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August 1st, 2016, 07:35 AM   #1
njk
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Sum over random vector values

I have the following expression:

$\sum_{\mathbf{x}}e^{\mathbf{x}}$,

where $\mathbf{x}$ is a random vector (i.e., $\mathbf{x} \in \mathbb{R}^D$ ), and $\sum_{\mathbf{x}}$ represents a sum over all possible values that $\mathbf{x}$ can assume.

Now, I'm trying to rewrite this expression more explicitly, but I'm a bit stuck and I'm not sure if I'm doing it correctly, here are two of my attempts:

$\sum_{\mathbf{x}}e^{\mathbf{x}} = \sum_{i}^{D} \int e^{x_i} \mathop{dx_i}$

or

$\sum_{\mathbf{x}}e^{\mathbf{x}} = \int e^{\sum_{i}^{D} x_i} \mathop{dx_i} = \prod_{i}^{D}\int e^{x_i} \mathop{dx_i}$

I'm more biased toward the second, but again, I'm not really sure that it is correct, and I'm even doubtful about the correctness of the notation that I'm using (because in the integral $x_i$ depends on $i$ which is the index of the summation, and this doesn't seem entirely right to me).
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August 1st, 2016, 03:02 PM   #2
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Your original expression doesn't make sense. The vector has a continuum of values and you can't sum. You might consider looking at it as some sort of integral.
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August 2nd, 2016, 01:19 AM   #3
njk
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Thanks mathman, I think you're right, the notation of my original expression is wrong in the first place.

However, considering that $\mathbf{x} \in \mathbb{R}^D$, if I had my original expression written directly like $\prod_{i}^{D}\int e^{x_i} \mathop{dx_i}$, i.e., with an explicit integral for each random variable and a product from $1$ to $D$, I think it should be correct, right?

To add more details, my intention was to use a more compact notation for the following expression:

$\int e^{x_1} \mathop{dx_1}\times\int e^{x_2} \mathop{dx_2}\times\dots\times\int e^{x_D} \mathop{dx_D}$

The only way I can think of is:

$\prod_{i}^{D}\int e^{x_i} \mathop{dx_i}$

Is there a better way to write the same?
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August 2nd, 2016, 03:01 PM   #4
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Not really. Just be careful for the product notation, i ranges from 1 to D.
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