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Yes, this is called the posterior predictive distribution. Mathematically, the histogram is approximating the following distribution $$p(\tilde{y} \vert y) = \int p(\tilde{y} \vert \theta) p(\theta \vert y) \, d\theta$$ You'll note the first part of the integrand is the likelihood and the second is the posterior. This integral is integrating over all $\... 4 Once you have the form of the PDF, there are various techniques for sampling. Some easy forms can be handled via Inverse Transform Sampling. Some special forms can be handled via methods special methods, e.g. sampling from normal distribution via Box-Müller. Other general methods exist for PDFs with non-easy/non-special forms (i.e. inverse transform sampling ... 4 For what you described, I cannot see any direct relationship with MCMC. What you needed is just a forward sampling. Here is how it works (suppose we have discrete binary random variables): Step 1. get a sample for$X_1$. In order to do this step, we need to have the distribution$P(X_1)$. (Something like$$P(X_1)=\left\{ \begin{array}... 1 I see no reason why this should be correct in general. The target distribution will then be a convex combination of$p$and of the stationary distribution of$q(\cdot,\cdot)$, if any. If$q(\cdot,\cdot)$is a transient or null recurrent Markov kernel, the chain does not even converge to a distribution. For instance, take$p$as a Uniform$(-1,1)$density ... 3 Usually, you set the tuning parameter once and evaluate the overall acceptance probability ex post once over all iterations of the sampler. Here, you aim for the ''golden acceptance ratio'' of 23,4%. In case your acceptance ratio is higher, this translates into the proposal variance being to small, leading to too many accepts as the posterior distribution is ... 2 So I think if I understand you correctly, what you are trying to do is to sample points from$B$according to the distribution of$A$. In which case I would take a different approach to what you have laid out here. First of all you talked about label encoding$A$and then doing a KDE. I don't think this makes sense unless your categorical variables can ... 0 There is an error in the paper, indeed. I think you state that the paper is wrong with the following "claim": There's really no proof or derivation of the expression. All they did was to plug the definition of$\bar \rho$on the same page into the definition of$r(x,y)$on p.5. Unfortunately, while doing so they messed up. Here's why. Both definitions are ... 2 Suppose$\nu_r$is the invariant measure. The claims made (preceding (5) and (6)) are that if$E_r > E_s$, then the flow from$r$to$s$is$\nu_r P_{rs}$the flow from$s$to$r$is$\nu_s P_{sr} \cdot \exp \left( - \left[ E_r - E_s \right] / kT \right)$In (5) and (6), the reasoning is that if these two terms are not equal, then the system is not at ... 2 There is a lot to address here, and while I appreciate you providing so much detail, I will only focus on some of the most important issues I see. So, the first thing I notice from looking at your model structure is that your model is perhaps misspecified. For one, the beta distribution is defined only for positive$\alpha, \beta$. However, your ... 3 The samples from the posterior allow you to compute expectations of the parameters. From Betancourt's A Conceptual Introduction to Hamiltonian Monte Carlo... Given sufficient time, the history of the Markov chain,$\{q_0,...,q_N\}\$, denoted samples generated by the Markov chain, becomes a convenient quantification of the typical set....