Difference between revisions of "Estimation of the log-likelihood"

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<div style="font-size:12pt; font-family:garamond">
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== Estimation using importance sampling ==
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\def\tqphii{\tilde{p}_{\phi_i}}
 
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The observed log-likelihood ${\llike}(\theta;\by)=\log({\like}(\theta;\by))$ can be estimated without requiring approximation of the model, using a Monte Carlo approach.
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\def\qypsith{p_{y,\psi,\theta}}
 
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\def\qypsithc{p_{y,\psi,\theta,c}}
 
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\def\qcypsiut{p_{y|\psi,u,t}}
 
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\def\qcpsithc{p_{\psi|\theta,c}}
 
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\def\qyth{p_{y,\theta}}
 
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\def\qcpsiiyi{p_{\psi_i|y_i}}
 
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\def\qz{p_z}
 
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\def\qw{p_w}
 
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\def\qcwz{p_{w|z}}
 
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\def\qw{p_w}
 
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\def\qcyipsii{p_{y_i|\psi_i}}
 
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\def\qyipsii{p_{y_i,\psi_i}}
 
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\def\qypsiij{p_{y_{ij}|\psi_{i}}}
 
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\def\qyipsi1{p_{y_{i1}|\psi_{i}}}
 
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\def\qtypsiij{p_{\transy(y_{ij})|\psi_{i}}}
 
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\def\qcyzipsii{p_{z_i,y_i|\psi_i}}
 
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\def\qczipsii{p_{z_i|\psi_i}}
 
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\def\qcyizpsii{p_{y_i|z_i,\psi_i}}
 
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\def\qcyijzpsii{p_{y_{ij}|z_{ij},\psi_i}}
 
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\def\qcyi1zpsii{p_{y_{i1}|z_{i1},\psi_i}}
 
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\def\qcypsiz{p_{y,\psi|z}}
 
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\def\qccypsiz{p_{y|\psi,z}}
 
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\def\qypsiz{p_{y,\psi,z}}
 
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\def\qcpsiz{p_{\psi|z}}
 
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\def\qeps{p_{\teps}}
 
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\def\monolix{\text{Monolix}}
 
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$
 
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−
The observed log-likelihood ${\llike}(\theta;\by)=\log({\like}(\theta;\by))$ can be estimated without any approximation of the model using a Monte-Carlo approach.
 
  
 
Since
 
Since
  
 
{{Equation1
 
{{Equation1
−
|equation=<math> \begin{eqnarray}
+
|equation=<math>\begin{eqnarray}
 
{\llike}(\theta;\by) &\eqdef& \log(\py(\by;\theta)) \\
 
{\llike}(\theta;\by) &\eqdef& \log(\py(\by;\theta)) \\
 
&=& \sum_{i=1}^{N} \log(\pyi(y_i;\theta)),
 
&=& \sum_{i=1}^{N} \log(\pyi(y_i;\theta)),
 
\end{eqnarray}</math> }}
 
\end{eqnarray}</math> }}
  
−
we will estimate the log-pdf $\log(\pyi(y_i;\theta))$ for each individual and derive an estimate of the log-likelihood as the sum of these individual log-likelihoods. We will  explain here how to estimate the log-pdf $\log(\pyi(y_i;\theta))$ for any individual $i$.
+
we can estimate $\log(\pyi(y_i;\theta))$ for each individual and derive an estimate of the log-likelihood as the sum of these individual log-likelihoods. We will now explain how to estimate $\log(\pyi(y_i;\theta))$ for any individual $i$.
  
−
Using the $\phi$-representation of the model, notice first that $\pyi(y_i;\theta)$ can be decomposed as follows
+
Using the $\phi$-representation of the model, notice first that $\pyi(y_i;\theta)$ can be decomposed as follows:
  
 
{{Equation1
 
{{Equation1
Line 285: Line 19:
 
\pyi(y_i;\theta) &=&
 
\pyi(y_i;\theta) &=&
 
\displaystyle{ \int \pyipsii(y_i,\phi_i;\theta)\,d\phi_i }\\
 
\displaystyle{ \int \pyipsii(y_i,\phi_i;\theta)\,d\phi_i }\\
−
&=& \displaystyle{ \int \pcyiphii(y_i {{!}} \phi_i;\theta)\pphii(\phi_i;\theta)\,d\phi_i }\\
+
&=& \displaystyle{\int \pcyiphii(y_i {{!}} \phi_i;\theta)\pphii(\phi_i;\theta)\,d\phi_i } \\
 
&=& \esps{\qphii}{\pcyiphii(y_i {{!}} \phi_i;\theta)}.
 
&=& \esps{\qphii}{\pcyiphii(y_i {{!}} \phi_i;\theta)}.
 
\end{eqnarray}</math> }}
 
\end{eqnarray}</math> }}
  
−
$\pyi(y_i;\theta)$ is expressed as a mean. It can therefore be approximated by an empirical mean using a Monte-Carlo procedure:
+
Thus, $\pyi(y_i;\theta)$ is expressed as a mean. It can therefore be approximated by an empirical mean using a Monte Carlo procedure:
  
  
 
<blockquote>
 
<blockquote>
−
1. draw $M$ independent realizations $\phi_i^{(1)}$, $\phi_i^{(2)}$, ..., $\phi_i^{(M)}$ of the normal distribution $\qphii(\, \cdot \, ; \theta)$,
+
1. Draw $M$ independent values $\phi_i^{(1)}$, $\phi_i^{(2)}$, ..., $\phi_i^{(M)}$ from the normal distribution $\qphii(\, \cdot \, ; \theta)$.
 
</blockquote>
 
</blockquote>
 
<blockquote>
 
<blockquote>
−
2. estimate $ \pyi(y_i;\theta)$ with
+
2. Estimate $ \pyi(y_i;\theta)$ with
 +
</blockquote>
  
 
{{Equation1
 
{{Equation1
 
|equation=<math>
 
|equation=<math>
−
\hat{p}_{i,M} = \displaystyle{ \frac{1}{M} } \sum_{m=1}^{M}\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta)
+
\hat{p}_{i,M} = \displaystyle{\frac{1}{M} }\sum_{m=1}^{M}\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) .
 
</math> }}
 
</math> }}
−
</blockquote>
 
  
  
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{{Equation1
 
{{Equation1
 
|equation=<math>\begin{eqnarray}
 
|equation=<math>\begin{eqnarray}
−
\esp{\hat{p}_{i,M}}&=& \esps{\qphii}{\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta)} \\
+
\esp{\hat{p}_{i,M} }&=& \esps{\qphii}{\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta)} \\
−
&=& \pyi(y_i;\theta)
+
&=& \pyi(y_i;\theta) .
 
\end{eqnarray}</math> }}
 
\end{eqnarray}</math> }}
  
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{{Equation1
 
{{Equation1
 
|equation=<math>
 
|equation=<math>
−
\var{\hat{p}_{i,M} }= \displaystyle{\frac{1}{M} }\vars{\qphii}{\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta)}.
+
\var{\hat{p}_{i,M} }= \displaystyle{ \frac{1}{M} }\vars{\qphii}{\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta)}.
 
</math> }}
 
</math> }}
  
−
We could be satisfied with this estimator, since we "only" have to select $M$ large enough to get an estimator with a small variance.
+
We could consider ourselves satisfied with this estimator since we "only" have to select $M$ large enough to get an estimator with a small variance. Nevertheless, we will see now that it is possible to improve the statistical properties of this estimator.
−
Nevertheless, we will see now that it is possible to improve the statistical properties of this estimator.
 
  
−
For any distribution $\tqphii$ absolutely continuous with respect to the marginal distribution $\qphii$, we can write
+
For any distribution $\tqphii$ that is absolutely continuous with respect to the marginal distribution $\qphii$, we can write
  
 
{{Equation1
 
{{Equation1
 
|equation=<math>\begin{eqnarray}
 
|equation=<math>\begin{eqnarray}
 +
\pyi(y_i;\theta) &=&
 
\displaystyle{ \int \pyiphii(y_i,\phi_i;\theta)\,d\phi_i }\\
 
\displaystyle{ \int \pyiphii(y_i,\phi_i;\theta)\,d\phi_i }\\
−
&=& \displaystyle{ \int \pcyiphii(y_i {{!}} \phi_i;\theta)\frac{\pphii(\phi_i;\theta)}{\tpphii(\phi_i;\theta)}\tpphii(\phi_i;\theta)\,d\phi_i }\\
+
&=& \displaystyle{\int \pcyiphii(y_i {{!}} \phi_i;\theta)\frac{\pphii(\phi_i;\theta)}{\tpphii(\phi_i;\theta)}\tpphii(\phi_i;\theta)\,d\phi_i } \\
 
&=& \esps{\tqphii}{\pcyiphii(y_i {{!}} \phi_i;\theta)\displaystyle{\frac{\pphii(\phi_i;\theta)}{\tpphii(\phi_i;\theta)} } }.
 
&=& \esps{\tqphii}{\pcyiphii(y_i {{!}} \phi_i;\theta)\displaystyle{\frac{\pphii(\phi_i;\theta)}{\tpphii(\phi_i;\theta)} } }.
 
\end{eqnarray}</math> }}
 
\end{eqnarray}</math> }}
  
−
We can now approximate $\pyi(y_i;\theta)$ via an ''Importance Sampling'' integration method using $\tqphii$ as a proposal distribution:
+
We can now approximate $\pyi(y_i;\theta)$ via an ''importance sampling'' integration method using $\tqphii$ as a proposal distribution:
  
  
 
<blockquote>
 
<blockquote>
−
1. draw $M$ independent realizations $\phi_i^{(1)}$, $\phi_i^{(2)}$, ..., $\phi_i^{(M)}$ of the proposal distribution $\tqphii(\, \cdot \, ; \theta)$,
+
1. Draw $M$ independent values $\phi_i^{(1)}$, $\phi_i^{(2)}$, ..., $\phi_i^{(M)}$ from the proposal distribution $\tqphii(\, \cdot \, ; \theta)$.
−
 
+
</blockquote>
−
2. estimate $ \pyi(y_i;\theta)$ with
+
<blockquote>
 +
2. Estimate $ \pyi(y_i;\theta)$ with
 +
</blockquote>
  
 
{{Equation1
 
{{Equation1
 
|equation=<math>
 
|equation=<math>
−
\hat{p}_{i,M} = \displaystyle{ \frac{1}{M} }\sum_{m=1}^{M}\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} }
+
\hat{p}_{i,M} = \displaystyle{\frac{1}{M} }\sum_{m=1}^{M}\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} }.
 
</math> }}
 
</math> }}
−
</blockquote>
 
  
  
−
By construction, this new estimator is also unbiased and its variance decreases also as $1/M$:
+
By construction, this new estimator is also unbiased and its variance also decreases as $1/M$:
  
 
{{Equation1
 
{{Equation1
 
|equation=<math>
 
|equation=<math>
−
\var{\hat{p}_{i,M} }= \displaystyle{\frac{1}{M} }\vars{\tqphii}{\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta)\displaystyle{ \frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} } }.
+
\var{\hat{p}_{i,M} }= \displaystyle{\frac{1}{M} }\vars{\tqphii}{\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} } }.
 
</math> }}
 
</math> }}
  
−
There exists an infinite number of possible proposal distributions $\tpphii$ which all provide the same rate of convergence $1/M$.  
+
There exists an infinite number of possible proposal distributions $\tpphii$ which all provide the same rate of convergence $1/M$.
−
The trick is to reduce the variance of the estimator by selecting a proposal such that the numerator is as small as possible.
+
The trick is to reduce the variance of the estimator by selecting a proposal distribution  so that the numerator is as small as possible.
  
−
Imagine that we use the conditional distribution $\qcphiiyi$ as proposal. Then,  for any $m=1,2,\ldots,M$,
+
Imagine that we use the conditional distribution $\qcphiiyi$ as the proposal. Then,  for any $m=1,2,\ldots,M$,
  
 
{{Equation1
 
{{Equation1
 
|equation=<math>\begin{eqnarray}
 
|equation=<math>\begin{eqnarray}
−
\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) \displaystyle{ \frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} }
+
\pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} }
−
&=& \pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) \displaystyle{ \frac{\pphii(\phi_i^{(m)};\theta)}{\pcphiiyi(\phi_i^{(m)} {{!}} y_i;\theta)} } \\
+
&=& \pcyiphii(y_i {{!}} \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\pcphiiyi(\phi_i^{(m)} {{!}} y_i;\theta)} }\\
−
&=& \pyi(y_i;\theta)
+
&=& \pyi(y_i;\theta),
 
\end{eqnarray}</math> }}
 
\end{eqnarray}</math> }}
  
−
which means that $\hat{p}_{i,M}=\pyi(y_i;\theta)$! Such an estimator is optimal since its variance is null and only one realization of $\qcphiiyi$ suffices for computing exactly $\pyi(y_i;\theta)$. The problem is that it is not possible to generate the $\phi_i^{(m)}$'s with this conditional distribution, since that would require to compute a normalizing constant ... which is precisely $\pyi(y_i;\theta)$.
+
which means that $\hat{p}_{i,M}=\pyi(y_i;\theta)$! Such an estimator is optimal since its variance is null and only one realization of $\qcphiiyi$ is required to exactly compute $\pyi(y_i;\theta)$. The problem is that it is not possible to generate the $\phi_i^{(m)}$ with this conditional distribution, since that would require to compute a normalizing constant, which here is precisely $\pyi(y_i;\theta)$.
  
−
Nevertheless, this conditional distribution can be estimated using the Metropolis-Hastings algorithm described in [[The Metropolis-Hastings algorithm for simulating the individual parameters|The Metropolis-Hastings algorithm]] section and a practical proposal "close" to the optimal proposal $\qcphiiyi$ can be derived. We can then expect to get a very accurate estimation with a relatively small Monte-Carlo size $M$.
+
Nevertheless, this conditional distribution can be estimated using the Metropolis-Hastings algorithm described in [[The Metropolis-Hastings algorithm for simulating the individual parameters]] section and a practical proposal "close" to the optimal proposal $\qcphiiyi$ can be derived. We can then expect to get a very accurate estimate with a relatively small Monte Carlo size $M$.
  
−
In $\monolix$, the  mean  and variance  of the conditional distribution $\qcphiiyi$ are estimated by M-H for each individual $i$. Then, the $\phi_i^{(m)}$'s are drawn with a non-central $t$ distribution:
+
In $\monolix$, the  mean  and variance  of the conditional distribution $\qcphiiyi$ are estimated by Metropolis-Hastings for each individual $i$. Then, the $\phi_i^{(m)}$ are drawn with a noncentral $t$ distribution:
  
 
{{Equation1
 
{{Equation1
−
|equation=<math>\phi_i^{(m)} = \mu_i + \sigma_i \times T_{i,m} </math> }}
+
|equation=<math>
 +
\phi_i^{(m)} = \mu_i + \sigma_i \times T_{i,m} ,
 +
</math> }}
  
−
where $\mu_i$ and $\sigma^2_i$ are estimates of $\esp{\phi_i|y_i;\theta}$ and $\var{\phi_i|y_i;\theta}$, and where $(T_{i,m})$ is a sequence of ''i.i.d.'' random variables distributed with a Student'$t-$distribution with $\nu$ degrees of freedom.  
+
where $\mu_i$ and $\sigma^2_i$ are estimates of $\esp{\phi_i|y_i;\theta}$ and $\var{\phi_i|y_i;\theta}$, and $(T_{i,m})$ is a sequence of i.i.d. random variables distributed with a Student's $t$ distribution with $\nu$ degrees of freedom.  
  
−
$\monolix$  uses the default value $\nu=5$. It is possible to use the default value $\nu=5$. It is also possible to automatically test different d.f in $\{2, 5, 10, 20\}$ and to select the one that provides the smallest empirical variance for $\widehat{ {\llike} }_M(\theta;\by) = \sum_{i=1}^{N}\log(\hat{p}_{i,M})$.
+
$\monolix$  uses the default value $\nu=5$. It is also possible to automatically test different degrees of freedom from the set $\{2, 5, 10, 20\}$ and to select the one that provides the smallest empirical variance for $\widehat{ {\llike} }_M(\theta;\by) = \sum_{i=1}^{N}\log(\hat{p}_{i,M})$.
  
  
 
{{Remarks
 
{{Remarks
−
|title=Remark
+
|title=Remarks
−
|text= Even if $\widehat{\like}_M(\theta;\by) = \prod_{i=1}^{N}\hat{p}_{i,M}$ is an unbiased estimator of ${\like}(\theta;\by)$,
+
|text=Even if $\widehat{ {\like} }_M(\theta;\by) = \prod_{i=1}^{N}\hat{p}_{i,M}$ is an unbiased estimator of ${\like}(\theta;\by)$, $\widehat{ {\llike} }_M(\theta;\by)$ is a biased estimator of $\log({\like}(\theta;\by))$. Indeed, by  Jensen's inequality, we have that
−
$\widehat{ {\llike} }_M(\theta;\by)$ is a biased estimator of $\log({\like}(\theta;\by))$. Indeed, by  Jensen's inequality, we have that
 
  
 
{{Equation1
 
{{Equation1
−
|equation=<math> \esp{\log(\widehat{ {\like} }_M(\theta;\by))} \leq \log \esp{\widehat{ {\like} }_M(\theta;\by)} = \log ({\like}(\theta;\by)). </math> }}
+
|equation=<math>
 +
\esp{\log(\widehat{ {\like} }_M(\theta;\by))} \leq \log \esp{\widehat{ {\like} }_M(\theta;\by)} = \log ( {\like}(\theta;\by)).
 +
</math> }}
  
−
But, ''i)'' the bias decreases as $M$ increases, ''ii)'' the bias is reduced if $\widehat{ {\like} }_M(\theta;\by)$ is close to ${\like}(\theta;\by)$. It is therefore highly recommended to use a proposal as close as possible to the conditional distribution $\qcphiiyi$, which means to estimate this conditional distribution before estimating the log-likelihood.
+
However, the bias decreases as $M$ increases, and also if $\widehat{ {\like} }_M(\theta;\by)$ is close to ${\like}(\theta;\by)$. It is therefore highly recommended to use a proposal as close as possible to the conditional distribution $\qcphiiyi$, which means having to estimate this conditional distribution before estimating the log-likelihood.
 
}}
 
}}
  
Line 394: Line 132:
 
{{Example
 
{{Example
 
|title=Example
 
|title=Example
−
|text=  
+
|text= A mixture model is used for this example. The goal here is not to provide details about the model and the formulas used for implementation. We only want to highlight  the importance of  the choice of the proposal distribution for estimating the likelihood in complex models. The three figures below display estimates of the deviance ($-2\times {\llike} (\theta;\by)$) by importance sampling using different proposal distributions. The estimated deviance is displayed as a function of the Monte Carlo size.
−
A mixture model is used for this example. We don't aim here to provide the details about the model and the formulas used for implementing the algorithm. We use this  example to highlight  the importance of  the choice of the proposal distribution for estimating the likelihood in complex models.
+
 
 +
In the first example, the conditional distributions of the individual parameters $\phi_i$ were estimated using the [[The Metropolis-Hastings algorithm for simulating the individual parameters|Metropolis-Hastings algorithm]], and noncentral $t$ distributions with 5 d.f. were used as proposal distributions. There is no bias and an accurate estimate is obtained with a small Monte Carlo size. The estimated deviance is 14386.8 (s.e. = 0.7). Here, only 100 Metropolis-Hastings iterations were required to correctly estimate the conditional mean and variance of the $\psi_i$.
  
−
The three graphics below display the estimation of the deviance ($-2\times {\llike}(\theta;\by)$) by Importance Sampling using different proposal distributions. The estimated deviance is displayed as a function of the Monte-Carlo size.
 
  
−
In this first example, the conditional distributions of the individual parameters $\phi_i$ were estimated using the M-H algorithm and non-central $t$ distributions with 5 $d.f.$ we used as proposal distributions. There is no bias and an accurate estimation is obtained with a small Monte-Carlo size. The estimated deviance is 14386.8 (s.e. = 0.7). Here, only 100 iterations of the M-H were enough for correctly estimating the conditional means and variances of the $\psi_i$.
+
{{ImageWithCaption|image=ll2.png|caption= }}
  
−
{{ImageWithCaption|image=ll2.png|size=700px|caption= }}
 
  
 +
In the second example, only the last 30 iterations of [[The SAEM algorithm for estimating population parameters|SAEM]] were used for estimating the conditional mean and variance of the $\phi_i$. Then, noncentral $t$ distributions with 5 d.f. we used as proposal distributions. Now there is bias which decreases very slowly with the Monte Carlo size and which is non-negligible even with $10^5$ simulations. The estimated deviance is 14397.0 (s.e. = 2.7).
  
−
In this second example, only the last 30 iterations of SAEM were used for estimating the conditional mean and variances of the $\phi_i$. Then, non-central $t$ distributions with 5 $d.f.$ we used as proposal distributions. There is a bias which decreases very slowly with the Monte-Carlo size and which is not negligible even with $10^5$ simulations: the estimated deviance is 14397.0 (s.e. = 2.7).
 
  
 +
{{ImageWithCaption|image=ll1.png|caption= }}
  
−
{{ImageWithCaption|image=ll1.png|size=700px|caption= }}
 
  
−
Normal distributions were used in this example as proposals. The parameters of these distributions  were the  means and variances of the conditional distributions of the $\phi_i$  estimated using the M-H algorithm. Results are similar to those obtained with a $t$ distribution(estimated deviance=14386.9, s.e. = 1) but more simulations are required for eliminating the bias.  
+
In the third example, normal distributions were used as proposals. The parameters of these distributions  were the  mean and variance of the conditional distributions of the $\phi_i$  estimated using Metropolis-Hastings. Results are similar to those obtained with a $t$ distribution (estimated deviance=14386.9, s.e. = 1) but here, more simulations are required to eliminate the bias.  
  
  
−
{{ImageWithCaption|image=ll3.png|size=700px|caption= }}
+
{{ImageWithCaption|image=ll3.png|caption= }}
 
}}
 
}}
  
  
 
<br>
 
<br>
−
==Estimation of the log-likelihood via linearization==
 
  
 +
== Estimation using linearization ==
 +
 +
For continuous data models, an alternative to the importance sampling approach is to use a linearization of the model like that proposed in the [[Estimation of the observed Fisher information matrix#Estimation using linearization of the model|Estimation of the F.I.M. using a linearization of the model]] chapter to approximate the observed Fisher Information Matrix. Indeed, the marginal distribution of a continuous vector of observations $y_i$ can be approximated by a normal distribution. It is then straightforward to derive the associated likelihood. All of these calculations are described in the [[Estimation of the observed Fisher information matrix#Estimation using linearization of the model|Estimation of the F.I.M. using a linearization of the model]] chapter.
  
−
For continuous data models, an alternative to the Importance Sampling approach is to use a linearization of the model, following what was proposed in [[Estimation of the observed Fisher information matrix#Estimation of the F.I.M. using a linearization of the model|Estimation of the F.I.M. using a linearization of the model]] section for approximating the observed Fisher Information Matrix. Indeed, the marginal distribution of a continuous vector of observations $y_i$ can be approximated by a normal distribution. It is then straightforward to derive the associated likelihood. All the calculations are detailed in [[#|????]] Section.
+
This method can be much faster than importance sampling. It should be used by  modelers for model selection purposes  during the initial runs, when the goal is to identify significant differences between models. Importance sampling should be used when a more precise evaluation of the log-likelihood is required.  
  
−
This method can be much faster than Importance Sampling. It should be used by the modeler for model selection purpose  during the first runs,  when the goal is to identify significant differences between models. Importance Sampling should be used when a more precise evaluation of the log-likelihood is required.
 
  
−
</div>
+
{{Back
 +
|link=Estimation of the observed Fisher information matrix }}

Latest revision as of 14:43, 3 June 2013

Estimation using importance sampling

The observed log-likelihood ${\llike}(\theta;\by)=\log({\like}(\theta;\by))$ can be estimated without requiring approximation of the model, using a Monte Carlo approach.

Since

\(\begin{eqnarray} {\llike}(\theta;\by) &\eqdef& \log(\py(\by;\theta)) \\ &=& \sum_{i=1}^{N} \log(\pyi(y_i;\theta)), \end{eqnarray}\)

we can estimate $\log(\pyi(y_i;\theta))$ for each individual and derive an estimate of the log-likelihood as the sum of these individual log-likelihoods. We will now explain how to estimate $\log(\pyi(y_i;\theta))$ for any individual $i$.

Using the $\phi$-representation of the model, notice first that $\pyi(y_i;\theta)$ can be decomposed as follows:

\(\begin{eqnarray} \pyi(y_i;\theta) &=& \displaystyle{ \int \pyipsii(y_i,\phi_i;\theta)\,d\phi_i }\\ &=& \displaystyle{\int \pcyiphii(y_i | \phi_i;\theta)\pphii(\phi_i;\theta)\,d\phi_i } \\ &=& \esps{\qphii}{\pcyiphii(y_i | \phi_i;\theta)}. \end{eqnarray}\)

Thus, $\pyi(y_i;\theta)$ is expressed as a mean. It can therefore be approximated by an empirical mean using a Monte Carlo procedure:


1. Draw $M$ independent values $\phi_i^{(1)}$, $\phi_i^{(2)}$, ..., $\phi_i^{(M)}$ from the normal distribution $\qphii(\, \cdot \, ; \theta)$.

2. Estimate $ \pyi(y_i;\theta)$ with

\( \hat{p}_{i,M} = \displaystyle{\frac{1}{M} }\sum_{m=1}^{M}\pcyiphii(y_i | \phi_i^{(m)};\theta) . \)


By construction, this estimator is unbiased:

\(\begin{eqnarray} \esp{\hat{p}_{i,M} }&=& \esps{\qphii}{\pcyiphii(y_i | \phi_i^{(m)};\theta)} \\ &=& \pyi(y_i;\theta) . \end{eqnarray}\)

Furthermore, it is consistent since its variance decreases as $1/M$:

\( \var{\hat{p}_{i,M} }= \displaystyle{ \frac{1}{M} }\vars{\qphii}{\pcyiphii(y_i | \phi_i^{(m)};\theta)}. \)

We could consider ourselves satisfied with this estimator since we "only" have to select $M$ large enough to get an estimator with a small variance. Nevertheless, we will see now that it is possible to improve the statistical properties of this estimator.

For any distribution $\tqphii$ that is absolutely continuous with respect to the marginal distribution $\qphii$, we can write

\(\begin{eqnarray} \pyi(y_i;\theta) &=& \displaystyle{ \int \pyiphii(y_i,\phi_i;\theta)\,d\phi_i }\\ &=& \displaystyle{\int \pcyiphii(y_i | \phi_i;\theta)\frac{\pphii(\phi_i;\theta)}{\tpphii(\phi_i;\theta)}\tpphii(\phi_i;\theta)\,d\phi_i } \\ &=& \esps{\tqphii}{\pcyiphii(y_i | \phi_i;\theta)\displaystyle{\frac{\pphii(\phi_i;\theta)}{\tpphii(\phi_i;\theta)} } }. \end{eqnarray}\)

We can now approximate $\pyi(y_i;\theta)$ via an importance sampling integration method using $\tqphii$ as a proposal distribution:


1. Draw $M$ independent values $\phi_i^{(1)}$, $\phi_i^{(2)}$, ..., $\phi_i^{(M)}$ from the proposal distribution $\tqphii(\, \cdot \, ; \theta)$.

2. Estimate $ \pyi(y_i;\theta)$ with

\( \hat{p}_{i,M} = \displaystyle{\frac{1}{M} }\sum_{m=1}^{M}\pcyiphii(y_i | \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} }. \)


By construction, this new estimator is also unbiased and its variance also decreases as $1/M$:

\( \var{\hat{p}_{i,M} }= \displaystyle{\frac{1}{M} }\vars{\tqphii}{\pcyiphii(y_i | \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} } }. \)

There exists an infinite number of possible proposal distributions $\tpphii$ which all provide the same rate of convergence $1/M$. The trick is to reduce the variance of the estimator by selecting a proposal distribution so that the numerator is as small as possible.

Imagine that we use the conditional distribution $\qcphiiyi$ as the proposal. Then, for any $m=1,2,\ldots,M$,

\(\begin{eqnarray} \pcyiphii(y_i | \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\tpphii(\phi_i^{(m)};\theta)} } &=& \pcyiphii(y_i | \phi_i^{(m)};\theta) \displaystyle{\frac{\pphii(\phi_i^{(m)};\theta)}{\pcphiiyi(\phi_i^{(m)} | y_i;\theta)} }\\ &=& \pyi(y_i;\theta), \end{eqnarray}\)

which means that $\hat{p}_{i,M}=\pyi(y_i;\theta)$! Such an estimator is optimal since its variance is null and only one realization of $\qcphiiyi$ is required to exactly compute $\pyi(y_i;\theta)$. The problem is that it is not possible to generate the $\phi_i^{(m)}$ with this conditional distribution, since that would require to compute a normalizing constant, which here is precisely $\pyi(y_i;\theta)$.

Nevertheless, this conditional distribution can be estimated using the Metropolis-Hastings algorithm described in The Metropolis-Hastings algorithm for simulating the individual parameters section and a practical proposal "close" to the optimal proposal $\qcphiiyi$ can be derived. We can then expect to get a very accurate estimate with a relatively small Monte Carlo size $M$.

In $\monolix$, the mean and variance of the conditional distribution $\qcphiiyi$ are estimated by Metropolis-Hastings for each individual $i$. Then, the $\phi_i^{(m)}$ are drawn with a noncentral $t$ distribution:

\( \phi_i^{(m)} = \mu_i + \sigma_i \times T_{i,m} , \)

where $\mu_i$ and $\sigma^2_i$ are estimates of $\esp{\phi_i|y_i;\theta}$ and $\var{\phi_i|y_i;\theta}$, and $(T_{i,m})$ is a sequence of i.i.d. random variables distributed with a Student's $t$ distribution with $\nu$ degrees of freedom.

$\monolix$ uses the default value $\nu=5$. It is also possible to automatically test different degrees of freedom from the set $\{2, 5, 10, 20\}$ and to select the one that provides the smallest empirical variance for $\widehat{ {\llike} }_M(\theta;\by) = \sum_{i=1}^{N}\log(\hat{p}_{i,M})$.


Remarks

Even if $\widehat{ {\like} }_M(\theta;\by) = \prod_{i=1}^{N}\hat{p}_{i,M}$ is an unbiased estimator of ${\like}(\theta;\by)$, $\widehat{ {\llike} }_M(\theta;\by)$ is a biased estimator of $\log({\like}(\theta;\by))$. Indeed, by Jensen's inequality, we have that

\( \esp{\log(\widehat{ {\like} }_M(\theta;\by))} \leq \log \esp{\widehat{ {\like} }_M(\theta;\by)} = \log ( {\like}(\theta;\by)). \)

However, the bias decreases as $M$ increases, and also if $\widehat{ {\like} }_M(\theta;\by)$ is close to ${\like}(\theta;\by)$. It is therefore highly recommended to use a proposal as close as possible to the conditional distribution $\qcphiiyi$, which means having to estimate this conditional distribution before estimating the log-likelihood.


Man02.jpg
Example


A mixture model is used for this example. The goal here is not to provide details about the model and the formulas used for implementation. We only want to highlight the importance of the choice of the proposal distribution for estimating the likelihood in complex models. The three figures below display estimates of the deviance ($-2\times {\llike} (\theta;\by)$) by importance sampling using different proposal distributions. The estimated deviance is displayed as a function of the Monte Carlo size.

In the first example, the conditional distributions of the individual parameters $\phi_i$ were estimated using the Metropolis-Hastings algorithm, and noncentral $t$ distributions with 5 d.f. were used as proposal distributions. There is no bias and an accurate estimate is obtained with a small Monte Carlo size. The estimated deviance is 14386.8 (s.e. = 0.7). Here, only 100 Metropolis-Hastings iterations were required to correctly estimate the conditional mean and variance of the $\psi_i$.


{{{2}}}


In the second example, only the last 30 iterations of SAEM were used for estimating the conditional mean and variance of the $\phi_i$. Then, noncentral $t$ distributions with 5 d.f. we used as proposal distributions. Now there is bias which decreases very slowly with the Monte Carlo size and which is non-negligible even with $10^5$ simulations. The estimated deviance is 14397.0 (s.e. = 2.7).


{{{2}}}


In the third example, normal distributions were used as proposals. The parameters of these distributions were the mean and variance of the conditional distributions of the $\phi_i$ estimated using Metropolis-Hastings. Results are similar to those obtained with a $t$ distribution (estimated deviance=14386.9, s.e. = 1) but here, more simulations are required to eliminate the bias.


{{{2}}}



Estimation using linearization

For continuous data models, an alternative to the importance sampling approach is to use a linearization of the model like that proposed in the Estimation of the F.I.M. using a linearization of the model chapter to approximate the observed Fisher Information Matrix. Indeed, the marginal distribution of a continuous vector of observations $y_i$ can be approximated by a normal distribution. It is then straightforward to derive the associated likelihood. All of these calculations are described in the Estimation of the F.I.M. using a linearization of the model chapter.

This method can be much faster than importance sampling. It should be used by modelers for model selection purposes during the initial runs, when the goal is to identify significant differences between models. Importance sampling should be used when a more precise evaluation of the log-likelihood is required.


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