Beyasian changepoint detection notation

Article is about detecting changepoint with bayesian approach. So we have vector with data $$Y = \{y_1, y_2,\ldots, y_n\}$$ and if there are $$k$$ changes in moments $$t_1,\ldots, t_k$$ we split $$Y$$ into $$(y_1,\ldots,y_{t_1-1}), (y_{t_1},\ldots,y_{t_2-1}),\ldots, (y_{t_k},...y_n)$$. And after that my problem begins. In the article we read

In a Bayesian formulation the joint posterior distribution for the latent changepoint indicator vector $$z$$ and segment parameters $$θ = \{θ_1,\ldots,θ_{k+1}\}$$ can be written as a product of the full segment likelihood $$(2.2)$$ and the priors for $$z$$ and $$θ$$, (...)

where $$\theta$$ is a parametr from likelihood estimation.

I'm not a native english speaker so I don't really understand that sentence and so on the next equation

equation from article

that $$\pi$$ function came from nothing (for me) and what does "$$\alpha$$" mean?

Greetings

• It seems that $\pi$ is "the joint posterior distribution for the latent changepoint indicator vector z and segment parameters θ" and $\propto$ means "is proportional to". Commented Dec 26, 2020 at 20:21
• If you want me to explain something further, please tell me. Commented Dec 26, 2020 at 20:28
• Thank you for ur answers. Still i don't really understand how can I find $\pi$ - the equation mentioned in post means $\pi ( z, \theta | y) = (\prod (...))$ right? $f(y_i |\theta_j)$ means density of $y$ until parametr $\theta_j$ fits to $y$ (so until the change occurs) so it should be known (if i have got $y$, and $\theta$ is its mean or variance. But further we have $\pi (\theta_j)$ and $\pi (z)$. How can i get those joint posterior distribution in Bayes approach? Commented Dec 27, 2020 at 18:40