The perturbations to the parameter space play several different roles. Firstly, they smooth out the likelihood surface, enabling the algorithm to overcome small-scale features of the likelihood during early stages of the global search. Secondly, Monte Carlo variation allows the search to escape from local minima. Thirdly, the iterated filtering update uses the perturbed parameter values to construct an approximation to the derivative of the log likelihood even though this quantity is not typically available in closed form. Fourthly, the parameter perturbations help to overcome numerical difficulties that can arise during sequential Monte Carlo.
Procedure: Iterated filtering (IF1)
:Input: A partially observed Markov model specified as above; Monte Carlo sample size J; number of iterations M; cooling parameters 0 and b; covariance matrix \Phi; initial parameter vector \theta^{(1)}
:for m^{}{}=1 to M^{}{}
::set X_F(t^{}_0,j)=h\big(\Theta_F(t^{}_0,j)\big) for j=1,\dots, J
::set \bar\theta(t^{}_0)=\theta^{(m)}
::for n^{}_{}=1 to N^{}_{}
:::draw \Theta_P(t^{}_n,j)\sim \mathrm{Normal}(\Theta_F(t^{}_{n-1},j), a^{m-1} \Phi) for j=1,\dots, J
:::set X_P(t^{}_n,j)=f(X_F(t^{}_{n-1},j),t^{}_{n-1},t_n,\Theta_P(t_{n},j),W) for j=1,\dots, J
:::set w(n,j) = g(y_n|X_P(t^{}_n,j),t^{}_n,\Theta_P(t_{n},j)) for j=1,\dots, J
:::draw k^{}_1,\dots,k^{}_J such that P(k^{}_j=i)=w(n,i)\big/{\sum}_\ell w(n,\ell)
:::set X_F(t^{}_n,j)=X_P(t^{}_n,k^{}_j) and \Theta_F(t^{}_n,j)=\Theta_P(t^{}_n,k^{}_j) for j=1,\dots, J
:::set \bar\theta_i^{}(t_n^{}) to the sample mean of \{\Theta_{F,i}^{}(t^{}_{n},j),j=1,\dots,J\}, where the vector \Theta^{}_F has components \{\Theta^{}_{F,i}\}
:::set V_i^{}(t_n^{}) to the sample variance of \{\Theta_{P,i}^{}(t^{}_{n},j),j=1,\dots,J\}
::set \theta_i^{(m+1)}= \theta_i^{(m)}+V_i(t_{1})\sum_{n=1}^N V_i^{-1}(t_{n})(\bar\theta_i(t_n)-\bar\theta_i(t_{n-1}))
:Output: Maximum likelihood estimate \hat\theta=\theta^{(M+1)}
## Variations
1. For IF1, parameters which enter the model only in the specification of the initial condition, X(t_0), warrant some special algorithmic attention since information about them in the data may be concentrated in a small part of the time series.
1. Theoretically, any distribution with the requisite mean and variance could be used in place of the normal distribution. It is standard to use the normal distribution and to reparameterise to remove constraints on the possible values of the parameters.
1. Modifications to the IF1 algorithm have been proposed to give superior asymptotic performance.
## Procedure: Iterated filtering (IF2)
:Input: A partially observed Markov model specified as above; Monte Carlo sample size J; number of iterations M; cooling parameter 0; covariance matrix \Phi; initial parameter vectors \{\Theta_j, j=1,\dots,J\}
:for m^{}_{}=1 to M^{}_{}
::set \Theta_F(t^{}_0,j) \sim \mathrm{Normal}(\Theta_j, a^{m-1} \Phi) for j=1,\dots, J
::set X_F(t^{}_0,j)=h\big(\Theta_F(t^{}_0,j)\big) for j=1,\dots, J
::for n^{}_{}=1 to N^{}_{}
:::draw \Theta_P(t^{}_n,j)\sim \mathrm{Normal}(\Theta_F(t^{}_{n-1},k^{}_j), a^{m-1} \Phi) for j=1,\dots, J
:::set X_P(t^{}_n,j)=f(X_F(t^{}_{n-1},j),t^{}_{n-1},t_n,\Theta_P(t_{n},j),W) for j=1,\dots, J
:::set w(n,j) = g(y_n|X_P(t^{}_n,j),t^{}_n,\Theta_P(t_{n},j)) for j=1,\dots, J
:::draw k^{}_1,\dots,k^{}_J such that P(k^{}_j=i)=w(n,i)\big/{\sum}_\ell w(n,\ell)
:::set X_F(t^{}_n,j)=X_P(t^{}_n,k^{}_j) and \Theta_F(t^{}_n,j)=\Theta_P(t^{}_n,k^{}_j) for j=1,\dots, J
::set \Theta_j=\Theta_F(t^{}_N,j) for j=1,\dots, J
:Output: Parameter vectors approximating the maximum likelihood estimate, \{\Theta_j, j=1,\dots, J \}
## Software
["pomp: statistical inference for observed Markov processes"](https://kingaa.github.io/pomp/) : R package.
## References
## References
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