Computing the Hastings ratio for multinomial distribution as a proposal distribution in Metropolis-Hastings accept-reject step I have a question concerning calculating the Hastings ratio in a specific case (multinomial proposal distribution).
I consider a discrete vector $M$ with integer values that sum up to some number $N$. In each step of the Metropolis-Hastings algorithm I need to propose a new value for $N$ and a new value of $M$.
Let's denote $q(x \mid y)$ as the proposal distribution, i.e., the conditional probability of proposing a new state $x$ given the previous value was equal to $y$.
We propose a new value $N^*$ according to:
$$
q_1\left(N^*\mid N^{\textrm{prev}} \right) \stackrel{d}{=}  \lfloor T \rfloor, \textrm{ where }
T \sim tNorm(N^{\textrm{prev}}, \sigma).
$$
For proposing a new value of $M$, we introduce:
$$
q_2\left( M^* \mid   N^{*}, M^{\textrm{prev}} \right) \sim Multinomial \left( N^*,\frac{ M^{\textrm{prev}} + s}{ \sum \limits_{i}  M^{\textrm{prev},\;i} + s} \right)
$$
where $s$ is a realisation of a random variable
$$
S \sim N(0, \sigma)
$$
and introduces an additional noise.
To perform a Metropolis-Hastings accept-reject step, we need to calculate the acceptance probability:
$$
r = \min \left(1, \frac{f\left(  x^*  \right) q\left( x^{\textrm{prev}} \mid x^* \right)}{f\left( x^{\textrm{prev}}\right) q\left( x^* \mid x^{\textrm{prev}}\right)} \right)
$$
where $x = [M,N]$.
$$ 
 \frac{q\left( x^{\textrm{prev}} \mid x^* \right)}{q\left( x^* \mid x^{\textrm{prev}}\right)} =
\frac{q_1( N^{\textrm{prev}}\mid N^{*} ) \;\; q_2\left( M^{\textrm{prev}}\mid  N^{\textrm{prev}}, \frac{ M^{*} + s}{ \sum \limits_{i}  (M^{*,i} + s)} \right)}{ q_1(  N^{*}\mid  N^{\textrm{prev}}  ) \;\; q_2 \left( M^{*}\mid  N^{*}, \frac{ M^{\textrm{prev}} + s}{ \sum \limits_{i}  (M^{\textrm{prev},i} + s)} \right) } =
$$
$$
= \frac{\text{dabsNorm}(N^{\textrm{prev}} \mid N^*, \sigma) }{\text{dabsNorm}(N^* \mid N^{\textrm{prev}}, \sigma)}
\frac{\text{dmultinom} \left( M^{\textrm{prev}}\mid  N^{\textrm{prev}}, \frac{ M^{*} + s}{ \sum \limits_{i}  (M^{*,t} + s)}  \right)}
{\text{dmultinom} \left( M^* \mid  N^*, \frac{ M^{\textrm{prev}} + s}{ \sum \limits_i  (M^{\textrm{prev},i} + s)}\right)},
$$
where $\text{dabsNorm}$ denotes density of absolute value of the floor of a random variable following the normal distribution and $\text{dmultinom}$ density of a random variable following the multinomial distribution.
Is the above calculation correct?~
Edit 1: (based on the first comment) I was missing randomness coming from $S$. Multinomial distribution should be integrated with respect to s, or at least $s^*$ and $s^\textrm{prev}$ should appear:
$$ 
 \frac{q\left( x^{\textrm{prev}} \mid x^* \right)}{q\left( x^* \mid x^{\textrm{prev}}\right)} =
\frac{q_1( N^{\textrm{prev}}\mid N^{*} ) \;\; q_2\left( M^{\textrm{prev}}\mid  N^{\textrm{prev}}, \frac{ M^{*} + s^*}{ \sum \limits_{i}  (M^{*,i} + s^*)} \right)}{ q_1(  N^{*}\mid  N^{\textrm{prev}}  ) \;\; q_2 \left( M^{*}\mid  N^{*}, \frac{ M^{\textrm{prev}} + s^{\textrm{prev}}}{ \sum \limits_{i}  (M^{\textrm{prev},i} + s^{\textrm{prev}})} \right) } =
$$
$$
= \frac{\text{dabsNorm}(N^{\textrm{prev}} \mid N^*, \sigma) }{\text{dabsNorm}(N^* \mid N^{\textrm{prev}}, \sigma)}
\frac{\text{dmultinom} \left( M^{\textrm{prev}}\mid  N^{\textrm{prev}}, \frac{ M^{*} + s^*}{ \sum \limits_{i}  (M^{*,t} + s^*)}  \right)}
{\text{dmultinom} \left( M^* \mid  N^*, \frac{ M^{\textrm{prev}} + s^{\textrm{prev}}}{ \sum \limits_i  (M^{\textrm{prev},i} + s^{\textrm{prev}})}\right)}.
$$
 A: The ratio of proposal densities$$\dfrac{\text{dabsNorm}(N^{\textrm{prev}} \mid N^*, \sigma) }{\text{dabsNorm}(N^* \mid N^{\textrm{prev}}, \sigma)}\times
\dfrac{\text{dmultinom} \left( M^{\textrm{prev}}\mid  N^{\textrm{prev}}, \frac{ M^{*} + s^*}{ \sum \limits_{i}  (M^{*,t} + s^*)}  \right)}
{\text{dmultinom} \left( M^* \mid  N^*, \frac{ M^{\textrm{prev}} + s^{\textrm{prev}}}{ \sum \limits_i  (M^{\textrm{prev},i} + s^{\textrm{prev}})}\right)}$$should be
$$\dfrac{\text{dabsNorm}(N^{\textrm{prev}} \mid N^*, \sigma) }{\text{dabsNorm}(N^* \mid N^{\textrm{prev}}, \sigma)}\times
\dfrac{\text{dmultinom} \left( M^{\textrm{prev}}\mid  N^{\textrm{prev}}, \frac{ M^{*} + s^{\textrm{prev}}}{ \sum \limits_{i}  (M^{*,t} + s^{\textrm{prev}})}  \right)}
{\text{dmultinom} \left( M^* \mid  N^*, \frac{ M^{\textrm{prev}} + s^{*}}{ \sum \limits_i  (M^{\textrm{prev},i} +s^{*} )}\right)}$$
and is justified as being a proper Metropolis-Hastings ratio by including the $s$'s within the Markov chain, that is, by setting
$$x=(N,M,s)$$
and by creating a pseudo-target on $s$ that is the
$$S∼N(0,σ)$$
distribution used in the proposal. (Hence its density cancels out in the overall Metropolis-Hastings ratio, being found in both target and proposal.)
