Understanding the Delayed Rejection Metropolis algorithm (Mira, 2001a) I'm having trouble understanding the algorithm as briefly described here, and I can't find the original paper by Mira since it seems to be from some obscure print journal (Metron Volume 59).
The first stage is typical,
$$\alpha_1(x,y) = \min\left(1,\frac{N_1}{D_1}\right)$$
Where,
$$N_1 = \pi(y)q_1(y,x)$$
$$D_1 = \pi(x)q_1(x,y)$$
$$x = \textrm{current value}$$
$$y = \textrm{proposed value}$$
$$\pi = \textrm{target distribution}$$
$$q_1(x,\cdot) = \textrm{distribution from which y is drawn}$$
The second stage is 
$$\alpha_2(x,y,z) = \min\left(1,\frac{N_2}{D_2}\right)$$
Where
$$N_2 = \pi(z)q_1(z,y)q_2(z,y,x)[1-\alpha_1(z,y)]$$
$$D_2 = \pi(x)q_1(x,y)q_2(x,y,z)[1-\alpha_1(x,y)]$$
Thanks to @Xi'an, I realized that I have to pay attention to the distributions $q_1$ and $q_2$ even if I assume the proposals are symmetric, because $q_1(z,y)$ is likely not equal to $q_1(x,y)$. The second thing that I realized is that "the second stage candidate is computed so that reversibility of the Markov Chain...is preserved", so I still don't get the intuition  but I understand how this form came about.
Questions:
If somebody has some intuition I would appreciate it. I'm playing around with toy distributions now, things like what if $z = x$ and what if $z = x\pm \epsilon$.
Also is $q_2(z,y,x)$ also not necessarily equal to $q_2(x,y,z)$ if we assume $q_1 \neq q_2$? The paper uses a smaller covariance for the second stage proposal. Is it like $q_1(y|z)q_2(x|y)$ vs. $q_1(y|x)q_2(z|y)$?
 A: The paper is available from ResearchGate through Google.
The validation of the delayed rejection algorithm is that, when starting with a realisation of the variable $X_t\sim\pi(x)$, the outcome of the Markov move to $X_{t+1}$ still remains distributed as $X_{t+1}\sim\pi(x)$. If the first step leads to an acceptance, the validation is the same as with the classical Metropolis-Hastings algorithm, hence using the same acceptance probability, $\alpha_1(x_t,\cdot)$ and the accepted value is distributed as $$q_1(x_t,y)\alpha(x_t,y)$$ If the value $Y$ at the first stage is rejected, it is distributed as $$q_1(x_t,y)[1-\alpha(x_t,y)]$$up to a normalising constant. The second stage value is then generated from the proposal $q_2(x_t,y,z)$ which means that $Z$ is generated conditional on $(x_t,y)$ [and hence that a better notation would be $q_2(z|x_t,y)$]. In this second stage, $y$ becomes an auxiliary variable, with the joint distribution of $(Y,Z)$ given $x_t$ and a first rejection being$$q_1(x_t,y)[1-\alpha(x_t,y)]q_2(x_t,y,z)$$which explains why this block appears in $\alpha_2(x_t,y,z)$. Another way to explain the representation of $\alpha_2(\cdot,\cdot,\cdot)$ is that the marginal distribution of $Z$ is (proportional to)$$\int_\mathcal{X} q_1(x_t,y)[1-\alpha(x_t,y)]q_2(x_t,y,z)\,\text{d}y$$ and that an alternative acceptance probability would be
\begin{align*}\alpha_2^\prime(x_t,z)=\min\bigg\{1,\,&\pi(z)\int_\mathcal{X} q_1(z,y)[1-\alpha(z,y)]q_2(z,y,x_t)\,\text{d}y\\&\left.\Big/\pi(x_t)\int_\mathcal{X} q_1(x_t,y)[1-\alpha(x_t,y)]q_2(x_t,y,z)\,\text{d}y\right\}\end{align*}
But this probability cannot be computed in most cases [except when $q_2(x_t,y,z)=q_2(x_t,z)$ and hence is replaced by a valid ratio of unbiased estimators of the integrals as in Andrieu & Roberts (2009) pseudo-marginal technique.
This is for the "intuition" part, but the validation proceeds by detailed balance, namely that the distribution of $X_{t+1}$ given $X_t=x_t$ writes as
$$q(x_t,x)\alpha_1(x_t,x)q_1(x_t,x)+\int_\mathcal{X} \alpha_2(x_t,y,x)\,q_1(x_t,y)[1-\alpha(x_t,y)]q_2(x_t,y,x)\,\text{d}y+\int_\mathcal{X^2} \alpha_2(x_t,y,x)\,q_1(x_t,y)[1-\alpha(x_t,y)]q_2(x_t,y,z)\,\text{d}y\,\text{d}z\,\delta_{x_t}(x)$$and that this density satisfies 
$$\pi(x_t)q(x_t,x)=\pi(x)q(x,x_t)$$
