Censored Label

A censored label reports a bound rather than an exact value. It appears when an assay, instrument, threshold, or reporting policy can only say that a value is above or below a limit.

Examples:

• $y<L$: value is below a lower detection limit.
• $y>U$: value is above an upper detection or assay limit.
• $y\le L$ or $y\ge U$: thresholded measurement.

The observation is not a point label:

$$y>U\nRightarrow y=U$$

The observed label should be represented as:

$$\tilde{y}=(b,d,L)$$

where $b$ is the bound value, $d\in \{<,\le ,>,\ge \}$ is the censoring direction, and $L$ records the assay or measurement context.

Likelihood View

For a regression target with density $p_{\theta }(y\mid x)$, a right-censored observation $y>U$ contributes probability mass:

$$P_{\theta }(Y>U\mid x)={\int }_U^{\infty }{p}_{\theta }(y\mid x)dy$$

A left-censored observation $y<L$ contributes:

$$P_{\theta }(Y<L\mid x)={\int }_{-\infty }^L{p}_{\theta }(y\mid x)dy$$

Treating censored values as exact points changes the training signal and can bias evaluation.

Point-Label Mistake

Replacing a censored value with its bound creates a biased pseudo-label.

Observation	Wrong point interpretation	Safer interpretation
$y>U$	$y=U$	value is somewhere above $U$
$y<L$	$y=L$	value is somewhere below $L$
inactive below threshold	exact low activity	thresholded or censored activity
capped score	exact maximum score	value reached or exceeded cap

This matters for regression metrics such as RMSE or MAE, because the true error cannot be computed exactly from a bound.

Modeling Choices

Approach	Use when	Risk
censored likelihood	distributional regression is available	requires modeling target distribution
interval loss	lower/upper bounds are known	implementation complexity
classification by threshold	decision boundary is the goal	loses continuous information
ranking constraints	relative order matters	needs enough comparable pairs
filter censored labels	few censored examples	may create selection bias

For a right-censored lower bound $y>U$, a prediction $\hat{y}$ should not be penalized the same way as an exact label unless the metric explicitly supports censoring.

Evaluation Boundary

Metric question	Need
exact regression error	exact labels only or censor-aware estimator
threshold decision	consistent threshold and censoring direction
ranking	valid pair construction under bounds
calibration	event definition that respects censored observations
assay comparison	harmonized units, thresholds, and reporting policy

Common Places

• Bioactivity measurements reported as greater-than or less-than values.
• Detection limits in assays and instruments.
• Thresholded active/inactive labels derived from continuous values.
• Clipped scores or capped ratings.

Checks

• Is the label exact, thresholded, clipped, censored, or rounded?
• Is the censoring direction recorded?
• Are units and transforms applied before or after censoring?
• Are censored values used in regression, classification, ranking, or filtering?
• Does the metric respect censoring, or does it treat bounds as exact targets?
• Are assay-specific censoring policies preserved in the dataset schema?
• Is the censoring direction stored separately from the numeric bound?
• Does label transformation happen before or after censoring, and is that documented?

Related

• Label semantics
• Label noise
• Missing data
• Data schema
• Assay harmonization
• Regression metrics
• Assay
• Bioactivity label
• Target-assay-label contract