Answers.Body: <p>It's the law of total probability, combined with the definition of conditional probability.</p>
  <p><span class="math-container">\begin{align*} 
  p(x_n) = \int_{x_{n-1}} p(x_n, x_{n-1}) = \int_{x_{n-1}} p(x_n \, | \, x_{n-1})p(x_{n-1})
  \end{align*}</span>
  First equality uses the law of total probability, second uses definition of conditional probability. The only difference in the paper, is that everything is additionally conditioned on <span class="math-container">$\mathcal{X}_t$</span>. (also, they are doing this for some <span class="math-container">$\tau$</span> steps, not just once.)</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-20T16:04:42.250Z
Answers.Text: So, what would the bounds of the integral be? Negative infinity to infinity? That seems to be the only answer that theoretically makes sense, but the paper implies that the integrals should be computable (but it should be not just infeasible as said, but impossible to begin with).
Answers.UserDisplayName: joelee
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2022-01-19T13:12:10.170Z
Answers.LastActivityDate: 2022-01-19T13:12:10.170Z
Answers.UserDisplayName: Taw
Body: <p>(My previous question regarded which theorem of probability was equation (2) referring to).</p>
  <p><a href="https://arxiv.org/pdf/2110.14392.pdf" rel="nofollow noreferrer">This paper</a> mentions that temporal forecasting is meant to solve an integral equation as denoted by (2) (found on page 3 under the section “3. PDE based Forecasting”), and that common auto-regressive approaches in the past have tried to approximate the integral equation.</p>
  <p>Where can I find information regarding these approaches? Unfortunately, the paper does not cite any of this, and the citations in the &quot;Related Work&quot; section seem to not make mention of the integral equation.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-19T12:33:49.100Z
Comments.Text: can you write the equation here, and please elaborate or write in more details the equation (2)? i just briefly read it and i can't find a clue what's the dots between the integral stands for.
Comments.UserDisplayName: Sanyou
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-20T15:44:24.480Z
Comments.Text: Yeah me neither, hence why I’m asking the question. The paper does not make it clear at all.
Comments.UserDisplayName: joelee
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-20T15:55:09.477Z
Comments.Text: My best guess is that the dots represent the integral signs in between, and the other set of dots represent the conditional probabilities in between. So basically the integral sign sequence and the probability sequence.
Comments.UserDisplayName: joelee
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-21T04:29:16.490Z
Comments.Text: oh, I thought you just don't understand where it came from or how it was formulated on question 1. question 1 and 2 is related, but I think question 3 can be a new separate question here.
Comments.UserDisplayName: Sanyou
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-21T23:25:56.013Z
Comments.Text: Thanks for leaving only one question. But can you please put your **specific** question in the title rather than writing "Question Regarding Early Approaches to Temporal Forecasting"?
Comments.UserDisplayName: nbro
Tags: <computer-vision><time-series>
Title: What Past Approaches is the "Taylor Swift" Paper Referring To?
UserDisplayName: joelee
ThreadText: # What Past Approaches is the "Taylor Swift" Paper Referring To?
  
  (Asked by: username_0 on 2022-01-18)
  
  (My previous question regarded which theorem of probability was equation (2) referring to).
  
  [This paper](https://arxiv.org/pdf/2110.14392.pdf) mentions that temporal forecasting is meant to solve an integral equation as denoted by (2) (found on page 3 under the section “3. PDE based Forecasting”), and that common auto-regressive approaches in the past have tried to approximate the integral equation.
  
  Where can I find information regarding these approaches? Unfortunately, the paper does not cite any of this, and the citations in the "Related Work" section seem to not make mention of the integral equation.
  
  **Tags:** computer-vision, time-series
  
  > **Comments:**
  > **username_1**: can you write the equation here, and please elaborate or write in more details the equation (2)? i just briefly read it and i can't find a clue what's the dots between the integral stands for.  
  > **username_0**: Yeah me neither, hence why I’m asking the question. The paper does not make it clear at all.  
  > **username_0**: My best guess is that the dots represent the integral signs in between, and the other set of dots represent the conditional probabilities in between. So basically the integral sign sequence and the probability sequence.  
  > **username_1**: oh, I thought you just don't understand where it came from or how it was formulated on question 1. question 1 and 2 is related, but I think question 3 can be a new separate question here.  
  > **username_2**: Thanks for leaving only one question. But can you please put your **specific** question in the title rather than writing "Question Regarding Early Approaches to Temporal Forecasting"?  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_3 on 2022-01-19. Score: 0 | Accepted answer
  
  It's the law of total probability, combined with the definition of conditional probability.
  
  \begin{align\*}
  p(x\_n) = \int\_{x\_{n-1}} p(x\_n, x\_{n-1}) = \int\_{x\_{n-1}} p(x\_n \, | \, x\_{n-1})p(x\_{n-1})
  \end{align\*}
  First equality uses the law of total probability, second uses definition of conditional probability. The only difference in the paper, is that everything is additionally conditioned on $\mathcal{X}\_t$. (also, they are doing this for some $\tau$ steps, not just once.)
  
  > **Comments:**
  > **username_0**: So, what would the bounds of the integral be? Negative infinity to infinity? That seems to be the only answer that theoretically makes sense, but the paper implies that the integrals should be computable (but it should be not just infeasible as said, but impossible to begin with).  
  
  ---
  
  