integrate() gives totally wrong number

Question

integrate() gives horribly wrong answer:

integrate(function (x) dnorm(x, -5, 0.07), -Inf, Inf, subdivisions = 10000L)
# 2.127372e-23 with absolute error <         


        

Answer 1

Interesting workaround: not too surprisingly, integrate does well when the values sampled (on (-Inf,Inf), no less) are closer to the "center" of the data. You can reduce this by using your function but hinting at a center:

Without adjustment:

t(sapply(-10:10, function(i) integrate(function (x) dnorm(x, i, 0.07), -Inf, Inf, subdivisions = 10000L)))
#       value        abs.error    subdivisions message call      
#  [1,] 0            0            1            "OK"    Expression
#  [2,] 1            4.611403e-05 10           "OK"    Expression
#  [3,] 6.619713e-19 1.212066e-18 2            "OK"    Expression
#  [4,] 7.344551e-71 0            2            "OK"    Expression
#  [5,] 3.389557e-06 6.086176e-06 3            "OK"    Expression
#  [6,] 2.127372e-23 3.849798e-23 2            "OK"    Expression
#  [7,] 1            3.483439e-05 8            "OK"    Expression
#  [8,] 1            6.338078e-07 11           "OK"    Expression
#  [9,] 1            3.408389e-06 7            "OK"    Expression
# [10,] 1            6.414833e-07 8            "OK"    Expression
# [11,] 1            7.578907e-06 3            "OK"    Expression
# [12,] 1            6.414833e-07 8            "OK"    Expression
# [13,] 1            3.408389e-06 7            "OK"    Expression
# [14,] 1            6.338078e-07 11           "OK"    Expression
# [15,] 1            3.483439e-05 8            "OK"    Expression
# [16,] 2.127372e-23 3.849798e-23 2            "OK"    Expression
# [17,] 3.389557e-06 6.086176e-06 3            "OK"    Expression
# [18,] 7.344551e-71 0            2            "OK"    Expression
# [19,] 6.619713e-19 1.212066e-18 2            "OK"    Expression
# [20,] 1            4.611403e-05 10           "OK"    Expression
# [21,] 0            0            1            "OK"    Expression

If we add a "centering" hint, though, we get more consistent results:

t(sapply(-10:10, function(i) integrate(function (x, offset) dnorm(x + offset, i, 0.07), -Inf, Inf, subdivisions = 10000L, offset = i)))
#       value abs.error    subdivisions message call      
#  [1,] 1     7.578907e-06 3            "OK"    Expression
#  [2,] 1     7.578907e-06 3            "OK"    Expression
#  [3,] 1     7.578907e-06 3            "OK"    Expression
#  [4,] 1     7.578907e-06 3            "OK"    Expression
#  [5,] 1     7.578907e-06 3            "OK"    Expression
#  [6,] 1     7.578907e-06 3            "OK"    Expression
#  [7,] 1     7.578907e-06 3            "OK"    Expression
#  [8,] 1     7.578907e-06 3            "OK"    Expression
#  [9,] 1     7.578907e-06 3            "OK"    Expression
# [10,] 1     7.578907e-06 3            "OK"    Expression
# [11,] 1     7.578907e-06 3            "OK"    Expression
# [12,] 1     7.578907e-06 3            "OK"    Expression
# [13,] 1     7.578907e-06 3            "OK"    Expression
# [14,] 1     7.578907e-06 3            "OK"    Expression
# [15,] 1     7.578907e-06 3            "OK"    Expression
# [16,] 1     7.578907e-06 3            "OK"    Expression
# [17,] 1     7.578907e-06 3            "OK"    Expression
# [18,] 1     7.578907e-06 3            "OK"    Expression
# [19,] 1     7.578907e-06 3            "OK"    Expression
# [20,] 1     7.578907e-06 3            "OK"    Expression
# [21,] 1     7.578907e-06 3            "OK"    Expression

I recognize this is mitigation for heuristics, presumes knowing something about your distribution before integration, and is not a perfect "generic" solution. Just offering another perspective.