Is it possible to merge acceptance probability with proposal distribution in Metropolis Hastings algorithm? For an ergodic Markov chain, it doesn't necessarily have to be $Detailed\ Balanced $ when it converges to stationary distribution, which means that:
$\pi(\theta)\ P(\theta^{\prime}|\theta) \neq \pi(\theta^{\prime})\ P(\theta|\theta^{\prime})$ ;
In the MH algorithm, however, by multiplying the acceptance rate $\alpha$ for both sides of the equation above, we can get a detailed balanced equation:
$\pi(\theta)\ P(\theta^{\prime}|\theta)\ \alpha(\theta^{\prime}|\theta) = \pi(\theta^{\prime})\ P(\theta|\theta^{\prime})\ \alpha(\theta|\theta^{\prime})$ ;
By forcing one of the $\alpha$ equal to 1, for instance, the one of RHS, we can derive the acceptance probability as:
$\alpha(\theta^{\prime}|\theta) = min\{1, \frac{\pi(\theta^{\prime})\ P(\theta|\theta^{\prime})}{\pi(\theta)\ P(\theta^{\prime}|\theta)}\}$;
So my questions are:


*

*Can I interpret the $\alpha$ as the "transition probability", for the posterior probability distribution, from $P(\theta|x)$ to $P(\theta^{\prime}|x)$?

*Is it possible to join the product of $\ P(\theta^{\prime}|\theta)\ \alpha(\theta^{\prime}|\theta)$ to make them one probability like $Q(\theta^{\prime}|\theta)$. So all we need to compute is $Q(\theta^{\prime}|\theta) =min\{1, \frac{\pi(\theta^{\prime})}{\pi(\theta)}\}$, without worrying about the proposal distribution? Besides, I think $Q(\theta^{\prime}|\theta)$ is more intuitively acceptable as a "transition probability".
 A: I think in the setting of MCMC the term transition probability refers to the probability of transitioning from one point to another, which should be the $P(\theta'|\theta)$ term in your fist equation and the $Q(\theta'|\theta)=P(\theta'|\theta)\alpha(\theta'|\theta)$ term in the second.
The goal of sampling algorithms is to generate samples from a known (unnormalized) distribution. As you said $Q(\theta'|\theta)$ should be easy to compute, however it's usually not easy to be sampled from. That's why we need a proposal distribution (to be easy to sample from) and an acceptance probability (to satisfy the detailed balance).
A: The condition for the probability distribution with density $\pi(\cdot)$ to be stationary for the Markov transition kernel $K$ with density $k(\cdot|\cdot)$ is that, for all measurable sets $A$
$$\pi(A)=\int_A \left\{\int\pi(\theta)k(\theta'|\theta)\text{d}\theta\right\}\text{d}\theta'$$
so indeed detailed balance does not have to hold for $K$ to be a correct transition kernel.
In the Metropolis-Hastings special case, the transition kernel $K$ is decomposed as $$K(\text{d}\theta'|\theta)=\alpha(\theta'|\theta)P(\theta'|\theta)\text{d}\theta'+(1-\varrho(\theta))\delta_{\theta}(\text{d}\theta')$$where $\text{d}\theta$ denotes the dominating measure of the target distribution (Lebesgue, counting &tc.), $\delta_\theta$ denotes the Dirac mass at $\theta$ and $$\varrho(\theta)=\int \alpha(\theta'|\theta)P(\theta'|\theta)\text{d}\theta'$$is the average acceptance probability when the Markov chain is standing at $\theta$.
With regard to your questions,


*

*$\alpha(\theta'|\theta)$ is customarily called the acceptance probability, not the transition probability as the Markov chain does not necessarily move from its current value $\theta$ to the next value $\theta'$, which can thus be rejected. The term transition probability is attached to the kernel $K$;

*Computing the product $\alpha(\theta'|\theta)P(\theta'|\theta)$ may prove handy to decide on the acceptance or rejection of the proposed value $\theta$ but in most situations the proposed value $\theta'$ must be simulated first. In any case, (i) the acceptance probability does not become$$\min\{1,\pi(\theta')/\pi(\theta)\}$$and (ii) the product $\alpha(\theta'|\theta)P(\theta'|\theta)$ is not a probability distribution. 


In a paper with Randal Douc, we devised an approach that relates to this product, called vanilla Rao-Blackwellisation by integrating out the rejection step. That is, we considered only the accepted values and weighted them by their estimated number of replications: since an MCMC estimator of $\mathbb{E}[h(\theta)]$ writes as
$$\hat{h}=\frac{1}{T}\sum_{t=1}^T h(\theta^T)$$
for a Markov chain $(\theta^t)$, it also writes as
$$\hat{h}=\frac{1}{T}\sum_{i=1}^{I_T} n_i h(\xi^i)$$
where the $\xi^i$'s are the distinct values in the sequence $\{\theta^1,\ldots,\theta^T\}$ and the $n_i$'s the numbers of times they are repeated. Since this number $n_i$ is a Geometric rv with probability $\varrho(\xi^i)$, its expectation is $1/\varrho(\xi^i)$, which can be unbiasedly estimated. This step improves the variance of the MCMC estimator, but does not modify running simulations from $P(\cdot|\theta)$.
There also exist other versions of the Metropolis-Hastings algorithm where the decision to reject is taken before the new value $\theta'$ is simulated from $P(\cdot|\theta)$, called delayed acceptance.
