I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it in a close form.

After starting with a Gaussian random variable and restricting it to a condition, I end up with a Gaussian random variable $X\sim \mathcal N(0, \Sigma)$, where $\Sigma$ is singular (say I end up with one dimension less). I guess, that one can make computations with density $$ \begin{align} \pi(x) &= \frac{1}{\left(2\pi^{c-1} {|\Sigma|}\right)^{\frac{1}{2}}} \exp \left( -\frac 1 2 x^\intercal \Sigma^- x\right), \end{align} $$ where $|\Sigma|$, $\Sigma^-$ denote the generalized determinant and generalized inverse respectively. What I have understand so far, is that $\pi$ is a density function but with respect to a degenerate measure (a sort of restriction of the Lebesgue measure). I tried working with the dicomposition theory, but didn't get any satisfactory solution so far. Since I can calculate the kernel of $\Sigma$, after doing a transformation with respect to the eigenvalues/eigenvectors of $\Sigma$, I can become another density $\widetilde \pi$ which ignores the first component. My guess would be that I can integrate then with respect to the Lebesgue measure in one dimension less. Does it make sense?

The main questions would be: With respect to which measure is this distribution defined and do it exist a close form for this measure?


1 Answer 1


$\newcommand\Si\Sigma\newcommand\si\sigma\newcommand\R{\mathbb R}\newcommand{\la}{\lambda}$ The calculations can be done as follows.

Consider the spectral decomposition of $\Si$: \begin{equation} \Si=QDQ^T,\tag{0} \end{equation} where $Q$ is an orthogonal matrix $n\times n$ matrix and $D$ is the diagonal matrix with diagonal entries $\si_1^2>0,\dots,\si_k^2>0,0,\dots,0$, so that $k$ is the rank of $\Si$. Let $Y:=Q^TX$, so that $X=QY$ and $Y=(Y_1,\dots,Y_k,0,\dots,0)\sim N(0,D)$. So, for any Borel-measurable function $g\colon\mathbb R^n\to\mathbb R$ such that $Eg(X)$ exists, we have \begin{equation} \begin{aligned} Eg(X)&=Eg(Q[Y_1,\dots,Y_k,0,\dots,0]^T) \\ &=\int_{\mathbb R^k}g(Q[y_1,\dots,y_k,0,\dots,0]^T) \\ &\times f_{\si_1}(y_1)\cdots f_{\si_k}(y_k)dy_1\cdots dy_k, \end{aligned} \tag{1} \end{equation} where $f_\si$ is the pdf of $N(0,\si^2)$.

Now you can compute the probabilities for $X$ as follows: $$P(X\in B)=E1(X\in B),$$ where $B$ is any Borel subset of $\mathbb R^n$.

For an illustration, here is a Mathematica notebook with the calculation of the probability $P(X_1>X_2+1)$ given $(X_1,X_2)\sim N\left((0,0),\left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \\ \end{array} \right)\right)$:

enter image description here

As you insist on writing the expectation $Eg(X)$ as an integral with respect to the Lebesgue measure on the support of the distribution of $X$, here it is.

Note that the support of the distribution of $X$ is the range(=column space) $\Si\R^n$ of $\Si$, which is the same as the range $q\R^k$ of $q$, where $q$ is the map \begin{equation*} \R^k\ni z=(z_1,\dots,z_k)\mapsto qz:=QJz\in q\R^k=\Si\R^n \end{equation*} and, in turn, $J$ is the map \begin{equation*} \R^k\ni z=(z_1,\dots,z_k)\mapsto Jz:=(z_1,\dots,z_k,0,\dots,0)\in\R^n. \end{equation*} The equality $\Si\R^n=q\R^k$ follows because, in view of (0), the range $\Si\R^n$ of $\Si$ is the same as that of $QD^{1/2}$, which is the same as that of $q$ (since the range of $D^{1/2}$ is the same as that of $J$). Here, we identify matrices with the corresponding linear transformations.

In view of (1), \begin{equation*} \begin{aligned} Eg(X)&=\int_{\R^k}g(qz)\nu(dz) =\int_{\Si\R^n}g(x)\mu(dx), \end{aligned} \tag{2} \end{equation*} where $\nu(dz):=\pi(z)\la_k(dz)$, $\pi(z):=f_{\si_1}(z_1)\cdots f_{\si_k}(z_k)$, $\la_k(dz)$ is the Lebesgue measure on $\R^k$, and $\mu(dx):=\nu(q^{-1}(dx))=\pi(q^{-1}(x))\la_k(q^{-1}(dx))$.

Since the map $q$ is an isometry, for the Lebesgue measure $\la_{\Si\R^n}$ over $\Si\R^n$ we have $\la_{\Si\R^n}(dx)=\la_k(q^{-1}(dx))$. So, by (2), \begin{equation*} \begin{aligned} Eg(X)&=\int_{\Si\R^n}g(x)\pi(q^{-1}(x))\la_{\Si\R^n}(dx). \end{aligned} \tag{3} \end{equation*} In particular, letting $g:=1_B$ for any Borel set $B\subseteq\Si\R^n$, we get \begin{equation*} \begin{aligned} P(X\in B)&=\int_B \pi(q^{-1}(x))\la_{\Si\R^n}(dx). \end{aligned} \tag{3a} \end{equation*}

I think formulas (3) and, in particular, (3a) are completely useless, though -- because you will actually compute $Eg(X)$ and $P(X\in B)$ (as in the above Mathematica notebook) by formula (1).

  • $\begingroup$ Thanks for your answer! This is what I guessed, when I meant using $\widetilde \pi$, since $$ \widetilde \pi(y) = f_{\sigma 1} (y_1) \cdots f_{\sigma k}(y_k). $$ Thus, $\widetilde \pi$ can be seen as the density function of $Q^\intercal X \sim \mathcal N (0,D)$ "in a sense" (avoiding the zero components). But, is there a way to express the density of $X$ (i.e., $\pi$ in my question) w.r.t. a (degenerate) measure?, i.e., $$P(X\in B) = \int \pi(x) d?, $$ The reason for this need, is that I want to apply later transformations, and applying already one, makes everything more confusing. $\endgroup$
    – Skull Soul
    Mar 15, 2021 at 17:27
  • $\begingroup$ @SkullSoul : I have to teach now. I'll respond to this later. $\endgroup$ Mar 15, 2021 at 17:39
  • $\begingroup$ @SkullSoul : I have added the measure stuff. $\endgroup$ Mar 15, 2021 at 20:27
  • $\begingroup$ @losifPinelis: Thanks a lot for the effort!! I understand what you mean and that your explanation help for numerical computations. But my question was slightly different as your answer. I formulate again with the same terminology: Given is a positiv-SEMI definite matrix $\Sigma$, $\mu = 0$ and $$ \pi(x) = \frac{1}{2 \pi^{c-1} |\Sigma| } e^{-\frac{1}{2} x^T \Sigma^- x},$$ $|\Sigma|, \Sigma^-$ are generalized determinant and inverse. Is there a way to write the degenerate measure $\zeta$ in a close form, i.e. if $$Pr(X \in B) = \int_B \pi(x) d\zeta,$$ then $$\zeta = ?$$ $\endgroup$
    – Skull Soul
    Mar 18, 2021 at 20:34
  • $\begingroup$ @SkullSoul : Of course, the probabilities $P(X\in B)$ can be expressed as integrals with respect to the Lebesgue measure on the support of the distribution of $X$. I have now added the corresponding formula (3a) -- which is a particular case of the previously given formula (3). Formula (3a) has the correct expression, $\pi(q^{-1}(\cdot))$, for the density of the distribution of $X$ with respect to the Lebesgue measure on the support of the distribution of $X$. $\endgroup$ Mar 18, 2021 at 21:06

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