Given $3$ real variables $x_1, x_2, x_3 \equiv \bf{x}$, consider their probability density function (PDF)
\begin{equation}
P({\bf x}) = C \, p(x_1) \cdots p(x_3) \exp[f({\bf x})],
\end{equation}
where $p(x)$ is a known one-dimensional PDF, $C$ a normalization constant, and $f$ is defined as follows. Consider three binary variables, or 'spins', $S_i = \pm 1, i=1,..,N$, their Hamiltonian
\begin{equation}
H_{\bf x}[{\bf S}] \equiv - (x_1 S_1 S_2 + x_2 S_1 S_3 + x_3 S_2 S_3),
\end{equation}
where the $x$s can be interpreted as the couplings between spin pairs. Let me denote by ${\bf S}^1_{{\bf x}}$ and ${\bf S}^{2}_{{\bf x}}$ the ground state and the first excited state of $H_{\bf x}$, respectively, and set
\begin{equation} f({\bf x}) = (H_{\bf x}[{\bf S}^1_{\bf x}]-H_{\bf x}[{\bf S}^2_{\bf x}])^2. \end{equation}
Do you know an efficient method to draw random samples from $P$, given the particular form above of $P$?
In particular, the form of $P$ above is such that random samples from the factorized distributions $p(x)$ can be easily drawn with inverse transform sampling.
I have tried the two following methods to solve my problem:
- theThe reweighting method, where one draws each variable $x_i$ from $p(x_i)$, and obtains a 'temporary' random sample ${\bf x}^1_{\rm t}$. One repeats $S\gg 1$ times and obtains $S$ temporary samples ${\bf x}^1_{\rm t}, \cdots, {\bf x}^S_{\rm t}$. One introduces the weight of each of these samples $w^s \propto \exp[f({\bf x^s_{\rm t}})]$ and resamples the temporary samples according to their weights, which will be then distributed according to $P$.:
- Consider a 'temporary' random sample ${\bf x}^1_{\rm t}$, where each of the three entries of ${\bf x}^1_{\rm t}$ is drawn independently from $p$.
- Repeat point 1 $S\gg 1$ times and obtain $S$ temporary samples ${\bf x}^1_{\rm t}, \cdots, {\bf x}^S_{\rm t}$.
- Introduce the weight of each of these samples \begin{equation} w^s \equiv \frac{e^{f({\bf x}^s_{\rm t})}}{\sum_{p=1}^S e^{f({\bf x}^p_{\rm t})}} \end{equation}
- Reweighting: draw a random number $r$ uniformly distributed in $[0,1)$, find the value of $s$ such that \begin{equation} \sum_{p=1}^s w^p < r < \sum_{p=1}^{s+1} w^p, \end{equation} and obtain sample ${\bf X}_1 \equiv {\bf x}^s_t$.
- Repeat point 4 $S\gg 1$ times and obtain samples ${\bf X}^1, \cdots, {\bf X}^S$, which are distributed according to $P$.
- the Markov Chain Monte Carlo method.
However, both methods are not efficient for my specific problem.
