Accelerated interference models

.github/workflows/ci.yml

@@ -65,3 +65,9 @@ jobs:
       run: |
         cd HiggsAnalysis/CombinedLimit
         make CONDA=1 -j 2
+    - name: Run tests
+      if: startsWith(matrix.python, '3.')
+      shell: bash -l {0}
+      run: |
+        cd HiggsAnalysis/CombinedLimit/test
+        python test_interference.py

Makefile

@@ -53,11 +53,12 @@ DICTNAME=$(LIBNAME)_xr
 # Linker and flags -------------------------------------------------------------
 LD = $(shell root-config --ld)
 ROOTLDFLAGS   = $(shell root-config --ldflags)
+ROOTLIBDIR = $(shell root-config --libdir)
 # OS x specific linkage
 DARWIN := $(shell uname|grep Darwin)
 ifdef DARWIN
-LDFLAGS       = $(ROOTLDFLAGS) -shared -install_name @rpath/$(SONAME) -fPIC
-EXELDFLAGS    = -Wl,-rpath,'@executable_path/../lib'
+LDFLAGS       = $(ROOTLDFLAGS) -g -shared -install_name @rpath/$(SONAME) -fPIC
+EXELDFLAGS    = -Wl,-rpath,'@executable_path/../lib' -Wl,-rpath,$(ROOTLIBDIR)
 else
 LDFLAGS       = $(ROOTLDFLAGS) -shared -Wl,-soname,$(SONAME) -Wl,-E -Wl,-z,defs -fPIC
 EXELDFLAGS    = 

docs/part2/physicsmodels.md

@@ -213,19 +213,19 @@ The `PhysicsModel` that encodes the signal model above is the [twoHypothesisHigg
 
 The two processes (S and S_ALT) will get different scaling parameters. The LEP-style likelihood for hypothesis testing can now be performed by setting **x** or **not_x** to 1 and 0 and comparing two likelihood evaluations.
 
-### Interference
+### Signal-background interference
 
-Since there are no such things as negative probability distribution functions, the recommended way to implement this is to start from the expression for the individual amplitudes and the parameter of interest $k$,
+Since there are no such things as negative probability distribution functions, the recommended way to implement this is to start from the expression for the individual amplitudes $A$ and the parameter of interest $k$,
 
 $$
-\mathrm{Yield} = (k * A_{s} + A_{b})^2
-= k^2 * A_{s}^2 + k * 2 A_{s} A_{b} + A_{b}^2
+\mathrm{Yield} = |k * A_{s} + A_{b}|^2
+= k^2 * |A_{s}|^2 + k * 2 \Re(A_{s}^* A_{b}) + |A_{b}|^2
 = \mu * S + \sqrt{\mu} * I + B
 $$
 
 where
 
-$\mu = k^2, ~S = A_{s}^2,~B = Ab^2$ and $ S+B+I = (As + Ab)^2$.
+$\mu = k^2, ~S = |A_{s}|^2,~B = |A_b|^2$ and $S+B+I = |A_s + A_b|^2$.
 
 With some algebra you can work out that,
 
@@ -244,3 +244,113 @@ An example of this scheme is implemented in a [HiggsWidth](https://gitlab.cern.c
  self.modelBuilder.factory_(  "expr::qqH_b_func(\"1-sqrt(@0)\", CMS_zz4l_mu)")
  self.modelBuilder.factory_(  "expr::qqH_sbi_func(\"sqrt(@0)\", CMS_zz4l_mu)")
 ```
+
+### Multi-process interference
+
+The above formulation can be extended to multiple parameters of interest
+(POIs). See
+[AnalyticAnomalousCoupling](https://github.com/amassiro/AnalyticAnomalousCoupling)
+for an example. However, the computational performance scales quadratically
+with the number of POIs, and can get extremely expensive for 10 or more, as may
+be encountered often with EFT analyses. To alleviate this issue, an accelerated
+interference modeling technique is implemented for template-based analyses via
+the `interferenceModel` physics model. In this model, each bin yield $y$ is parameterized
+$$y(\theta) = y_0 (\theta^\top M \theta)$$
+as a function of the POI vector $\theta$, a nominal template $y_0$, and a scaling matrix $M$.
+To see how this parameterization relates to that of the previous section, we can define:
+
+$$
+y_0 = A_b^2, \qquad
+M = \frac{1}{A_b^2} \begin{bmatrix}
+ |A_s|^2 & \Re(A_s^* A_b) \\
+ \Re(A_s A_b^*) & |A_b|^2
+ \end{bmatrix}, \qquad \theta = \begin{bmatrix}
+ \sqrt{\mu} \\
+ 1
+ \end{bmatrix}
+$$
+
+which leads to the same parameterization. At present, this technique only works with
+`CMSHistFunc`-based workspaces, as these are the most common workspace types
+encountered and the default when using
+[autoMCStats](https://cms-analysis.github.io/HiggsAnalysis-CombinedLimit/part2/bin-wise-stats).
+To use this model, for each bin find $y_0$ and put it into the datacard as a signal process, then find $M$ and
+save the lower triangular component as an array in a `scaling.json` file with a
+syntax as follows:
+
+```json
+[
+  {
+    "channel": "my_channel",
+    "process": "my_nominal_process",
+    "parameters": ["sqrt_mu[1,0,2]", "Bscaling[1]"],
+    "scaling": [
+      [0.5, 0.1, 1.0],
+      [0.6, 0.2, 1.0],
+      [0.7, 0.3, 1.0]
+    ]
+  }
+]
+```
+
+where the parameters are declared using RooFit's [factory
+syntax](https://root.cern.ch/doc/v622/classRooWorkspace.html#a0ddded1d65f5c6c4732a7a3daa8d16b0)
+and each row of the `scaling` field represents the scaling information of a bin, e.g. if $y_0 = |A_b|^2$
+then each row would contain three entries:
+$$|A_s|^2 / |A_b|^2,\quad \Re(A_s^* A_b)/|A_b|^2,\quad 1$$
+
+For several coefficients, one would enumerate as follows:
+```python
+scaling = []
+for ibin in range(nbins):
+    binscaling = []
+    for icoef in range(ncoef):
+        for jcoef in range(icoef + 1):
+            binscaling.append(amplitude_squared_for(ibin, icoef, jcoef))
+    scaling.append(binscaling)
+```
+
+Then, to construct the workspace, run
+
+```bash
+text2workspace.py card.txt -P HiggsAnalysis.CombinedLimit.InterferenceModels:interferenceModel \
+    --PO verbose --PO scalingData=scaling.json
+```
+For large amounts of scaling data, you can optionally use gzipped json (`.json.gz`) or pickle (`.pkl.gz`)
+files with 2D numpy arrays for the scaling coefficients instead of lists. The function `numpy.tril_indices(ncoef)`
+is helpful for extracting the lower triangle of a square matrix.
+
+You could pick any
+nominal template, and adjust the scaling as appropriate. Generally it is
+advisable to use a nominal template corresponding to near where you expect the
+POIs to land so that the shape systematic effects are well-modeled in that
+region.
+
+It may be the case that the relative contributions of the terms are themselves
+a function of the POIs. For example, in VBF di-Higgs production, BSM
+modifications to the production rate can be parameterized in the "kappa"
+framework via three diagrams, with scaling coefficients $\kappa_V \kappa_\lambda$,
+$\kappa_V^2$, and $\kappa_{2V}$, respectively, that interfere.  In
+that case, you can declare formulas with the factory syntax to represent each
+amplitude as follows:
+
+```json
+[
+  {
+    "channel": "a_vbf_channel",
+    "process": "VBFHH",
+    "parameters": ["expr::a0('@0*@1', kv[1,0,2], kl[1,0,2])", "expr::a1('@0*@0', kv[1,0,2])", "k2v[1,0,2]"],
+    "scaling": [
+      [3.30353674666415, -8.54170982038222, 22.96464188467882, 4.2353483207128, -11.07996258835088, 5.504469544697623],
+      [2.20644332142891, -7.076836641962523, 23.50989689214267, 4.053185685866683, -13.08569222837996, 7.502346155380032]
+    ]
+  }
+]
+```
+
+However, you will need to manually specify what the POIs should be when creating the workspace using the `POIs=` physics option, e.g.
+
+```bash
+text2workspace.py card.txt -P HiggsAnalysis.CombinedLimit.InterferenceModels:interferenceModel \
+  --PO scalingData=scaling.json --PO 'POIs=kl[1,0,2]:kv[1,0,2]:k2v[1,0,2]'
+```

interface/CMSExternalMorph.h

@@ -0,0 +1,45 @@
+#ifndef CMSExternalMorph_h
+#define CMSExternalMorph_h
+#include <vector>
+
+#include "RooAbsReal.h"
+#include "RooRealVar.h"
+#include "RooRealProxy.h"
+
+class CMSExternalMorph : public RooAbsReal {
+  public:
+    CMSExternalMorph();
+    /*
+     * All subclasses need to provide an edges array of length nbins+1
+     * of the observable (x)
+     * TODO: CMSHistFunc and CMSHistSum do not check the binning is compatible
+     * with their binning other than having the correct length
+     */
+    CMSExternalMorph(
+        const char* name,
+        const char* title,
+        RooRealVar& x,
+        const std::vector<double>& edges
+    );
+    CMSExternalMorph(CMSExternalMorph const& other, const char* name = 0);
+    virtual ~CMSExternalMorph();
+
+    /* Batch accessor for CMSHistFunc / CMSHistSum, to be overriden by concrete
+     * implementations. hasChanged() should indicate whether or not
+     * batchGetBinValues() would return a new vector, given the state of
+     * any dependent variables. 
+     */
+    virtual bool hasChanged() const = 0;
+    virtual const std::vector<double>& batchGetBinValues() const = 0;
+
+  protected:
+    RooRealProxy x_;
+    std::vector<double> edges_;
+
+    double evaluate() const;
+
+  private:
+    ClassDef(CMSExternalMorph, 1)
+};
+
+#endif // CMSExternalMorph_h

interface/CMSHistFunc.h

@@ -15,6 +15,7 @@
 #include "CMSHistV.h"
 #include "FastTemplate_Old.h"
 #include "SimpleCacheSentry.h"
+#include "CMSExternalMorph.h"
 
 class CMSHistFuncWrapper;
 
@@ -85,7 +86,7 @@ class CMSHistFunc : public RooAbsReal {
   virtual TObject* clone(const char* newname) const {
     return new CMSHistFunc(*this, newname);
   }
-  virtual ~CMSHistFunc() {}
+  virtual ~CMSHistFunc();
 
   void addHorizontalMorph(RooAbsReal& hvar, TVectorD hpoints);
 
@@ -148,6 +149,7 @@ class CMSHistFunc : public RooAbsReal {
   friend class CMSHistV<CMSHistFunc>;
   friend class CMSHistSum;
 
+  void injectExternalMorph(CMSExternalMorph& morph);
   /*
 
   – RooAbsArg::setVerboseEval(Int_t level) • Level 0 – No messages
@@ -190,6 +192,9 @@ class CMSHistFunc : public RooAbsReal {
 
   static bool enable_fast_vertical_; //! not to be serialized
 
+  // This is an "optional proxy", i.e. a list with either zero or one entry
+  RooListProxy external_morph_;
+
  private:
   void initialize() const;
   void setGlobalCache() const;
@@ -214,7 +219,7 @@ class CMSHistFunc : public RooAbsReal {
 
   void applyRebin() const;
 
-  ClassDef(CMSHistFunc, 1)
+  ClassDef(CMSHistFunc, 2)
 };
 
 

interface/CMSHistSum.h

@@ -73,9 +73,13 @@ class CMSHistSum : public RooAbsReal {
   RooArgList const& coefList() const { return coeffpars_; }
   // RooArgList const& funcList() const { return funcs_; }
 
+  RooAbsReal const& getXVar() const { return x_.arg(); }
+
   static void EnableFastVertical();
   friend class CMSHistV<CMSHistSum>;
 
+  void injectExternalMorph(int idx, CMSExternalMorph& morph);
+
  protected:
   RooRealProxy x_;
 
@@ -128,6 +132,9 @@ class CMSHistSum : public RooAbsReal {
   mutable int fast_mode_; //! not to be serialized
   static bool enable_fast_vertical_; //! not to be serialized
 
+  RooListProxy external_morphs_;
+  std::vector<int> external_morph_indices_;
+
   inline int& morphField(int const& ip, int const& iv) {
     return vmorph_fields_[ip * n_morphs_ + iv];
   }
@@ -143,7 +150,7 @@ class CMSHistSum : public RooAbsReal {
 
 
  private:
-  ClassDef(CMSHistSum,1)
+  ClassDef(CMSHistSum,2)
 };
 
 #endif

interface/CMSInterferenceFunc.h

@@ -0,0 +1,55 @@
+#ifndef CMSInterferenceFunc_h
+#define CMSInterferenceFunc_h
+#include "RooListProxy.h"
+#include "SimpleCacheSentry.h"
+
+#include "CMSExternalMorph.h"
+
+class _InterferenceEval;
+
+class CMSInterferenceFunc : public CMSExternalMorph {
+  public:
+    CMSInterferenceFunc();
+    CMSInterferenceFunc(CMSInterferenceFunc const& other, const char* name = 0);
+    /*
+     * For a coefficients list of length n and edges array of length b+1,
+     * the binscaling nested vector should have b entries (for b bins) with
+     * each being of length n*(n+1)/2, corresponding to the lower triangular
+     * elements of the scaling matrix, i.e. (m_00, m_10, m_11, m_20, m_21, m_22, ...)
+     */
+    CMSInterferenceFunc(
+        const char* name,
+        const char* title,
+        RooRealVar& x,
+        const std::vector<double>& edges,
+        const RooArgList& coefficients,
+        const std::vector<std::vector<double>> binscaling
+        );
+    virtual ~CMSInterferenceFunc();
+
+    virtual TObject* clone(const char* newname) const override {
+      return new CMSInterferenceFunc(*this, newname);
+    };
+
+    void printMultiline(
+        std::ostream& os, Int_t contents, Bool_t verbose, TString indent
+        ) const override;
+
+    bool hasChanged() const override { return !sentry_.good(); };
+    const std::vector<double>& batchGetBinValues() const override;
+
+  protected:
+    RooListProxy coefficients_;
+    std::vector<std::vector<double>> binscaling_;
+
+    mutable SimpleCacheSentry sentry_; //!
+    mutable std::unique_ptr<_InterferenceEval> evaluator_; //!
+
+  private:
+    void initialize() const;
+    void updateCache() const;
+
+    ClassDefOverride(CMSInterferenceFunc, 1)
+};
+
+#endif // CMSInterferenceFunc_h