Bayesian online changepoint detection (modeling assumptions in recursive derivation) I am reading Bayesian Online Changepoint Detection (https://arxiv.org/pdf/0710.3742.pdf), and I do not understand one step in the derivation of Equation $3$. For completeness, this is my derivation:
$$
\require{cancel}
\begin{align}
p(r_t, \mathbf{x}_{1:t})
&= \sum_{r_{t-1}} p(r_t, r_{t-1}, \mathbf{x}_{1:t})
\\
&= \sum_{r_{t-1}} p(r_t, x_t \mid r_{t-1}, \mathbf{x}_{1:t-1}) p(r_{t-1}, \mathbf{x}_{1:t-1})
\\
&= \sum_{r_{t-1}} p(x_t \mid r_t, r_{t-1}, \mathbf{x}_{1:t-1}) p(r_t \mid r_{t-1}, \mathbf{x}_{1:t-1}) p(r_{t-1}, \mathbf{x}_{1:t-1})
\\
&= \sum_{r_{t-1}} p(x_t \mid r_{t-1}, \mathbf{x}_{1:t-1}) p(r_t \mid r_{t-1}) p(r_{t-1}, \mathbf{x}_{1:t-1})
\end{align}
$$
The only way this works if is two independence assumptions are made:


*

*$p(r_t \mid r_{t-1}, \mathbf{x}_{1:t-1}) = p(r_t \mid r_{t-1})$

*$p(x_t \mid r_t, r_{t-1}, \mathbf{x}_{1:t-1}) = p(x_t \mid r_{t-1}, \mathbf{x}_{1:t-1})$
The first assumption makes sense. $r_t$ is conditionally independent from the data if we know $r_{t-1}$. In other words, nothing about the data tells us about whether or not a changepoint will or will not occur. This is just prior knowledge we need to encode into our model.
What I don't understand is the second assumption. Why isn't it
$$
p(x_t \mid r_t, r_{t-1}, \mathbf{x}_{1:t-1}) = p(x_t \mid r_t, \mathbf{x}_{1:t-1})\tag{$\star$}
$$
(Condition on $r_t$ rather than $r_{t-1}$.) I have seen a couple resources write Equation $\star$ when explaining BOCD, but the paper is pretty consistent in writing this predictive distribution as conditioned on the previous $r_{t-1}$.
 A: Before I start writing the answer I have to say that I love this topic but I do not like this paper you are referring to: In my opinion it is a particularly bad example of how NOT to do mathematics, statistics and science in general.
How to properly do statistics:


*

*Define the random variables / their densities / the symbols, i.e. all the ingredients in a very precise way.

*Claim things about them, cleanly prove them using rigorous mathematical equations that even a "child in kindergarden" that always keeps on asking 'why is that so?' understands.

*Explain the results in human like words in order to make it easier to follow the weird/complicated equations.


How they do science (at least in the paper referred to above):


*

*Informally describe what you want to model but do not precisly define a single symbol in a formal way.

*Redefine some of the symbols in a way that does not at all match the informal description before and do NOT check mathematically that these two potentially different definitions actually coincide.

*'''Prove''' complicated facts about the relations between the symbols by arguing why it should be true and/or without even giving any proof at all.


What I want to say: We do not have any reason to believe one of these sources (since there is no math in it)!

The first assumption makes sense. $r_t$ is conditionally independent from the data if we know $r_{t-1}$.

I definitely do not believe that statement if I cannot see the maths behind that!

Why isn't it $p(x_t|r_t, r_{t-1}, x_{...}) = p(x_t|r_t,x_{...})$?

Again, we have no reason to believe that either one (the one you state or the one in the paper) is actually true! This is a complicated relation that needs mathematical proof.
I have thought about this for quite a while (a few months ago) and I think I have figured out what they actually want to do, i.e. I have translated the second version of how not to do statistics into the first version here: https://ufile.io/2gdjvx5a. Thm. 10 states that
$$ p(r_t, x_t|r_{t-1}, x_{0:t-1}) = p(r_t|r_{t-1}) p(x_t|\underbrace{r_t}_{\text{still here!!!}}, r_{t-1}, x_{t-1-r_{t-1}:t-1}) $$
while they claim
$$ p(r_t, x_t|r_{t-1}, x_{0:t-1}) = p(r_t|r_{t-1}) p(x_t|\underbrace{ }_{\text{???}} r_{t-1}, x_{t-1-r_{t-1}:t-1}) $$
which (I guess) is simply a mistake in the paper. I tried to contact the authors but I think unfortunately, one of them already passed away and the other one did not reply. So due to the fact that they work in a "non scientific" mode, we will probably never know (we can't even know for sure what they mean with '$R_t$' because they never actually define it in a clean way!)... 
A: I am struggling with the Equation 3. too.
Here is my derivation:
$$
\begin{aligned}
P(r_t, x_{1:t}) &= \sum_{r_{t-1}}P(r_t, r_{t-1}, x_{1:t}) \\
&=\sum_{r_{t-1}}P(r_t, x_t|r_{t-1}, x_{1:t-1})P(r_{t-1},x_{1:t-1}) \\
&\underset{a}{=} \sum_{r_{t-1}}P(r_t|x_t, r_{t-1}, x_{1:t-1})P(x_t|r_{t-1},x_{1:t-1})P(r_{t-1},x_{1:t-1}) \\
&=\sum_{r_{t-1}}P(r_t|r_{t-1},x_{1:t})P(x_t|r_{t-1},x_{1:t-1})P(r_{t-1},x_{1:t-1})
\end{aligned}
$$
Here is the explanation of a:
$$
P(A,B)=P(A|B)P(B) \\
all\ condition\ on\ event\ C,\ D, \\
P(A, B|C, D) = P(A | B, C, D)P(B|C,D)
$$
so:
$$
P(r_t, x_t|r_{t-1}, x_{1:t-1}) = P(r_t | x_t, r_{t-1}, x_{1:t-1})P(x_t|r_{t-1}, x_{1:t-1})
$$
What I don't understand is how come
$$
P(r_t | x_t, r_{t-1}, x_{1:t-1}) \underset{???}{=} P(r_t|r_{t-1})
$$
Updated:
In https://stats.stackexchange.com/a/234981/227553

it is assumed* that whether a changepoint occurs at +1 (between  and +1) does not depend on the history of , we have (+1∣,())=(+1∣)

Then it all makes sense.
