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A. Mulitivariate normal distribution case with latent variable $X$

Mapping from the low-dimensional space $X$ in Q-dimensional space (Q=2) to the high-dimensional space of $Y$ in D-dimensional space (say D=10) can be considered as

$$Y=WX+\mu+\epsilon$$

Distribution of $Y$ given $X$ taking isotropic noise $\epsilon$ approximately as $N(0,\sigma^2I)$ can be defined as $$P(Y|X)=(2\pi\sigma^2)^{-\frac{D}{2}} exp(-\frac{1}{2\sigma^2}||Y-WX-\mu||^2),$$ Were $W$ is taken as $D$ by $Q$ weight matrix.

If I take the Gaussian Prior over the latent variable $X$ with zero mean and unit variance $$P(X)=(2\pi)^{-Q/2}exp(-\frac{1}{2}X^TX)$$

Then the marginal distribution over $Y$ can be define as

$$P(Y)=\int P(Y|X)P(X)dX$$ $$=(2\pi)^{-\frac{D}{2}} |C|^{-\frac{1}{2}}exp(-\frac{1}{2}(Y-\mu)^TC^{-1}(Y-\mu)),$$ where $C=\sigma^2I+WW^T$

B. Mulitivariate normal distribution case with latent variable $X$

Now here comes my question if I define distribution of $P(Y|X)$ as log normal when taking isotropic noise $\epsilon$ approximately $LogN(0,\sigma^2I)$

$$P(Y|X)=(2\pi\sigma^2)^{-\frac{D}{2}}(\prod_{i=1}^{D}Y_{i}^{-1}) exp(-\frac{1}{2\sigma^2}||ln(Y)-WX-\mu||^2),$$ Where $W$ is taken as $D$ by $Q$ weight matrix.

If I take the lognormal Prior over the latent variable $X$ with zero mean and unit variance $$P(X)=(2\pi)^{-Q/2}(\prod_{i=1}^{Q}X_{i}^{-1})exp(-\frac{1}{2}ln(X)^Tln(X))$$

Then the marginal distribution over $Y$ can be define as

$$P(Y)=\int P(Y|X)P(X)dX$$ $$=(2\pi)^{-\frac{D}{2}} (\prod_{i=1}^{D}Y_{i}^{-1})|C|^{-\frac{1}{2}}exp(-\frac{1}{2}(ln(Y)-\mu)^TC^{-1}(ln(Y)-\mu)),$$ where $C=\sigma^2I+WW^T$

Questions:

I have two parts of my question: The first "is the product of two multivariate lognormal distribution is also multivariate lognormal distribution (as seen in equation of $p(Y)$ in the lognormal case)" If it is how can I prove this property?

and the second part of the question is "how can I perform marginalisation over $Y$ in case of lognormal case and will the marginalising out the latent variable $X$ will also lead to the lognormal distribution" and how can I prove this property

Assuming the same properties to that of normal distribution I have re-written the lognormal distribution is the expression for the PDF of $Y$ written above is true.

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I have came across a paper where they stated that: "The product of independent log-normal quantities also follows a log-normal distribution", pp-345. It also have very rich understanding of the Lognormal Distribution. You can download the Article from here: http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

As for the Second question, If I come across any solution, i will let you know.

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