots, X_N\), the **predictive distribution** of \(X_{N+1}\) is a scaled and shifted *Student’s t-distribution*:

\[ X_{N+1} \sim \mathrm{Student}_{N-1}\left(\bar{X},\; s \cdot \sqrt{1 + \frac{1}{N}} \right) \]

This formula reflects that:
The new measurement shares the same variance \(S^2\),
But \(S\) is unknown and replaced by the sample \(s\),

And both estimation of the mean and future randomness contribute to uncertainty.

Therefore, the Expert should expect the next measurement to fluctuate around \(\bar{X}\), with a spread wider than either \(s\) or \(c_N\).

This helps avoid **overconfidence** in forecasting — especially with small \(N\) — and acknowledges that even “true value” \(X_0\) being fixed doesn’t guarantee low variance in future data.

Practical Significance

The consideration above:

is defined within classical probability theory,

requires no any specific ideology,

corrects the naive standard error \(\displaystyle \ c_N = \frac s{\sqrt N} \ \), which underestimates the uncertainty at small \(N\).

Originality

The formulas above are not original.

Various authors mention the possible misleading and misinterpretations related to qualification of precision of estimate of the mean value from the set of independent measurements .

However we tried to compile the result in the most closed and compact, but still correct and self-consistent form.

Conclusion

This construction shows how statistical estimation can be framed as probabilistically coherent predictive inference, even within a non-Bayesian or fully deterministic worldview.

It provides a way for Experts to answer vague or ill-posed questions ("How accurate is your estimate?") in a rigorous, quantitative manner that is transparent to both the Expert and the Boss from the example at the top of this article.

Warning

Neither deduction nor proof of the formulas above is presented in this article.

However, Editor and ChatGPT applied all the efforts to catch and to correct all possible mistakes, misprints.

If you see at least one mistake that is not yet corrected here, then, please let Editor know.

References

Keywords

«Bayesian ideology», «Bayesian statistics», «Central Limit Theorem» (CLT), «ChatGPT», «Confidence interval», «Credible interval», «Duration5», «Expectation and variance», «Frequentist ideology», «Independence and conditional probability», «Law of Large Numbers» (LLN), «Maximum likelihood estimation» (MLE), «Mean square deviation», «Mean value», «Normal distribution», «Probability», «Probability Density Function», «Random variable», «Sampling distributions», «Standard deviation», «Student Distribution», «Theory of Probability»,