between observations and the Markovian nature of the latent state. We now consider “cloning” the data $y$, i.e. we choose a positive integer $K$, produce $K$ copies of $y$ and stack them in $y^{(K)}=(y,y,...,y)$ where $y$ is replicated $K$ times. Now, we generate $K$ independent vectors for process $\{X_t\}_{t\geq t_0}$ from $g(\cdot)$ at times $\{t_0,...,t_n\}$, say $X^{(1)},X^{(2)},...,X^{(K)}$, all simulated conditionally on the same value of $\theta$. Therefore we set $X^{(k)}=(X_{t_0}^{(k)},...,X_{t_n}^{(k)}\}$ for a generic $k$ ($k=1,...,K$). All vectors in $y^{(K)}$ are assumed to be conditionally independent of each other, given their individual latent state $X^{(k)}$, for example we could imagine a series of $K$ independent experiments leading to exactly the same result. Then the likelihood function for the cloned data $y^{(K)}$ results in $$L(\theta;y^{(K)})=\int \prod_{k=1}^K \{f(y|X^{(k)},\theta)p(X^{(k)}|\theta)\}dX^{(1)}\cdots dX^{(K)}
\label{eq:cloned-likelihood}$$ where we denote with $p(X^{(k)}|\cdot)$ the joint density of vector $X^{(k)}$, hence $p(X^{(k)}|\theta)=\prod_{i=1}^n g(X^{(k)}_i|X^{(k)}_{i-1},\theta)$. Now, for each $k$ a given term in the product in depends on $X^{(k)}$ and not on terms having different indices $k'$ ($k'\neq k$). Therefore is a product of $K$ integrals, each returning the likelihood function based on $y$, and we can write $$L(\theta;y^{(K)})=\prod_{k=1}^K \int f(y|X^{(k)},\theta)p(X^{(k)}|\theta)dX^{(k)} = (L(\theta;y))^K.
\label{eq:cloned-likelihood-bis}$$ Therefore the likelihood function for the cloned data is the 