How to sample a conditional distribution

Question

Let $X=(X_1,...,X_n)$ denote the multinormal distribution with covariance matrix $\Sigma$. Then $$\texttt{RandomVariate}[\texttt{MultinormalDistribution}[\Sigma]]$$ samples this distribution.

Question: Say we want to condition the random variable on something, for instance that $X_{i_k}=x_k$ is a particular number for $k=1,...,r$. How do I sample $\texttt{MultinormalDistribution}[\Sigma]$ conditionally on these restrictions?

Answer 1

If you're willing to take it on faith that the distribution of the remaining random variables given $X_1=x_1$ is a multivariate normal, then the following approach should work. (And I've replaced the conditioned value $x_1$ with $z$.)

r = 3;
Σ = Table[σ[Min[i, j], Max[i, j]], {i, r}, {j, r}];
dist = MultinormalDistribution[{{σ11, σ12, σ13}, {σ12, σ22, σ23}, {σ13, σ23, σ33}}];
μGivenx1[i_] := Expectation[x[i] \[Conditioned] x[1] == z, Table[x[k], {k, r}] \[Distributed] dist]
covGivenx1[i_, j_] := Expectation[(x[i] - μGivenx1[i]) (x[j] - μGivenx1[j]) \[Conditioned] x[1] == z, 
  Table[x[k], {k, r}] \[Distributed] dist]
distGivenx1 = MultinormalDistribution[Table[μGivenx1[i], {i, 2, r}],
  Table[covGivenx1[i, j], {i, 2, r}, {j, 2, r}]];

Then just assign values to the terms in $\Sigma$ and use RandomVariate on distGivenx1.

Here is a specific example:

r = 3;
z = 1.2;
Σ = {{1, .5, .3}, {0.5, 3, 0.2}, {0.3, 0.2, 2}};
dist = MultinormalDistribution[Σ];
μGivenx1[i_] := Expectation[x[i] \[Conditioned] x[1] == z, 
  Table[x[k], {k, r}] \[Distributed] dist]
covGivenx1[i_, j_] := Expectation[(x[i] - μGivenx1[i]) (x[j] - μGivenx1[j]) \[Conditioned] x[1] == z, 
  Table[x[k], {k, r}] \[Distributed] dist]
distGivenx1 = MultinormalDistribution[Table[μGivenx1[i], {i, 2, r}], 
   Table[covGivenx1[i, j], {i, 2, r}, {j, 2, r}]];
SeedRandom[12345];
RandomVariate[distGivenx1, 4]
(* {{-1.03966, -1.9972}, {3.58599, 0.964509}, {-0.450731, 0.0574758}, {1.09308, -0.670069}} *)