Output uniformly random guess at frequency distribution when reach is too small

src/main/kotlin/org/wfanet/measurement/measurementconsumer/stats/LiquidLegions.kt

@@ -253,20 +253,32 @@ object LiquidLegions {
     sketchParams: LiquidLegionsSketchParams,
     collisionResolution: Boolean,
     frequencyNoiseVariance: Double,
-    totalReach: Long,
-    reachRatio: Double,
-    frequencyMeasurementParams: FrequencyMeasurementParams,
-    multiplier: Int,
+    relativeFrequencyMeasurementVarianceParams: RelativeFrequencyMeasurementVarianceParams,
   ): Double {
+    val (
+      totalReach: Long,
+      reachMeasurementVariance: Double,
+      reachRatio: Double,
+      frequencyMeasurementParams: FrequencyMeasurementParams,
+      multiplier: Int) =
+      relativeFrequencyMeasurementVarianceParams
+
     val expectedRegisterNum =
       expectedNumberOfNonDestroyedRegisters(
         sketchParams,
         collisionResolution,
         totalReach,
         frequencyMeasurementParams.vidSamplingInterval.width,
       )
-    if (expectedRegisterNum < 1.0) {
-      return 0.0
+
+    // When reach is too small, we have little info to estimate frequency, and thus the estimate of
+    // relative frequency is equivalent to a uniformly random guess of a probability in [0, 1].
+    if (
+      isReachTooSmallForComputingRelativeFrequencyVariance(totalReach, reachMeasurementVariance) ||
+        expectedRegisterNum < 1.0
+    ) {
+      return if (frequencyMeasurementParams.maximumFrequency == multiplier) 0.0
+      else VARIANCE_OF_UNIFORMLY_RANDOM_PROBABILITY
     }
 
     val registerNumVariance =

src/main/kotlin/org/wfanet/measurement/measurementconsumer/stats/MeasurementStatistics.kt

@@ -16,6 +16,7 @@
 
 package org.wfanet.measurement.measurementconsumer.stats
 
+import kotlin.math.sqrt
 import org.wfanet.measurement.eventdataprovider.noiser.DpParams
 
 /** Noise mechanism enums. */
@@ -72,6 +73,43 @@ data class FrequencyMeasurementVarianceParams(
   val measurementParams: FrequencyMeasurementParams,
 )
 
+/**
+ * The parameters used to compute the variance of a reach ratio at a certain frequency in a relative
+ * frequency measurement.
+ */
+data class RelativeFrequencyMeasurementVarianceParams(
+  val totalReach: Long,
+  val reachMeasurementVariance: Double,
+  val reachRatio: Double,
+  val measurementParams: FrequencyMeasurementParams,
+  val multiplier: Int,
+)
+
+/**
+ * A reach result is considered too small when computing variances of relative frequency if the 95%
+ * confidence interval of the reach covers 0 or negative values. The 95% confidence interval =
+ * reach_result +/- 1.96 * reach_std.
+ */
+private const val REACH_THRESHOLD_CONSTANT_FOR_RELATIVE_FREQUENCY_VARIANCE = 1.96
+
+/**
+ * A uniformly random number from [0, 1] has a variance equal to 1 / 12
+ * (en.wikipedia.org/wiki/Continuous_uniform_distribution).
+ */
+const val VARIANCE_OF_UNIFORMLY_RANDOM_PROBABILITY = 1.0 / 12.0
+
+/** Determines if a reach is too small for computing relative frequency variance. */
+fun isReachTooSmallForComputingRelativeFrequencyVariance(
+  reach: Long,
+  reachVariance: Double,
+): Boolean {
+  // A reach result is considered too small for computing variances of relative frequency if the
+  // confidence interval lower bound of the reach <= 0.
+  val reachConfidenceIntervalLowerBound =
+    reach - REACH_THRESHOLD_CONSTANT_FOR_RELATIVE_FREQUENCY_VARIANCE * sqrt(reachVariance)
+  return reachConfidenceIntervalLowerBound <= 0
+}
+
 /** The parameters used to compute the variance of an impression measurement. */
 data class ImpressionMeasurementVarianceParams(
   val impression: Long,

src/main/kotlin/org/wfanet/measurement/measurementconsumer/stats/Variances.kt

@@ -146,11 +146,26 @@ object VariancesImpl : Variances {
    * Different types of frequency histograms have different values of [multiplier].
    */
   private fun deterministicFrequencyRelativeVariance(
-    totalReach: Long,
-    reachRatio: Double,
-    measurementParams: FrequencyMeasurementParams,
-    multiplier: Int,
+    relativeFrequencyMeasurementVarianceParams: RelativeFrequencyMeasurementVarianceParams
   ): Double {
+
+    val (
+      totalReach: Long,
+      reachMeasurementVariance: Double,
+      reachRatio: Double,
+      measurementParams: FrequencyMeasurementParams,
+      multiplier: Int) =
+      relativeFrequencyMeasurementVarianceParams
+
+    // When reach is too small, we have little info to estimate frequency, and thus the estimate of
+    // relative frequency is equivalent to a uniformly random guess at probability.
+    if (
+      isReachTooSmallForComputingRelativeFrequencyVariance(totalReach, reachMeasurementVariance)
+    ) {
+      return if (measurementParams.maximumFrequency == multiplier) 0.0
+      else VARIANCE_OF_UNIFORMLY_RANDOM_PROBABILITY
+    }
+
     val frequencyNoiseVariance: Double =
       computeNoiseVariance(measurementParams.dpParams, measurementParams.noiseMechanism)
     val varPart1 =
@@ -257,22 +272,16 @@ object VariancesImpl : Variances {
     sketchParams: LiquidLegionsSketchParams,
     measurementParams: FrequencyMeasurementParams,
   ): (
-    totalReach: Long,
-    reachRatio: Double,
-    measurementParams: FrequencyMeasurementParams,
-    multiplier: Int,
+    relativeFrequencyMeasurementVarianceParams: RelativeFrequencyMeasurementVarianceParams
   ) -> Double {
     val frequencyNoiseVariance: Double =
       computeNoiseVariance(measurementParams.dpParams, measurementParams.noiseMechanism)
-    return { totalReach, reachRatio, freqParams, multiplier ->
+    return { relativeFrequencyMeasurementVarianceParams ->
       LiquidLegions.liquidLegionsFrequencyRelativeVariance(
         sketchParams = sketchParams,
         collisionResolution = true,
         frequencyNoiseVariance = frequencyNoiseVariance,
-        totalReach = totalReach,
-        reachRatio = reachRatio,
-        frequencyMeasurementParams = freqParams,
-        multiplier = multiplier,
+        relativeFrequencyMeasurementVarianceParams = relativeFrequencyMeasurementVarianceParams,
       )
     }
   }
@@ -326,23 +335,17 @@ object VariancesImpl : Variances {
     sketchParams: LiquidLegionsSketchParams,
     measurementParams: FrequencyMeasurementParams,
   ): (
-    totalReach: Long,
-    reachRatio: Double,
-    measurementParams: FrequencyMeasurementParams,
-    multiplier: Int,
+    relativeFrequencyMeasurementVarianceParams: RelativeFrequencyMeasurementVarianceParams
   ) -> Double {
     val frequencyNoiseVariance: Double =
       computeDistributedNoiseVariance(measurementParams.dpParams, measurementParams.noiseMechanism)
 
-    return { totalReach, reachRatio, freqParams, multiplier ->
+    return { relativeFrequencyMeasurementVarianceParams ->
       LiquidLegions.liquidLegionsFrequencyRelativeVariance(
         sketchParams = sketchParams,
         collisionResolution = false,
         frequencyNoiseVariance = frequencyNoiseVariance,
-        totalReach = totalReach,
-        reachRatio = reachRatio,
-        frequencyMeasurementParams = freqParams,
-        multiplier = multiplier,
+        relativeFrequencyMeasurementVarianceParams = relativeFrequencyMeasurementVarianceParams,
       )
     }
   }
@@ -387,10 +390,7 @@ object VariancesImpl : Variances {
     params: FrequencyMeasurementVarianceParams,
     frequencyRelativeVarianceFun:
       (
-        totalReach: Long,
-        reachRatio: Double,
-        measurementParams: FrequencyMeasurementParams,
-        multiplier: Int,
+        relativeFrequencyMeasurementVarianceParams: RelativeFrequencyMeasurementVarianceParams
       ) -> Double,
     frequencyCountVarianceFun:
       (
@@ -415,20 +415,26 @@ object VariancesImpl : Variances {
     val relativeVariances: Map<Int, Double> =
       (1..maximumFrequency).associateWith { frequency ->
         frequencyRelativeVarianceFun(
-          params.totalReach,
-          params.relativeFrequencyDistribution.getOrDefault(frequency, 0.0),
-          params.measurementParams,
-          1,
+          RelativeFrequencyMeasurementVarianceParams(
+            params.totalReach,
+            params.reachMeasurementVariance,
+            params.relativeFrequencyDistribution.getOrDefault(frequency, 0.0),
+            params.measurementParams,
+            1,
+          )
         )
       }
 
     val kPlusRelativeVariances: Map<Int, Double> =
       (1..maximumFrequency).associateWith { frequency ->
         frequencyRelativeVarianceFun(
-          params.totalReach,
-          kPlusRelativeFrequencyDistribution.getValue(frequency),
-          params.measurementParams,
-          maximumFrequency - frequency + 1,
+          RelativeFrequencyMeasurementVarianceParams(
+            params.totalReach,
+            params.reachMeasurementVariance,
+            kPlusRelativeFrequencyDistribution.getValue(frequency),
+            params.measurementParams,
+            maximumFrequency - frequency + 1,
+          )
         )
       }