Output uniformly random guess at frequency distribution when reach is…

… too small (#1498)

The variance calculation of frequency distribution will output `NaN` when reach is zero. Moreover, the estimated variance is not accurate when reach is impractically small. The solution is to check whether the reach is too small using its confidence interval. If the confidence interval of the reach contains values <= 0, we claim the reach is too small for an accurate variance estimate of frequency distribution, and output the variance of uniformly random draw from [0, 1].

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,
+          )
         )
       }