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probability or statistics - Why does conditional NExpectation fail for a multivariate distribution?


To give an example for a multiparametric distribution let us take a binormal distribution:


binormalDist = With[
    {
       μ = { 0, 0 },
       σ = { 1, 1 },
       ρ = 7/10

    },
    BinormalDistribution[ μ, σ, ρ ]
];

Calculating the conditional expectation in simple cases works out:


NExpectation[2 x + 3 \[Conditioned] x < 1., {x, y} \[Distributed] binormalDist ]


2.4248




and so does this too:


NExpectation[2 x + 3 \[Conditioned] y == 2., {x, y} \[Distributed] binormalDist ]


5.8



but


NExpectation[2 x + 3 \[Conditioned] (x < 1 && y == 2.), {x, y} \[Distributed] binormalDist ]

is returned unevaluated (as is Expectation with identical args)




NExpectation[ 3+ 2 x [Conditioned] x < 1 && y == 2., {x, y} [Distributed] BinormalDistribution[{0, 0}, {1, 1}, 7/10]]



Why? And how can it be solved?



Answer



Since


Probability[x < 1 && y == 2,
            {x, y} \[Distributed] BinormalDistribution[{0, 0}, {1, 1}, 7/10]]
   0


and the PDF of the purported conditional distribution has that expression as a denominator, this may be the reason why your attempt remains unevaluated.



Using Method -> "Trace", we see that NExpectation[] is attempting to evaluate the following expression internally:


NIntegrate[(3 + 2 x)
           PDF[Statistics`Library`ConditionalDistribution[x < 1 && y == 2,
               {x, y} \[Distributed] BinormalDistribution[{0, 0}, {1, 1}, 7/10]], {x, y}],
           {x, -∞, ∞}, {y, -∞, ∞}, AccuracyGoal -> ∞, Compiled -> Automatic,
           PrecisionGoal -> Automatic, WorkingPrecision -> MachinePrecision,
           MinRecursion -> 1]


where we see that Statistics`Library`ConditionalDistribution[] is used to represent the distribution induced by the conditional expectation. Notice that both PDF[Statistics`Library`ConditionalDistribution[x < 1, {x, y} \[Distributed] BinormalDistribution[{0, 0}, {1, 1}, 7/10]], {x, y}] and PDF[Statistics`Library`ConditionalDistribution[y == 2, {x, y} \[Distributed] BinormalDistribution[{0, 0}, {1, 1}, 7/10]], {x, y}] evaluate to expressions, but PDF[Statistics`Library`ConditionalDistribution[x < 1 && y == 2, {x, y} \[Distributed] BinormalDistribution[{0, 0}, {1, 1}, 7/10]], {x, y}] does not.


If you think Mathematica should be able to evaluate combined conditions like in the OP, you could try writing a suggestion to support.


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