diff --git a/docs/CHANGELOG.asciidoc b/docs/CHANGELOG.asciidoc
index 454d67135b..cd3f2e182a 100644
--- a/docs/CHANGELOG.asciidoc
+++ b/docs/CHANGELOG.asciidoc
@@ -28,6 +28,13 @@
 
 //=== Regressions
 
+== {es} version 7.2.1
+
+=== Bug Fixes
+
+* Fix an edge case causing spurious anomalies (false positives) if the variance in the count of events
+changed significantly throughout the period of a seasonal quantity. (See {ml-pull}489[#489].)
+
 == {es} version 7.2.0
 
 === Enhancements
diff --git a/lib/maths/COneOfNPrior.cc b/lib/maths/COneOfNPrior.cc
index f969336601..b77170f37e 100644
--- a/lib/maths/COneOfNPrior.cc
+++ b/lib/maths/COneOfNPrior.cc
@@ -54,7 +54,7 @@ double logn(std::size_t n) {
 const double DERATE = 0.99999;
 const double MINUS_INF = DERATE * boost::numeric::bounds<double>::lowest();
 const double INF = DERATE * boost::numeric::bounds<double>::highest();
-const double MAXIMUM_LOG_BAYES_FACTOR = std::log(1e6);
+const double MAXIMUM_LOG_BAYES_FACTOR = std::log(1e8);
 const double MAXIMUM_UNPENALISED_RELATIVE_VARIANCE_ERROR = 9.0;
 const double MINIMUM_SIGNIFICANT_WEIGHT = 0.01;
 const double MAXIMUM_RELATIVE_ERROR = 1e-3;
diff --git a/lib/maths/CTimeSeriesModel.cc b/lib/maths/CTimeSeriesModel.cc
index e3ca68169a..7437579e98 100644
--- a/lib/maths/CTimeSeriesModel.cc
+++ b/lib/maths/CTimeSeriesModel.cc
@@ -1619,6 +1619,11 @@ void CUnivariateTimeSeriesModel::reinitializeStateGivenNewComponent(
     // re-weight so that the total sample weight corresponds to the sample
     // weight the model receives from a fixed (shortish) time interval.
 
+    // We can't properly handle periodicity in the variance of the rate if
+    // using a Poisson process so remove it from model detectio if we detect
+    // seasonality.
+    m_ResidualModel->removeModels(
+        maths::CPrior::CModelFilter().remove(maths::CPrior::E_Poisson));
     m_ResidualModel->setToNonInformative(0.0, m_ResidualModel->decayRate());
 
     if (initialValues.size() > 0) {
@@ -1648,16 +1653,6 @@ void CUnivariateTimeSeriesModel::reinitializeStateGivenNewComponent(
         }
     }
 
-    // Note we can't reinitialize this from the initial values because we do
-    // not expect these to be at the bucket length granularity.
-    if (m_MultibucketFeature != nullptr) {
-        m_MultibucketFeature->clear();
-    }
-    if (m_MultibucketFeatureModel != nullptr) {
-        m_MultibucketFeatureModel->setToNonInformative(
-            0.0, m_MultibucketFeatureModel->decayRate());
-    }
-
     if (m_Correlations != nullptr) {
         m_Correlations->clearCorrelationModels(m_Id);
     }
@@ -1670,6 +1665,18 @@ void CUnivariateTimeSeriesModel::reinitializeStateGivenNewComponent(
             controller.reset();
         }
     }
+
+    // Note we can't reinitialize this from the initial values because we do
+    // not expect these to be at the bucket length granularity.
+    if (m_MultibucketFeature != nullptr) {
+        m_MultibucketFeature->clear();
+    }
+    if (m_MultibucketFeatureModel != nullptr) {
+        m_MultibucketFeatureModel->removeModels(
+            maths::CPrior::CModelFilter().remove(maths::CPrior::E_Poisson));
+        m_MultibucketFeatureModel->setToNonInformative(0.0, m_ResidualModel->decayRate());
+    }
+
     if (m_AnomalyModel != nullptr) {
         m_AnomalyModel->reset();
     }
@@ -2987,16 +2994,6 @@ void CMultivariateTimeSeriesModel::reinitializeStateGivenNewComponent(
         }
     }
 
-    // Note we can't reinitialize this from the initial values because we do
-    // not expect these to be at the bucket length granularity.
-    if (m_MultibucketFeature != nullptr) {
-        m_MultibucketFeature->clear();
-    }
-    if (m_MultibucketFeatureModel != nullptr) {
-        m_MultibucketFeatureModel->setToNonInformative(
-            0.0, m_MultibucketFeatureModel->decayRate());
-    }
-
     if (m_Controllers != nullptr) {
         m_ResidualModel->decayRate(m_ResidualModel->decayRate() /
                                    (*m_Controllers)[E_ResidualControl].multiplier());
@@ -3008,6 +3005,17 @@ void CMultivariateTimeSeriesModel::reinitializeStateGivenNewComponent(
             controller.reset();
         }
     }
+
+    // Note we can't reinitialize this from the initial values because we do
+    // not expect these to be at the bucket length granularity.
+    if (m_MultibucketFeature != nullptr) {
+        m_MultibucketFeature->clear();
+    }
+
+    if (m_MultibucketFeatureModel != nullptr) {
+        m_MultibucketFeatureModel->setToNonInformative(0.0, m_ResidualModel->decayRate());
+    }
+
     if (m_AnomalyModel != nullptr) {
         m_AnomalyModel->reset();
     }
diff --git a/lib/model/unittest/CEventRateModelTest.cc b/lib/model/unittest/CEventRateModelTest.cc
index fdba81e19a..fba69bef76 100644
--- a/lib/model/unittest/CEventRateModelTest.cc
+++ b/lib/model/unittest/CEventRateModelTest.cc
@@ -592,7 +592,7 @@ void CEventRateModelTest::testProbabilityCalculation() {
             for (int j = 0; j < 2; ++j) {
                 double multiBucketImpact = minProbabilities[j].third.s_MultiBucketImpact;
                 LOG_DEBUG(<< "multi_bucket_impact = " << multiBucketImpact);
-                CPPUNIT_ASSERT(multiBucketImpact > expectedMultiBucketImpactThresholds[j]);
+                CPPUNIT_ASSERT(multiBucketImpact >= expectedMultiBucketImpactThresholds[j]);
                 CPPUNIT_ASSERT(multiBucketImpact <= CAnomalyDetectorModelConfig::MAXIMUM_MULTI_BUCKET_IMPACT_MAGNITUDE);
             }
         }
diff --git a/lib/model/unittest/CMetricAnomalyDetectorTest.cc b/lib/model/unittest/CMetricAnomalyDetectorTest.cc
index ef29b01be8..c011081e1a 100644
--- a/lib/model/unittest/CMetricAnomalyDetectorTest.cc
+++ b/lib/model/unittest/CMetricAnomalyDetectorTest.cc
@@ -444,7 +444,7 @@ void CMetricAnomalyDetectorTest::testExcludeFrequent() {
 
         // expect there to be 1 anomaly
         CPPUNIT_ASSERT_EQUAL(std::size_t(1), highAnomalyTimes.size());
-        CPPUNIT_ASSERT_DOUBLES_EQUAL(9.0, highAnomalyFactors[0], 2.0);
+        CPPUNIT_ASSERT_DOUBLES_EQUAL(12.0, highAnomalyFactors[0], 2.0);
     }
 }