From 8846e0544a5919a4c429796ed9cc37602ab6732a Mon Sep 17 00:00:00 2001
From: wdli <wdli@umich.edu>
Date: Sun, 24 Dec 2023 00:08:30 -0500
Subject: [PATCH 1/3] change to \epsilon and table format

fix table

fix table

fix table

fix table:

fix table:

fix table
---
 README.md | 72 ++++++++++++++++++++++++++++++++++---------------------
 1 file changed, 45 insertions(+), 27 deletions(-)

diff --git a/README.md b/README.md
index 59417fbf..b1264e27 100644
--- a/README.md
+++ b/README.md
@@ -29,35 +29,35 @@ $$ x_0 \overset{q(x_1 | x_0)}{\rightarrow} x_1 \overset{q(x_2 | x_1)}{\rightarro
 
 This process is a markov chain, $x_t$ only depends on $x_{t-1}$. $q(x_{t} | x_{t-1})$ adds Gaussian noise at each time step $t$, according to a known variance schedule $β_{t}$ 
 
-$$ x_t = \sqrt{1-β_t}\times x_{t-1} + \sqrt{β_t}\times ϵ_{t} $$
+$$ x_t = \sqrt{1-β_t}\times x_{t-1} + \sqrt{β_t}\times \epsilon_{t} $$
 
 * $β_t$ is not constant at each time step $t$. In fact one defines a so-called "variance schedule", which can be linear, quadratic, cosine, etc. 
 
 $$ 0 < β_1 < β_2 < β_3 < \dots < β_T < 1 $$
 
-* $ϵ_{t}$ Gaussian noise, sampled from standard normal distribution.
+* $\epsilon_{t}$ Gaussian noise, sampled from standard normal distribution.
 
 
 
 
 
-$$ x_t = \sqrt{1-β_t}\times x_{t-1} + \sqrt{β_t} \times ϵ_{t} $$
+$$ x_t = \sqrt{1-β_t}\times x_{t-1} + \sqrt{β_t} \times \epsilon_{t} $$
 
 Define $a_t = 1 - β_t$
 
-$$ x_t = \sqrt{a_{t}}\times x_{t-1} +  \sqrt{1-a_t} \times ϵ_{t} $$
+$$ x_t = \sqrt{a_{t}}\times x_{t-1} +  \sqrt{1-a_t} \times \epsilon_{t} $$
 
 ### 2.1 Relationship between $x_t$ and $x_{t-2}$
 
-$$ x_{t-1} = \sqrt{a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t-1}} \times ϵ_{t-1}$$ 
+$$ x_{t-1} = \sqrt{a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t-1}} \times \epsilon_{t-1}$$ 
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}} (\sqrt{a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t-1}} ϵ_{t-1}) +  \sqrt{1-a_t} \times ϵ_t $$
+$$ x_t = \sqrt{a_{t}} (\sqrt{a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t-1}} \epsilon_{t-1}) +  \sqrt{1-a_t} \times \epsilon_t $$
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{a_{t}(1-a_{t-1})} ϵ_{t-1} +  \sqrt{1-a_t} \times ϵ_t $$
+$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{a_{t}(1-a_{t-1})} \epsilon_{t-1} +  \sqrt{1-a_t} \times \epsilon_t $$
 
 <details><summary>Because $N(\mu_{1},\sigma_{1}^{2}) + N(\mu_{2},\sigma_{2}^{2}) = N(\mu_{1}+\mu_{2},\sigma_{1}^{2} + \sigma_{2}^{2})$</summary>
 <p>
@@ -66,52 +66,54 @@ $$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{a_{t}(1-a_{t-1})} ϵ_{t-1} +
 </details>
 
 
-$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{a_{t}(1-a_{t-1}) + 1-a_t} \times ϵ $$
+$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{a_{t}(1-a_{t-1}) + 1-a_t} \times \epsilon $$
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t}a_{t-1}} \times ϵ $$
+$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t}a_{t-1}} \times \epsilon $$
 
 ### 2.2 Relationship between $x_t$ and $x_{t-3}$
 
-$$ x_{t-2} = \sqrt{a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t-2}} \times ϵ_{t-2} $$
+$$ x_{t-2} = \sqrt{a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t-2}} \times \epsilon_{t-2} $$
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}a_{t-1}}(\sqrt{a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t-2}} ϵ_{t-2}) +  \sqrt{1-a_{t}a_{t-1}}\times ϵ $$
+$$ x_t = \sqrt{a_{t}a_{t-1}}(\sqrt{a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t-2}} \epsilon_{t-2}) +  \sqrt{1-a_{t}a_{t-1}}\times \epsilon $$
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{a_{t}a_{t-1}(1-a_{t-2})} ϵ_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times ϵ $$
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{a_{t}a_{t-1}(1-a_{t-2})} \epsilon_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times \epsilon $$
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}} ϵ_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times ϵ $$
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}} \epsilon_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times \epsilon $$
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{(a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}) + 1-a_{t}a_{t-1}} \times ϵ $$
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{(a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}) + 1-a_{t}a_{t-1}} \times \epsilon $$
 
 $$ \Downarrow  $$
 
-$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}} \times ϵ $$
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}} \times \epsilon $$
 
 ### 2.3 Relationship between $x_t$ and $x_0$
 
-* $x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times ϵ$
-* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}}\times ϵ$
-* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times x_{t-k} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times ϵ$
-* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times x_{0} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times ϵ$
+* $x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times \epsilon$
+* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}}\times \epsilon$
+* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times x_{t-k} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times \epsilon$
+* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times x_{0} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times \epsilon$
 
 
 $$\bar{a}_{t} := a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}$$
 
-$$x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times ϵ , ϵ \sim N(0,I) $$
+
+$$x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times \epsilon , \epsilon \sim N(0,I) $$
 
 $$ \Downarrow  $$
 
 $$ q(x_{t}|x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}   \right ) } $$
 
+
 # 3.Reverse Process $p$
 
 ![](denoise.jpg)
@@ -120,7 +122,7 @@ Because $P(A|B) = \frac{ P(B|A)P(A) }{ P(B) }$
 
 $$ p(x_{t-1}|x_{t},x_{0}) = \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{q(x_{t}|x_0)} $$
 
-<table>
+<!-- <table>
   <tbody>
     <tr>
       <td>
@@ -157,7 +159,26 @@ $$ p(x_{t-1}|x_{t},x_{0}) = \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{
     </tr>
   </tbody>
   
-</table>
+</table> -->
+
+
+|  $$x_{t} = \sqrt{a_t}x_{t-1}+\sqrt{1-a_t}\times \epsilon$$  | ~ | $$N(\sqrt{a_t}x_{t-1}, 1-a_{t})$$ |
+| -------------------------------------------------- | - | --------------------------------- |
+|  $x_{t-1} = \sqrt{\bar{a}_{t-1}}x_0 + \sqrt{1-\bar{a}_{t-1}}\times \epsilon$ | ~ | $ N( \sqrt{\bar{a}_{t-1}}x_0, 1-\bar{a}_{t-1}) $ |
+|  $x_{t} = \sqrt{\bar{a}_{t}}x_0+ \sqrt{1-\bar{a}_{t}}\times \epsilon$ | ~ | $N( \sqrt{\bar{a}_{t}}x_0, 1-\bar{a}_{t})$ |
+
+| Equation                                               |  | Distribution                              |
+| ------------------------------------------------------- | --- | ------------------------------------------ |
+| $$x_{t} = \sqrt{a_t}x_{t-1}+\sqrt{1-a_t}\times \epsilon$$ | ~   | $$N(\sqrt{a_t}x_{t-1}, 1-a_{t})$$         |
+| $$x_{t-1} = \sqrt{\bar{a}_{t-1}}x_0 + \sqrt{1-\bar{a}_{t-1}}\times \epsilon$$ | ~ | $$N( \sqrt{\bar{a}_{t-1}}x_0, 1-\bar{a}_{t-1})$$ |
+| $$x_{t} = \sqrt{\bar{a}_{t}}x_0+ \sqrt{1-\bar{a}_{t}}\times \epsilon$$ | ~ | $$N( \sqrt{\bar{a}_{t}}x_0, 1-\bar{a}_{t})$$ |
+
+| Equation                                               |  | Distribution                              |
+| ------------------------------------------------------- | --- | ------------------------------------------ |
+| $x_{t} = \sqrt{a_t}x_{t-1}+\sqrt{1-a_t}\times \epsilon$ | ~   | $N(\sqrt{a_t}x_{t-1}, 1-a_{t})$            |
+| $x_{t-1} = \sqrt{\bar{a}_{t-1}}x_0 + \sqrt{1-\bar{a}_{t-1}}\times \epsilon$ | ~ | $N( \sqrt{\bar{a}_{t-1}}x_0, 1-\bar{a}_{t-1})$ |
+| $x_{t} = \sqrt{\bar{a}_{t}}x_0+ \sqrt{1-\bar{a}_{t}}\times \epsilon$ | ~ | $N( \sqrt{\bar{a}_{t}}x_0, 1-\bar{a}_{t})$   |
+
 
 
 $$ q(x_{t}|x_{t-1},x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-a_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}}   \right ) } $$
@@ -219,9 +240,6 @@ exp \left[  -\frac{1}{2}
 
 
 
-
-
-
 $$ \Downarrow  $$
 
 $$ \frac{1}{\sqrt{2\pi} \left ( {\color{Red} \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}}}  \right ) }  
@@ -250,13 +268,13 @@ $$ p(x_{t-1}|x_{t}) \sim N\left(
 
 
 
-Because $x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times ϵ$, $x_0 = \frac{x_t - \sqrt{1-\bar{a}_t}\times ϵ}{\sqrt{\bar{a}_t}}$. Substitute $x_0$ with this formula.
+Because $x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times \epsilon$, $x_0 = \frac{x_t - \sqrt{1-\bar{a}_t}\times \epsilon}{\sqrt{\bar{a}_t}}$. Substitute $x_0$ with this formula.
 
 
 $$ p(x_{t-1}|x_{t}) \sim N\left( 
       {\color{Purple} \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
       +
-      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}\times \frac{x_t - \sqrt{1-\bar{a}_t}\times ϵ}{\sqrt{\bar{a}_t}} } ,
+      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}\times \frac{x_t - \sqrt{1-\bar{a}_t}\times \epsilon}{\sqrt{\bar{a}_t}} } ,
        {\color{Red} \frac{ \beta_{t} (1-\bar{a}_{t-1}) } { 1-\bar{a}_{t}}} 
  \right) $$
 

From c64ac1301a3af0330e8bfadfef832d5c3c8cfa73 Mon Sep 17 00:00:00 2001
From: wdli <wdli@umich.edu>
Date: Sun, 24 Dec 2023 00:32:07 -0500
Subject: [PATCH 2/3] add

add

add

add

add

add

add

add

add

add

update README.md

delete README
---
 README.md     | 185 ++++++++++++++++++++---------------
 README.raw.md | 266 ++++++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 375 insertions(+), 76 deletions(-)
 create mode 100644 README.raw.md

diff --git a/README.md b/README.md
index b1264e27..d7cecf09 100644
--- a/README.md
+++ b/README.md
@@ -122,7 +122,7 @@ Because $P(A|B) = \frac{ P(B|A)P(A) }{ P(B) }$
 
 $$ p(x_{t-1}|x_{t},x_{0}) = \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{q(x_{t}|x_0)} $$
 
-<!-- <table>
+<table>
   <tbody>
     <tr>
       <td>
@@ -159,38 +159,22 @@ $$ p(x_{t-1}|x_{t},x_{0}) = \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{
     </tr>
   </tbody>
   
-</table> -->
-
-
-|  $$x_{t} = \sqrt{a_t}x_{t-1}+\sqrt{1-a_t}\times \epsilon$$  | ~ | $$N(\sqrt{a_t}x_{t-1}, 1-a_{t})$$ |
-| -------------------------------------------------- | - | --------------------------------- |
-|  $x_{t-1} = \sqrt{\bar{a}_{t-1}}x_0 + \sqrt{1-\bar{a}_{t-1}}\times \epsilon$ | ~ | $ N( \sqrt{\bar{a}_{t-1}}x_0, 1-\bar{a}_{t-1}) $ |
-|  $x_{t} = \sqrt{\bar{a}_{t}}x_0+ \sqrt{1-\bar{a}_{t}}\times \epsilon$ | ~ | $N( \sqrt{\bar{a}_{t}}x_0, 1-\bar{a}_{t})$ |
-
-| Equation                                               |  | Distribution                              |
-| ------------------------------------------------------- | --- | ------------------------------------------ |
-| $$x_{t} = \sqrt{a_t}x_{t-1}+\sqrt{1-a_t}\times \epsilon$$ | ~   | $$N(\sqrt{a_t}x_{t-1}, 1-a_{t})$$         |
-| $$x_{t-1} = \sqrt{\bar{a}_{t-1}}x_0 + \sqrt{1-\bar{a}_{t-1}}\times \epsilon$$ | ~ | $$N( \sqrt{\bar{a}_{t-1}}x_0, 1-\bar{a}_{t-1})$$ |
-| $$x_{t} = \sqrt{\bar{a}_{t}}x_0+ \sqrt{1-\bar{a}_{t}}\times \epsilon$$ | ~ | $$N( \sqrt{\bar{a}_{t}}x_0, 1-\bar{a}_{t})$$ |
-
-| Equation                                               |  | Distribution                              |
-| ------------------------------------------------------- | --- | ------------------------------------------ |
-| $x_{t} = \sqrt{a_t}x_{t-1}+\sqrt{1-a_t}\times \epsilon$ | ~   | $N(\sqrt{a_t}x_{t-1}, 1-a_{t})$            |
-| $x_{t-1} = \sqrt{\bar{a}_{t-1}}x_0 + \sqrt{1-\bar{a}_{t-1}}\times \epsilon$ | ~ | $N( \sqrt{\bar{a}_{t-1}}x_0, 1-\bar{a}_{t-1})$ |
-| $x_{t} = \sqrt{\bar{a}_{t}}x_0+ \sqrt{1-\bar{a}_{t}}\times \epsilon$ | ~ | $N( \sqrt{\bar{a}_{t}}x_0, 1-\bar{a}_{t})$   |
-
+</table>
 
 
 $$ q(x_{t}|x_{t-1},x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-a_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}}   \right ) } $$
 
+<p>
 $$ q(x_{t-1}|x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t-1}}} e^{\left (  -\frac{1}{2}\frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}}   \right ) } $$
+</p>
 
+<p>
 $$ q(x_{t}|x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}   \right ) } $$
+</p>
 
 
 
-
-
+<p>
 $$ \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{q(x_{t}|x_0)} = \left [
   \frac{1}{\sqrt{2\pi} \sqrt{1-a_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}}   \right ) } 
 \right ] * 
@@ -200,86 +184,135 @@ $$ \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{q(x_{t}|x_0)} = \left [
 \left [ 
   \frac{1}{\sqrt{2\pi} \sqrt{1-\bar{a}_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}   \right ) }
 \right ]  $$
+</p>
 
 $$ \Downarrow  $$
 
-$$ \frac{\sqrt{2\pi} \sqrt{1-\bar{a}_{t}}}{\sqrt{2\pi} \sqrt{1-a_{t}} \sqrt{2\pi} \sqrt{1-\bar{a}_{t-1}} }
-e^{\left [  -\frac{1}{2}
-\left (
- \frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}} +
- \frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}} -
- \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
- \right )
-    \right ] } $$
+
+<p>
+  $$
+  \frac{\sqrt{2\pi} \sqrt{1-\bar{a}_{t}}}{\sqrt{2\pi} \sqrt{1-a_{t}} \sqrt{2\pi} \sqrt{1-\bar{a}_{t-1}} }
+  e^{\left [ -\frac{1}{2}
+  \left (
+    \frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}} +
+    \frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}} -
+    \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
+  \right )
+  \right]}
+  $$
+</p>
+
 
 $$ \Downarrow  $$
 
-$$ \frac{1}{\sqrt{2\pi} \left ( \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}} \right ) }
-exp{\left [  -\frac{1}{2}
-\left (
- \frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_t} +
- \frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}} -
- \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
- \right )
-    \right ] } $$
+
+<p>
+  $$\frac{1}{\sqrt{2\pi} \left( 
+    \frac{\sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}}}{\sqrt{1-\bar{a}_{t}}}
+  \right)}
+  \exp \left[ -\frac{1}{2}
+  \left(
+    \frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_t} +
+    \frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}} -
+    \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
+  \right)
+  \right] $$
+</p>
+
 
 $$ \Downarrow  $$
 
-$$ \frac{1}{\sqrt{2\pi} \left ( \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}} \right ) }
-exp \left[  -\frac{1}{2}
-\left (
- \frac{
-   x_{t}^2-2\sqrt{a_t}x_{t}x_{t-1}+{a_t}x_{t-1}^2
- }{1-a_t} +
- \frac{
-   x_{t-1}^2-2\sqrt{\bar{a}_{t-1}}x_0x_{t-1}+\bar{a}_{t-1}x_0^2
-  }{1-\bar{a}_{t-1}} -
- \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
-\right)
-\right] $$
 
+<p>
+  $$ \frac{1}{\sqrt{2\pi} \left( 
+    \frac{\sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}}}{\sqrt{1-\bar{a}_{t}}}
+  \right)}
+  \exp \left[ -\frac{1}{2}
+  \left (
+    \frac{
+      x_{t}^2-2\sqrt{a_t}x_{t}x_{t-1}+{a_t}x_{t-1}^2
+    }{1-a_t} +
+    \frac{
+      x_{t-1}^2-2\sqrt{\bar{a}_{t-1}}x_0x_{t-1}+\bar{a}_{t-1}x_0^2
+    }{1-\bar{a}_{t-1}} -
+    \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
+  \right)
+  \right] $$
+</p>
 
 
 $$ \Downarrow  $$
 
-$$ \frac{1}{\sqrt{2\pi} \left ( {\color{Red} \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}}}  \right ) }  
-exp \left[
--\frac{1}{2}
-\frac{
-  \left(
-    x_{t-1} - \left(
-      {\color{Purple} \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
-      +
-      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}x_0} 
+
+<p>
+  $$\frac{1}{\sqrt{2\pi} \left( 
+      \frac{\sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}}}{\sqrt{1-\bar{a}_{t}}}
+  \right)}
+  \exp \left[ -\frac{1}{2}
+  \frac{
+    \left(
+      x_{t-1} - \left(
+          \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
+          +
+          \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}x_0
       \right)
-  \right) ^2
-} {   \left( {\color{Red} \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}}}  \right)^2 }
-\right] $$
+    \right)^2
+  }{ \left( 
+        \frac{\sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}}}{\sqrt{1-\bar{a}_{t}}}
+    \right)^2
+  } \right]$$
+</p>
+
 
 
 $$ \Downarrow  $$
 
-$$ p(x_{t-1}|x_{t}) \sim N\left( 
-      {\color{Purple} \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
+
+<p>
+  <span style="color: Purple;">
+    $$ p(x_{t-1}|x_{t}) \sim N\left( 
+      \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
       +
-      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}x_0}  ,
-      \left( {\color{Red} \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}}}  \right)^2
- \right) $$
+      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}x_0,
+    </span>
+  </span>
+  <span style="color: Red;">
+    \left( \frac{\sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}}}{\sqrt{1-\bar{a}_{t}}} \right)^2
+  \right) $$
+  </span>
+</p>
 
 
 
 Because $x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times \epsilon$, $x_0 = \frac{x_t - \sqrt{1-\bar{a}_t}\times \epsilon}{\sqrt{\bar{a}_t}}$. Substitute $x_0$ with this formula.
 
 
-$$ p(x_{t-1}|x_{t}) \sim N\left( 
-      {\color{Purple} \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
+ <p>
+  <span style="color: Purple;">
+    $$ p(x_{t-1}|x_{t}) \sim N\left( 
+      \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
       +
-      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}\times \frac{x_t - \sqrt{1-\bar{a}_t}\times \epsilon}{\sqrt{\bar{a}_t}} } ,
-       {\color{Red} \frac{ \beta_{t} (1-\bar{a}_{t-1}) } { 1-\bar{a}_{t}}} 
- \right) $$
-
-
-
+      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}\times \frac{x_t - \sqrt{1-\bar{a}_t}\times \epsilon}{\sqrt{\bar{a}_t}},
+    </span>
+  </span>
+  <span style="color: Red;">
+    \frac{\beta_{t} (1-\bar{a}_{t-1})}{1-\bar{a}_{t}} 
+  \right) $$
+  </span>
+</p>
 
 
 
+<p>
+  <span style="color: Purple;">
+     $$ p(x_{t-1}|x_{t}) \sim N\left( 
+      \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
+      +
+      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}\times \frac{x_t - \sqrt{1-\bar{a}_t}\times \epsilon}{\sqrt{\bar{a}_t}},
+    </span>
+  </span>
+  <span style="color: Red;">
+    \frac{\beta_{t} (1-\bar{a}_{t-1})}{1-\bar{a}_{t}} 
+  \right) $$
+  </span>
+</p>
diff --git a/README.raw.md b/README.raw.md
new file mode 100644
index 00000000..44ff83a7
--- /dev/null
+++ b/README.raw.md
@@ -0,0 +1,266 @@
+# Diffusion model in web browser
+
+
+[Demo Site](https://wangjia184.github.io/diffusion_model/)
+[哔哩哔哩 | 大白话AI | 扩散模型 ](https://www.bilibili.com/video/BV1tz4y1h7q1/)(Chinese)
+
+
+[Video Explaination](https://www.youtube.com/watch?v=zEZOYZeIPUs&ab_channel=%E5%A4%A7%E7%99%BD%E8%AF%9DAI)(Chinese)
+
+
+## 1. DDPM Introduction
+
+![](https://huggingface.co/blog/assets/78_annotated-diffusion/diffusion_figure.png)
+
+* $q$ - a fixed (or predefined) **forward** diffusion process of adding Gaussian noise to an image gradually, until ending up with pure noise
+* $p_θ$ - a learned **reverse** denoising diffusion process, where a neural network is trained to gradually denoise an image starting from pure noise, until ending up with an actual image.
+
+Both the forward and reverse process indexed by $t$ happen for some number of finite time steps $T$ (the DDPM authors use $T$=1000). You start with $t=0$ where you sample a real image $x_0$ from your data distribution, and the forward process samples some noise from a Gaussian distribution at each time step $t$, which is added to the image of the previous time step. Given a sufficiently large $T$ and a well behaved schedule for adding noise at each time step, you end up with what is called an *isotropic Gaussian distribution* at $t=T$ via a gradual process
+
+
+
+
+## 2. Forward Process $q$
+
+![](chain.png)
+
+$$ x_0 \overset{q(x_1 | x_0)}{\rightarrow} x_1 \overset{q(x_2 | x_1)}{\rightarrow} x_2 \rightarrow \dots  \rightarrow x_{T-1} \overset{q(x_{t} | x_{t-1})}{\rightarrow} x_T $$
+
+
+This process is a markov chain, $x_t$ only depends on $x_{t-1}$. $q(x_{t} | x_{t-1})$ adds Gaussian noise at each time step $t$, according to a known variance schedule $β_{t}$ 
+
+$$ x_t = \sqrt{1-β_t}\times x_{t-1} + \sqrt{β_t}\times ϵ_{t} $$
+
+* $β_t$ is not constant at each time step $t$. In fact one defines a so-called "variance schedule", which can be linear, quadratic, cosine, etc. 
+
+$$ 0 < β_1 < β_2 < β_3 < \dots < β_T < 1 $$
+
+* $ϵ_{t}$ Gaussian noise, sampled from standard normal distribution.
+
+
+
+
+
+$$ x_t = \sqrt{1-β_t}\times x_{t-1} + \sqrt{β_t} \times ϵ_{t} $$
+
+Define $a_t = 1 - β_t$
+
+$$ x_t = \sqrt{a_{t}}\times x_{t-1} +  \sqrt{1-a_t} \times ϵ_{t} $$
+
+### 2.1 Relationship between $x_t$ and $x_{t-2}$
+
+$$ x_{t-1} = \sqrt{a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t-1}} \times ϵ_{t-1}$$ 
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}} (\sqrt{a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t-1}} ϵ_{t-1}) +  \sqrt{1-a_t} \times ϵ_t $$
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{a_{t}(1-a_{t-1})} ϵ_{t-1} +  \sqrt{1-a_t} \times ϵ_t $$
+
+<details><summary>Because $N(\mu_{1},\sigma_{1}^{2}) + N(\mu_{2},\sigma_{2}^{2}) = N(\mu_{1}+\mu_{2},\sigma_{1}^{2} + \sigma_{2}^{2})$</summary>
+<p>
+<a href="https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables" target="_blank">Proof</a>
+</p>
+</details>
+
+
+$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{a_{t}(1-a_{t-1}) + 1-a_t} \times ϵ $$
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t}a_{t-1}} \times ϵ $$
+
+### 2.2 Relationship between $x_t$ and $x_{t-3}$
+
+$$ x_{t-2} = \sqrt{a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t-2}} \times ϵ_{t-2} $$
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}a_{t-1}}(\sqrt{a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t-2}} ϵ_{t-2}) +  \sqrt{1-a_{t}a_{t-1}}\times ϵ $$
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{a_{t}a_{t-1}(1-a_{t-2})} ϵ_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times ϵ $$
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}} ϵ_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times ϵ $$
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3}  +  \sqrt{(a_{t}a_{t-1}-a_{t}a_{t-1}a_{t-2}) + 1-a_{t}a_{t-1}} \times ϵ $$
+
+$$ \Downarrow  $$
+
+$$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}} \times ϵ $$
+
+### 2.3 Relationship between $x_t$ and $x_0$
+
+* $x_t = \sqrt{a_{t}a_{t-1}}\times x_{t-2} +  \sqrt{1-a_{t}a_{t-1}}\times ϵ$
+* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}}\times ϵ$
+* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times x_{t-k} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{t-(k-2)}a_{t-(k-1)}}\times ϵ$
+* $x_t = \sqrt{a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times x_{0} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}}\times ϵ$
+
+
+$$\bar{a}_{t} := a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}$$
+
+$$x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times ϵ , ϵ \sim N(0,I) $$
+
+$$ \Downarrow  $$
+
+$$ q(x_{t}|x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}   \right ) } $$
+
+# 3.Reverse Process $p$
+
+![](denoise.jpg)
+
+Because $P(A|B) = \frac{ P(B|A)P(A) }{ P(B) }$
+
+$$ p(x_{t-1}|x_{t},x_{0}) = \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{q(x_{t}|x_0)} $$
+
+<table>
+  <tbody>
+    <tr>
+      <td>
+         $$x_{t} = \sqrt{a_t}x_{t-1}+\sqrt{1-a_t}\times ϵ$$
+      </td>
+      <td>
+        ~
+      </td>
+      <td>
+        $N(\sqrt{a_t}x_{t-1}, 1-a_{t})$
+      </td>
+    </tr>
+    <tr>
+      <td>
+        $$x_{t-1} = \sqrt{\bar{a}_{t-1}}x_0+ \sqrt{1-\bar{a}_{t-1}}\times ϵ$$
+      </td>
+      <td>
+        ~
+      </td>
+      <td>
+        $N( \sqrt{\bar{a}_{t-1}}x_0, 1-\bar{a}_{t-1})$
+      </td>
+    </tr>
+    <tr>
+      <td>
+        $$x_{t} = \sqrt{\bar{a}_{t}}x_0+ \sqrt{1-\bar{a}_{t}}\times ϵ$$
+      </td>
+      <td>
+        ~
+      </td>
+      <td>
+        $N( \sqrt{\bar{a}_{t}}x_0, 1-\bar{a}_{t})$
+      </td>
+    </tr>
+  </tbody>
+  
+</table>
+
+
+$$ q(x_{t}|x_{t-1},x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-a_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}}   \right ) } $$
+
+$$ q(x_{t-1}|x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t-1}}} e^{\left (  -\frac{1}{2}\frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}}   \right ) } $$
+
+$$ q(x_{t}|x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}   \right ) } $$
+
+
+
+
+
+$$ \frac{ q(x_{t}|x_{t-1},x_{0})\times q(x_{t-1}|x_0)}{q(x_{t}|x_0)} = \left [
+  \frac{1}{\sqrt{2\pi} \sqrt{1-a_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}}   \right ) } 
+\right ] * 
+\left [ 
+\frac{1}{\sqrt{2\pi} \sqrt{1-\bar{a}_{t-1}}} e^{\left (  -\frac{1}{2}\frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}}   \right ) }  
+\right ] \div
+\left [ 
+  \frac{1}{\sqrt{2\pi} \sqrt{1-\bar{a}_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}   \right ) }
+\right ]  $$
+
+$$ \Downarrow  $$
+
+$$ \frac{\sqrt{2\pi} \sqrt{1-\bar{a}_{t}}}{\sqrt{2\pi} \sqrt{1-a_{t}} \sqrt{2\pi} \sqrt{1-\bar{a}_{t-1}} }
+e^{\left [  -\frac{1}{2}
+\left (
+ \frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_{t}} +
+ \frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}} -
+ \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
+ \right )
+    \right ] } $$
+
+$$ \Downarrow  $$
+
+$$ \frac{1}{\sqrt{2\pi} \left ( \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}} \right ) }
+exp{\left [  -\frac{1}{2}
+\left (
+ \frac{(x_{t}-\sqrt{a_t}x_{t-1})^2}{1-a_t} +
+ \frac{(x_{t-1}-\sqrt{\bar{a}_{t-1}}x_0)^2}{1-\bar{a}_{t-1}} -
+ \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
+ \right )
+    \right ] } $$
+
+$$ \Downarrow  $$
+
+$$ \frac{1}{\sqrt{2\pi} \left ( \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}} \right ) }
+exp \left[  -\frac{1}{2}
+\left (
+ \frac{
+   x_{t}^2-2\sqrt{a_t}x_{t}x_{t-1}+{a_t}x_{t-1}^2
+ }{1-a_t} +
+ \frac{
+   x_{t-1}^2-2\sqrt{\bar{a}_{t-1}}x_0x_{t-1}+\bar{a}_{t-1}x_0^2
+  }{1-\bar{a}_{t-1}} -
+ \frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}
+\right)
+\right] $$
+
+
+
+
+
+
+$$ \Downarrow  $$
+
+$$ \frac{1}{\sqrt{2\pi} \left ( {\color{Red} \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}}}  \right ) }  
+exp \left[
+-\frac{1}{2}
+\frac{
+  \left(
+    x_{t-1} - \left(
+      {\color{Purple} \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
+      +
+      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}x_0} 
+      \right)
+  \right) ^2
+} {   \left( {\color{Red} \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}}}  \right)^2 }
+\right] $$
+
+
+$$ \Downarrow  $$
+
+$$ p(x_{t-1}|x_{t}) \sim N\left( 
+      {\color{Purple} \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
+      +
+      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}x_0}  ,
+      \left( {\color{Red} \frac{ \sqrt{1-a_t} \sqrt{1-\bar{a}_{t-1}} } {\sqrt{1-\bar{a}_{t}}}}  \right)^2
+ \right) $$
+
+
+
+Because $x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times ϵ$, $x_0 = \frac{x_t - \sqrt{1-\bar{a}_t}\times ϵ}{\sqrt{\bar{a}_t}}$. Substitute $x_0$ with this formula.
+
+
+$$ p(x_{t-1}|x_{t}) \sim N\left( 
+      {\color{Purple} \frac{\sqrt{a_t}(1-\bar{a}_{t-1})}{1-\bar{a}_t}x_t
+      +
+      \frac{\sqrt{\bar{a}_{t-1}}(1-a_t)}{1-\bar{a}_t}\times \frac{x_t - \sqrt{1-\bar{a}_t}\times ϵ}{\sqrt{\bar{a}_t}} } ,
+       {\color{Red} \frac{ \beta_{t} (1-\bar{a}_{t-1}) } { 1-\bar{a}_{t}}} 
+ \right) $$
+
+
+
+
+

From a0b1d2e5ad209577466409b0732a46e804e95468 Mon Sep 17 00:00:00 2001
From: wdli <wdli@umich.edu>
Date: Sun, 24 Dec 2023 02:36:49 -0500
Subject: [PATCH 3/3] update README.md

---
 README.md | 7 ++++++-
 1 file changed, 6 insertions(+), 1 deletion(-)

diff --git a/README.md b/README.md
index d7cecf09..a680ac1c 100644
--- a/README.md
+++ b/README.md
@@ -106,12 +106,15 @@ $$ x_t = \sqrt{a_{t}a_{t-1}a_{t-2}}\times x_{t-3} +  \sqrt{1-a_{t}a_{t-1}a_{t-2}
 
 $$\bar{a}_{t} := a_{t}a_{t-1}a_{t-2}a_{t-3}...a_{2}a_{1}$$
 
-
+<p>
 $$x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times \epsilon , \epsilon \sim N(0,I) $$
+</p>
 
 $$ \Downarrow  $$
 
+<p>
 $$ q(x_{t}|x_{0}) = \frac{1}{\sqrt{2\pi } \sqrt{1-\bar{a}_{t}}} e^{\left (  -\frac{1}{2}\frac{(x_{t}-\sqrt{\bar{a}_{t}}x_0)^2}{1-\bar{a}_{t}}   \right ) } $$
+</p>
 
 
 # 3.Reverse Process $p$
@@ -316,3 +319,5 @@ Because $x_{t} = \sqrt{\bar{a}_t}\times x_0+ \sqrt{1-\bar{a}_t}\times \epsilon$,
   \right) $$
   </span>
 </p>
+
+**Note:** This `README.md` is intended solely for previewing on the Github page. If you wish to view the rendered page locally, please consult `README.raw.md`.
\ No newline at end of file