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Made KDE more boring

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Markus Bullmann
2018-10-16 17:15:01 +02:00
parent 6c3604da16
commit 2ce0da80d8
1 changed files with 2 additions and 2 deletions
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tex_review/chapters/estimation.tex
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@@ -30,7 +30,7 @@ In the case of particle filters the MMSE estimate equals to the weighted-average
where $W_t=\sum_{i=1}^{N}w^i_t$ is the sum of all weights. where $W_t=\sum_{i=1}^{N}w^i_t$ is the sum of all weights.
While producing an overall good result in many situations, it fails when the posterior is multimodal. While producing an overall good result in many situations, it fails when the posterior is multimodal.
In these situations the weighted-average estimate will find the estimate somewhere between the modes. In these situations the weighted-average estimate will find the estimate somewhere between the modes.
Clearly, such a position between modes is extremely unlikely the position of the pedestrian. \del{Clearly}\add{It is expected that}, such a position between modes is extremely unlikely the position of the pedestrian.
The real position is more likely to be found at the position of one of the modes, but virtually never somewhere between. The real position is more likely to be found at the position of one of the modes, but virtually never somewhere between.
In the case of a multimodal posterior the system should estimate the position based on the highest mode. In the case of a multimodal posterior the system should estimate the position based on the highest mode.
@@ -39,7 +39,7 @@ A straightforward approach is to select the particle with the highest weight.
However, this is in fact not necessarily a valid MAP estimate, because only the weight of the particle is taken into account. However, this is in fact not necessarily a valid MAP estimate, because only the weight of the particle is taken into account.
In order to compute the true MAP estimate the local density of the particles needs to be considered as well \cite{cappe2007overview}. In order to compute the true MAP estimate the local density of the particles needs to be considered as well \cite{cappe2007overview}.
\del{It is obvious,} A computation of the probability density function of the posterior could solve the above, but finding such an analytical solution is clearly an intractable problem, which is the reason for applying a sample representation in the first place. \del{It is obvious,} A computation of the probability density function of the posterior could solve the above, but finding such an analytical solution is \del{clearly} an intractable problem, which is the reason for applying a sample representation in the first place.
A feasible alternative is to estimate the parameters of a specific parametric model based on the sample set, assuming that the unknown distribution is approximately a parametric distribution or a mixture of parametric distributions, \eg{} Gaussian mixture distributions. A feasible alternative is to estimate the parameters of a specific parametric model based on the sample set, assuming that the unknown distribution is approximately a parametric distribution or a mixture of parametric distributions, \eg{} Gaussian mixture distributions.
Given the estimated parameters the most probable state can be obtained from the parameterised density function. Given the estimated parameters the most probable state can be obtained from the parameterised density function.
%In the case of multi-modalities several parametric distributions can be combined into a mixture distribution. %In the case of multi-modalities several parametric distributions can be combined into a mixture distribution.