Return NaN for negative ModeResult variance estimates (#2471)

* Return NaN for negative ModeResult variance estimates

* Apply suggestions from mhauru

Co-authored-by: Markus Hauru <markus@mhauru.org>

* Add doc to StatsBase.coefTable(::ModeResult, ...) for numerrors_warnonly

* Add test for same coeftable with/without numerrors_warnonly

* Add a test for coeftable with negative variance

---------

Co-authored-by: Markus Hauru <markus@mhauru.org>
Co-authored-by: Xianda Sun <5433119+sunxd3@users.noreply.github.com>
Co-authored-by: Markus Hauru <mhauru@turing.ac.uk>

src/optimisation/Optimisation.jl

@@ -16,6 +16,7 @@ using Printf: Printf
 using ForwardDiff: ForwardDiff
 using StatsAPI: StatsAPI
 using Statistics: Statistics
+using LinearAlgebra: LinearAlgebra
 
 export maximum_a_posteriori, maximum_likelihood
 # The MAP and MLE exports are only needed for the Optim.jl interface.
@@ -228,11 +229,61 @@ end
 
 # Various StatsBase methods for ModeResult
 
-function StatsBase.coeftable(m::ModeResult; level::Real=0.95)
+"""
+    StatsBase.coeftable(m::ModeResult; level::Real=0.95, numerrors_warnonly::Bool=true)
+
+
+Return a table with coefficients and related statistics of the model. level determines the
+level for confidence intervals (by default, 95%).
+
+In case the `numerrors_warnonly` argument is true (the default) numerical errors encountered
+during the computation of the standard errors will be caught and reported in an extra
+"Error notes" column.
+"""
+function StatsBase.coeftable(m::ModeResult; level::Real=0.95, numerrors_warnonly::Bool=true)
     # Get columns for coeftable.
     terms = string.(StatsBase.coefnames(m))
     estimates = m.values.array[:, 1]
-    stderrors = StatsBase.stderror(m)
+    # If numerrors_warnonly is true, and if either the information matrix is singular or has
+    # negative entries on its diagonal, then `notes` will be a list of strings for each
+    # value in `m.values`, explaining why the standard error is NaN.
+    notes = nothing
+    local stderrors
+    if numerrors_warnonly
+        infmat = StatsBase.informationmatrix(m)
+        local vcov
+        try
+            vcov = inv(infmat)
+        catch e
+            if isa(e, LinearAlgebra.SingularException)
+                stderrors = fill(NaN, length(m.values))
+                notes = fill("Information matrix is singular", length(m.values))
+            else
+                rethrow(e)
+            end
+        else
+            vars = LinearAlgebra.diag(vcov)
+            stderrors = eltype(vars)[]
+            if any(x -> x < 0, vars)
+                notes = []
+            end
+            for var in vars
+                if var >= 0
+                    push!(stderrors, sqrt(var))
+                    if notes !== nothing
+                        push!(notes, "")
+                    end
+                else
+                    push!(stderrors, NaN)
+                    if notes !== nothing
+                        push!(notes, "Negative variance")
+                    end
+                end
+            end
+        end
+    else
+        stderrors = StatsBase.stderror(m)
+    end
     zscore = estimates ./ stderrors
     p = map(z -> StatsAPI.pvalue(Distributions.Normal(), z; tail=:both), zscore)
 
@@ -244,7 +295,7 @@ function StatsBase.coeftable(m::ModeResult; level::Real=0.95)
     level_ = 100 * level
     level_percentage = isinteger(level_) ? Int(level_) : level_
 
-    cols = [estimates, stderrors, zscore, p, ci_low, ci_high]
+    cols = Vector[estimates, stderrors, zscore, p, ci_low, ci_high]
     colnms = [
         "Coef.",
         "Std. Error",
@@ -253,6 +304,10 @@ function StatsBase.coeftable(m::ModeResult; level::Real=0.95)
         "Lower $(level_percentage)%",
         "Upper $(level_percentage)%",
     ]
+    if notes !== nothing
+        push!(cols, notes)
+        push!(colnms, "Error notes")
+    end
     return StatsBase.CoefTable(cols, colnms, terms)
 end
 

test/optimisation/Optimisation.jl

@@ -635,6 +635,65 @@ using Turing
             maximum_a_posteriori(m; adtype=adbackend)
         end
     end
+
+    @testset "Collinear coeftable" begin
+        xs = [-1.0, 0.0, 1.0]
+        ys = [0.0, 0.0, 0.0]
+
+        @model function collinear(x, y)
+            a ~ Normal(0, 1)
+            b ~ Normal(0, 1)
+            return y ~ MvNormal(a .* x .+ b .* x, 1)
+        end
+
+        model = collinear(xs, ys)
+        mle_estimate = Turing.Optimisation.estimate_mode(model, MLE())
+        tab = coeftable(mle_estimate)
+        @assert isnan(tab.cols[2][1])
+        @assert tab.colnms[end] == "Error notes"
+        @assert occursin("singular", tab.cols[end][1])
+    end
+
+    @testset "Negative variance" begin
+        # A model for which the likelihood has a saddle point at x=0, y=0.
+        # Creating an optimisation result for this model at the x=0, y=0 results in negative
+        # variance for one of the variables, because the variance is calculated as the
+        # diagonal of the inverse of the Hessian.
+        @model function saddle_model()
+            x ~ Normal(0, 1)
+            y ~ Normal(x, 1)
+            Turing.@addlogprob! x^2 - y^2
+            return nothing
+        end
+        m = saddle_model()
+        ctx = Turing.Optimisation.OptimizationContext(DynamicPPL.LikelihoodContext())
+        optim_ld = Turing.Optimisation.OptimLogDensity(m, ctx)
+        vals = Turing.Optimisation.NamedArrays.NamedArray([0.0, 0.0])
+        m = Turing.Optimisation.ModeResult(vals, nothing, 0.0, optim_ld)
+        ct = coeftable(m)
+        @assert isnan(ct.cols[2][1])
+        @assert ct.colnms[end] == "Error notes"
+        @assert occursin("Negative variance", ct.cols[end][1])
+    end
+
+    @testset "Same coeftable with/without numerrors_warnonly" begin
+        xs = [0.0, 1.0, 2.0]
+
+        @model function extranormal(x)
+            mean ~ Normal(0, 1)
+            return x ~ Normal(mean, 1)
+        end
+
+        model = extranormal(xs)
+        mle_estimate = Turing.Optimisation.estimate_mode(model, MLE())
+        warnonly_coeftable = coeftable(mle_estimate; numerrors_warnonly=true)
+        no_warnonly_coeftable = coeftable(mle_estimate; numerrors_warnonly=false)
+        @assert warnonly_coeftable.cols == no_warnonly_coeftable.cols
+        @assert warnonly_coeftable.colnms == no_warnonly_coeftable.colnms
+        @assert warnonly_coeftable.rownms == no_warnonly_coeftable.rownms
+        @assert warnonly_coeftable.pvalcol == no_warnonly_coeftable.pvalcol
+        @assert warnonly_coeftable.teststatcol == no_warnonly_coeftable.teststatcol
+    end
 end
 
 end