Confidence Value Sampler (CVS) Estimation

• The paper presents CVS as an expectation estimation framework that computes parameter‐weighted estimates using data-derived confidence measures to eliminate reliance on prior assumptions.
• The methodology leverages a defined confidence level function and discrete approximations to construct weights that remain invariant under smooth reparameterizations.
• CVS is validated on canonical models like the Normal and Binomial, offering robust, posterior-like estimates compared to traditional Bayesian and frequentist techniques.

The Confidence Value Sampler (CVS) is an expectation estimation methodology that constructs parameter‐weighted estimates based on data‐derived confidence measures, circumventing explicit prior assumptions typical in Bayesian and frequentist approaches. The CVS framework, introduced in "Equal confidence weighted expectation value estimates" (Pijlman, 2017), leverages the definition of a confidence level function $\alpha(D, \theta)$ quantifying, for each parameter value $\theta$, the volume of possible datasets more probable than the observed $D$ under $p(\cdot \mid \theta)$. This lead to invariant estimation procedures under reparameterization, producing equal‐confidence expectations for observables without external prior inputs.

1. Problem Formulation and Motivation

Given observed data $D$ with a parametric model $p(D \mid \theta)$, the standard objective is to estimate the expectation of a function $f(\theta)$ reflecting uncertainty in $\theta$. Classical methods—Bayesian, least squares, and maximum likelihood—require specification of priors or rely implicitly on likelihood maximization heuristics. The CVS methodology was devised to address these limitations by producing estimates solely anchored to confidence intervals derived directly from data likelihoods, obviating any need for hypotheses on underlying probability distributions, prior beliefs, or subjective parameterization choices.

2. Confidence Level Definition

Central to CVS is the confidence level $\alpha(D,\theta)$,

$$\alpha(D,\theta)\;\equiv\;\int_{\{y:\;p(y\mid\theta)>p(D\mid\theta)\}} p(y\mid\theta)\,\mathrm{d}y,$$

where $\alpha$ measures the total probability under $p(\cdot \mid \theta)$ of data more likely than $D$. $\alpha(D,\theta)$ is small if $D$ falls in a tail of the model at $\theta$, large if $D$ is typical. This confidence level is applied to parameter weighting, motivating the following principle: parameter values contributing equal increments to $\alpha$ should be identically weighted in the final expectation.

3. Derivation of Equal‐Confidence Weights

The expectation according to CVS is constructed as

$$\langle f\rangle_{\rm CVS} \;=\; \frac{1}{K} \int_{\Theta} f(\theta) \Bigl|\partial_{\theta}\,\alpha(D,\theta)\Bigr| \mathrm{d}\theta,$$

with normalization $K=\int_\Theta c(\theta)\,\mathrm{d}\theta$. The unnormalized confidence weight is $c(\theta) = |\partial_\theta \alpha(D, \theta)|$, generalizable to multivariate settings by replacing the derivative with the norm of the gradient or the determinant of the Jacobian of $\alpha$ with respect to $\theta$. This construction ensures invariance under smooth changes of variables, formalized by demonstrating the transformation properties of $c(\theta)$ and the associated $w(\theta)$.

4. Algorithmic Realization and Discrete Approximations

For practical computation, CVS is applied by discretizing parameter space via grids or samples $\{\theta_i\}_{i=1}^M$. The algorithm proceeds as follows:

1. Evaluate likelihoods $L_i=p(D\mid\theta_i)$ for all $\theta_i$.
2. Compute $\alpha_i=\alpha(D,\theta_i)$: integrate $p(y\mid\theta_i)$ over data-space where $p(y\mid\theta_i) > L_i$.
3. Numerically approximate derivatives (e.g., finite differences for 1D):

$$c_i \approx \frac{|\alpha(D,\theta_{i+1}) - \alpha(D,\theta_{i-1})|}{\theta_{i+1} - \theta_{i-1}}$$

1. Normalize to get $w_i = c_i / \sum_j c_j$.
2. Estimate observables and variances:

$$E_{\rm CVS}[f] = \sum_i f(\theta_i) w_i, \qquad \mathrm{Var}_{\rm CVS}[f] = \sum_i (f(\theta_i) - E_{\rm CVS}[f])^2 w_i.$$

5. Analytical Forms and Application to Canonical Models

Normal Model (Unknown Mean)

For data $\{x_i\}$ and model $p(x_i \mid \mu)$ with known $\sigma$: $$\alpha(D, \mu) = 1-\frac{1}{\Gamma(\tfrac n2)} \Gamma\left(\tfrac n2,\;\tfrac1{2\sigma^2}\sum_i(x_i-\mu)^2\right)$$

$$c(\mu) = \frac{(\sqrt{2\pi}\,\sigma)^n}{2\sqrt2\,\sigma\,\Gamma(n/2)} \left[\sum_i\tfrac{(x_i-\mu)^2}{2\sigma^2}\right]^{\tfrac n2-\tfrac12} p(D\mid\mu)$$

For $n=1$, CVS weight reduces to the likelihood—coinciding with Bayesian estimation using a uniform prior.

Binomial Model

For $p(k|n,p) = \binom{n}{k}p^k(1-p)^{n-k}$: $$\alpha(k, n, p) = \sum_{\ell=0}^{n} p(\ell \mid n, p) \mathbf{1}\{p(\ell \mid n,p) > p(k \mid n,p)\}$$ Weights $w(p) \propto |\mathrm{d}\alpha/\mathrm{d}p|$ are normalized and expectations formed as

$$E_{\rm CVS}[\,\#\!\text{failures}\mid k\,] = \int_0^1 w(p)\,n p\,dp.$$

Empirical results show CVS estimates are slightly more conservative (higher predicted failures) than MLE in regimes with small $k \ll n$.

6. Invariance Properties and Multivariate Estimation

The CVS confidence weight is coordinate-free: $$c_\phi(\phi) = |\partial_\phi \alpha(D, \theta(\phi))| = |\partial_\theta \alpha|\;|\partial\theta/\partial\phi| \Rightarrow w_\phi(\phi)\,d\phi = w_\theta(\theta)\,d\theta$$ Thus, observable expectations are not affected by smooth reparameterization.

In multivariate cases, such as linear regression with $y=ax+b$, one computes iso-$\alpha$ contours and performs line integrals for weight assignments: $$\langle V\rangle_{\rm CVS} = \frac{1}{K} \int_0^1 d\alpha \left[ \frac{\oint_{\alpha(a,b)=\alpha} V(b)\,dl} {\oint_{\alpha(a,b)=\alpha} dl} \right], \qquad V(b) = (1+|b|)^2$$ Numerical implementation involves grid evaluation, contour extraction, and Riemann summation. CVS results have shown close alignment with Bayesian estimates using uninformative priors.

7. Algorithm Summary and Practical Considerations

The CVS procedure consists of:

1. Grid or sample point selection in parameter space.
2. Likelihood evaluation for each parameter.
3. Confidence level computation as data-space integrals.
4. Derivative estimation for confidence weights.
5. Normalization to obtain weights.
6. Estimation of observables and associated variance.

No external prior is required, and the procedure is robust to reparameterization. CVS can be implemented with standard numerical and scientific packages providing likelihood evaluation and multidimensional integration or contour extraction functionality. The resulting estimates reflect “equal-confidence” weighting and produce posterior-like results where classical methods invoke arbitrary priors, yet remain distinct in general parameter regimes (Pijlman, 2017).