# The bootstrap particle filter

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## Outline

• The following topics will be covered in this lecture:
• A sampling approach to nonlinear estimation
• Empirical estimates of statistics
• Importance sampling
• The bootstrap filter and resampling strategies

## Motivation

• We have now begun to introduce some of the key concepts that bridge the estimation problem to nonlinear models.

• Rather than the usual Gauss-Markov model, we may generally consider a system of equations

\begin{align} \pmb{x}_k &= \mathcal{M}_k (\pmb{x}_{k-1}) + \pmb{w}_k \\ \pmb{y}_k &= \mathcal{H}_k (\pmb{x}_k) + \pmb{v}_k \end{align} where

• $$\mathcal{M}_k:\mathbb{R}^{N_x} \rightarrow \mathbb{R}^{N_x}$$ is a nonlinear mapping that encompasses the equations of motion in time;
• $$\mathcal{H}_k: \mathbb{R}^{N_x} \rightarrow \mathbb{R}^{N_y}$$ is a nonlinear mapping that transmits the hidden modeled states to the observed variables; and
• $$\{\pmb{w}_k\}_{k=1}^\infty$$ and $$\{ \pmb{v}_k\}_{k=1}^\infty$$ are mutually independent, white-in-time error sequences, with arbitrary distributions.
• Note, even if the error distributions are Gaussian, the nonlinearity of the process model and observation model will deform the forecast and posterior distributions;

• therefore, in the presence of nonlinearity, the Kalman filter approach (in all of its variants) is a sub-optimal estimator, relying on biased assumptions.
• While we have seen that the Gauss-Markov model can be used as an approximation within certain restrictions,

• it is of interest to consider, how can we perform fully nonlinear estimation?

### Motivation

• Recall the highly general hidden Markov model, without the simplification of the linear-Gaussian restriction.

\begin{align} \pmb{x}_k &= \mathcal{M}_k (\pmb{x}_{k-1}) + \pmb{w}_k \\ \pmb{y}_k &= \mathcal{H}_k (\pmb{x}_k) + \pmb{v}_k \end{align}

• It is possible to define pure Bayesian estimators for the general configuration, with very few assumptions.

• This type of estimator has had a long history in physics and engineering, where they are broadly known as particle filters / smoothers.
• These types of estimators are very robust in terms of estimating the nonlinear evolution, and can identify highly-skewed and multi-modal distributions.

• The high generality of this type of estimator means that there is very little bias by its construction;

• however, these estimators also tend to be of extremely high variance, due to the lack of any prescribed form like a Gaussian.
• This tradeoff means that in order to gain robust estimates as above, the sample size needs to be very large.

• For a small dimensional process / observation model, the large sample size is feasible, and these estimators are very good choices.
• However, for too large a system, the computation of standard particle methods becomes unfeasible, unless additional forms of bias are introduced to the estimator.

## A sampling approach to nonlinear estimation

• Recall that in our discussion of the Kalman filter, we derived that

\begin{align} p(\pmb{x}_k|\pmb{y}_{k:1})\propto p(\pmb{y}_k|\pmb{x}_k) p(\pmb{x}_k | \pmb{y}_{k-1:1}), \end{align}

• so that we interpret:

• Given the last posterior density $$p(\pmb{x}_{k-1}|\pmb{y}_{k-1:1})$$, we can apply the Markov transition kernel

\begin{align} p(\pmb{x}_{k}|\pmb{y}_{k-1:1}) = \int p(\pmb{x}_k|\pmb{x}_{k-1}) p(\pmb{x}_{k-1}|\pmb{y}_{k-1:1})\mathrm{d}\pmb{x}_{k-1} \end{align} to obtain the forecast density for $$\pmb{x}_{k|k-1}$$.

• We condition based on the likelihood of the observed data, $$\pmb{y}_k$$ by multiplication

\begin{align} p(\pmb{y}_k|\pmb{x}_k) p(\pmb{x}_k | \pmb{y}_{k-1:1}); \end{align} and

• after re-normalization by a constant, we obtain the posterior $$p(\pmb{x}_k | \pmb{y}_{k:1})$$.
• All of these steps are implicitly encoded in the Kalman filter equations for the recursive conditional mean and covariance.

• However, the above derivation actually never made use of any linear-Gaussian model assumptions;

• indeed, this only required the Markov assumptions and independence of the observation and model errors.

### A sampling approach to nonlinear estimation

• Suppose that somehow we had access to the posterior density $$p(\pmb{x}_L\vert \pmb{y}_{L:1})$$.
• If we draw $$N_e$$ independent, identically distributed (iid) ensemble members from this distribution, $$\{\pmb{x}_L^i\}_{i=1}^{N_e}$$,
• an empirical representation of the distribution (probability measure) is given by \begin{align} \mathcal{P}_{N_e}(\pmb{x}_L\vert \pmb{y}_{L:1}) = \frac{1}{N_e} \sum_{i=1}^{N_e} \boldsymbol{\delta}_{\pmb{x}^i_L}\left(\mathrm{d}\pmb{x}_L\right), \end{align} where
• $$\boldsymbol{\delta}_{\pmb{x}^i_L}$$ – this the Dirac delta measure centered at the ensemble member $$\pmb{x}_L^i$$;
• i.e., we write, \begin{align} \int f \boldsymbol{\delta}_{\pmb{x}^i_L} (\mathrm{d}\pmb{x}_L) = f\left(\pmb{x}^i_L\right). \end{align}
• In the above, the denominator $$\frac{1}{N_e}$$ represents that all point volumes have equal mass or weights, so that the measure integrates to one.
• Then for any statistic $$f$$ of the posterior, we can recover an estimate of its expected value directly as \begin{align} \mathbb{E}_{\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})} \left[f \right] := \int f \mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1} ) &\approx \int f \mathcal{P}_{N_e}(\pmb{x}_L\vert \pmb{y}_{L:1})\\ &= \frac{1}{N_e}\sum_{i=1}^{N_e} \int f(\pmb{x})\delta_{\pmb{x}^i_L}(\mathrm{d}\pmb{x}_L) = \frac{1}{N_e} \sum_{i=1}^{N_e} f\left(\pmb{x}_L^i\right), \end{align}
• that is, by taking the empirical average of the statistic evaluated over the ensemble members.

## Empirical estimates

• The empirical estimate, $\mathbb{E}_{\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})} \left[f \right] \approx \frac{1}{N_e} \sum_{i=1}^{N_e} f\left(\pmb{x}_L^i\right),$ is also an unbiased estimator of the statistic $$f$$ .
• If the posterior variance of $$f(\pmb{x})$$ satisfies, \begin{align} \sigma^2_f = \mathbb{E}_{\mathcal{P}(\pmb{x}_L \vert \pmb{y}_{L:1})}\left[f^2(\pmb{x}_L)\right] - \left(\mathbb{E}_{\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})}\left[f(\pmb{x}_L)\right]\right)^2 < \infty; \end{align}
• then the variance of the empirical estimate, \begin{align} \mathrm{var}\left(\mathbb{E}_{\mathcal{P}_{N_e}(\pmb{x}\vert \pmb{y})}\left[f\right]\right) = \frac{\sigma_f^2}{N_e}, \end{align} where the variance is understood as taken over the possible sample outcomes, $$\pmb{x}_L^i \sim \mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})$$.
• For $\sigma_f^2 < \infty$ as above, the Central Limit Theorem tells us \begin{align} \lim_{N_e\rightarrow +\infty}\sqrt{N_e}\left\{ \mathbb{E}_{\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})}\left[f\right] - \mathbb{E}_{\mathcal{P}_N(\pmb{x}_L\vert \pmb{y}_{L:1})}\left[f\right]\right\} =N(0, \sigma_f^2), \end{align}
• i.e., the empirical distribution converges to the true distribution in the weak sense as the ensemble size $$N_e$$ gets sufficiently large.
• In particular, under very general conditions, the empirical probability measure will produce statistics that converge to the statistics of the true posterior in expected value.

## Importance sampling

• In practice, we often cannot sample the posterior directly but we may need to sample some other distribution that shares its support.
• This idea of sampling another distribution with shared support is known as importance sampling.
• Importance sampling is a broadly used Bayesian technique which can be given its own detailed treatment in a course of Bayesian estimation.
• We will only go over a high-level view of this concept as it relates to particle filters and smoothers.
• We will suppose that we have access, perhaps not to $$\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})$$ but instead $$\mathcal{Q}(\pmb{x}_{L}\vert \pmb{y}_{L:1})$$ such that $$\mathcal{P} \ll \mathcal{Q}$$,
• i.e., by contraposition, $$\mathcal{P}(A\vert \pmb{y}_{L:1})>0 \Rightarrow \mathcal{Q}(A\vert \pmb{y}_{L:1})>0$$.
• This above assumption allows us to take the Radon-Nikodym derivative of the true posterior $$\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})$$ with respect to the proposal distribution $$\mathcal{Q}(\pmb{x}_L \vert \pmb{y}_{L:1})$$.
• The key innovation to the last formulation is that this allows us to evaluate a statistic of the posterior by point volumes but with non-equal importance weights.
• For each of the probabilities $$\mathcal{P} / \mathcal{Q}$$ define the associated densities as $$p / q$$.
• Let us define the importance weight function $$w(\pmb{x}_L \vert \pmb{y}_{L:1}) := \frac{p(\pmb{x}_L\vert \pmb{y}_{L:1})}{q(\pmb{x}_L\vert \pmb{y}_{L:1})}$$.
• We will suppose that the weights can be computed even if $$\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})$$ is not available – this will be explained shortly.

### Importance sampling

• Given importance weights, we can re-write the expected value of some statistic $$f$$ of the posterior density as, \begin{align} \mathbb{E}_{\mathcal{P}(\pmb{x}_L\vert \pmb{y}_{L:1})}[f]= \int f(\pmb{x}_L)p(\pmb{x}_L\vert \pmb{y}_{L:1})\mathrm{d}\pmb{x}_L & = \frac{\int f(\pmb{x}_L)p(\pmb{x}_L\vert \pmb{y}_{L:1})\mathrm{d}\pmb{x}_L}{\int p(\pmb{x}_L\vert \pmb{y}_{L:1})\mathrm{d}\pmb{x}_L}\\ &=\frac{\int f(\pmb{x}_{L})w(\pmb{x}_L\vert \pmb{y}_{L:1})q(\pmb{x}_L\vert \pmb{y}_{L:1})\mathrm{d}\pmb{x}_L}{\int w(\pmb{x}_L\vert \pmb{y}_{L:1})q(\pmb{x}_L\vert \pmb{y}_{L:1})\mathrm{d}\pmb{x}_L} , \end{align}
• The above is shown by the definition of the importance weights above and because for any density, $\int p(\pmb{x}_L\vert \pmb{y}_{L:1})\mathrm{d}\pmb{x}_L =1.$
• The benefit for sampling techniques is therefore to draw iid $$\pmb{x}^i_L \sim \mathcal{Q}(\pmb{x}_L\vert \pmb{y}_{L:1})$$ which we assume that we can sample.
• The empirical measure is thus, \begin{align} \mathcal{Q}_{N_e} := \frac{1}{N_e} \sum_{i=1}^{N_e} \pmb{\delta}_{\pmb{x}^i_{L}} \end{align}
• Using the importance weights, then one defines the empirical expected value of $$f$$ as, \begin{align} \mathbb{E}_{\mathcal{P}_{N_e}(\pmb{x}_L \vert \pmb{y}_{L:1})}[f] = \frac{ \frac{1}{N_e} \sum_{i=1}^{N_e} f(\pmb{x}^i_L)w(\pmb{x}^i_L\vert \pmb{y}_{L:1})}{\frac{1}{N_e} \sum_{i=1}^{N_e} w(\pmb{x}^i_L \vert \pmb{y}_{L:1})} = \sum_{i=1}^{N_e} f(\pmb{x}^i_L) \tilde{w}^i_L. \end{align}
• Here, the $$\tilde{w}^i_L := \frac{w(\pmb{x}^i_L \vert \pmb{y}_{L:1})}{\sum_{i=1}^{N_e} w(\pmb{x}^i_L \vert \pmb{y}_{L:1})}$$ are defined as the normalized importance weights, ensuring integration to one.

### Importance sampling

• We will write our empirical estimate of the posterior as, \begin{align} \mathcal{P}_{N_e}(\pmb{x}_L \vert \pmb{y}_{L:1}) := \sum_{i=1}^{N_e} \tilde{w}^i_L\delta_{\pmb{x}^i_L}(\mathrm{d}\pmb{x}_L ). \end{align}
• Notice, if we take $$f(\pmb{x}) = 1$$, the expected value is indeed one.
• Using this formulation, we have an extremely flexible view of the posterior as combination of positions and weights.
• In a hidden Markov model, positions correspond to initial conditions for, e.g., a nonlinear, stochastic differential equation $$\mathcal{M} + \pmb{w}_k$$.
• Therefore, we can evolve all the point volumes, keeping their weights, and construct a forward-in-time density.
• For particle filters in physical process models, we have a natural choice of how to find the next prior from the last posterior;
• we simply evolve the points within the process model, carrying the importance weights with them.
• When we condition on new information, the goal then is to find new appropriate weights, and / or resample positions.
• Despite the heavy mathematical machinery, this is an elegant algorithm that has a very natural, empirical interpretation.
• The next key is in how we update the weights to condition the sample appropriately

### Sequential importance sampling

• Using Bayes' Law, we can derive up to proportionality, \begin{align} \tilde{w}^i_L \propto \tilde{w}^i_{L-1} \frac{ {\color{#d24693} {p\left(\pmb{y}_L \vert \pmb{x}^i_{L}\right) } } {\color{#1b9e77} {p\left(\pmb{x}^i_L \vert \pmb{y}_{L-1:1}\right)}} }{q\left(\pmb{x}_L^i \vert \pmb{y}_{L-1:1} \right)}. \end{align}
• The key again, is how we define a proposal density as above that can be sampled even when the posterior cannot be sampled directly.
• As one special choice, we can choose a proposal of $$\mathcal{Q}$$ as the forecast-prior distribution $${\color{#1b9e77} {\mathcal{P}(\pmb{x}_{L}\vert \pmb{y}_{L-1:1})} }$$;
• in this case, $$p=q$$ and we have the weights given recursively by $\tilde{w}^i_L \propto \tilde{w}^i_{L-1} {\color{#d24693} {p(\pmb{y}_L \vert \pmb{x}_L^i)} }.$
• The proportionality statement says that:
• Suppose we have knowledge of the normalized weights $$\tilde{w}^i_{L-1}$$ at time $$t_{L-1}$$;
• we can generate a forecast for each position $$\pmb{x}_{L-1}^i$$ at time $$t_L$$ via the model, $\pmb{x}_L^i = {\color{#1b9e77} {\mathcal{M}_{L}(}} \pmb{x}_{L-1}^i {\color{#1b9e77} {) + \pmb{w}_L} }.$
• Prior to obtaining the new observation, the forecast weight will remain as $$\tilde{w}_{L-1}^i$$;
• to condition the weights on the new observation, we apply Bayes' Law, $\tilde{w}^i_L \propto \tilde{w}^i_{L-1} {\color{#d24693} {p(\pmb{y}_L \vert \pmb{x}_L^i)} }.$ such that we need only compute $$\tilde{w}^i_{L-1} {\color{#d24693} {p(\pmb{y}_L \vert \pmb{x}_L^i)} }$$ for each $$i$$ and then re-normalize the weights so they sum to $$1$$.
• This technique is known in the stochastic filtering literature as a sequential importance sampling (SIS) particle filter.

### Sequential importance sampling

• SIS particle filters are extremely flexible and makes few assumptions on the form of the problem whatsoever;

• however, the primary issue is that the importance weights become extremely skewed very quickly, leading to all the probability mass landing on a single point after only a few iterations.
• This is one example of a concept more broadly known in nonlinear filtering as ensemble collapse / filter divergence.

• In effect, the empirical estimate becomes overly self-certain, and will no longer be receptive to new data.

• Because a single point mass cannot represent the spread of the data, this also cannot be used to represent any of the variation in the estimate.

• Finding a method for handling the degeneracy of the weights is explicitly the motivation for the bootstrap particle filter, and implicitly one of the motivations for the ensemble Kalman filter.

• In particular, the weights tend to be of too high variance in the particle filter generally, and one rectification is to impose a bias in the estimate.
• We will return to the idea of the ensemble Kalman filter later, but for now consider the classical particle filter rectification.

## The bootstrap filter

• The method of the bootstrap filter essentially proposes to eliminate the degeneracy of the weights by eliminating ensemble members with weights close to zero and resampling.
• At the point of the Bayesian update and re-weighting, one:
1. eliminates all ensemble members with weights $$\tilde{w}^i < W$$ where $$W\ll 1$$ will be some threshold for the weights;
2. then, make replicates of the higher weighted ensemble members and reset the importance weights all equal to $$\frac{1}{N_e}$$;
3. then the new empirical posterior is then given by, \begin{align} \mathcal{P}_{N_e}(\pmb{x}_{L}\vert \pmb{y}_{L:1}) = \frac{1}{N_e} \sum_{i=1}^{N} N^i \delta_{\pmb{x}^i_L}(\mathrm{d}\pmb{x}_L), \end{align} where $$N^i$$ is the number of replicates $$\left(N^i\in[0, N_e]\right)$$ of sample $$\pmb{x}^i_L$$ such that $$\sum_{i=1}^{N} N^i =N_e$$
4. If the first prior sample is drawn iid, the first weights $$w_0^i \equiv \frac{1}{N_e}$$; this gives a complete algorithm for a general hidden Markov model.
• Note, how the number of replicates $$N^i$$ is chosen is the basis of several different approaches to particle filters.

### Basic resampling algorithm

• The basic resampling algorithm simply utilizes the strategy of the inverse CDF transformation of a uniform sample.

• In particular, the weights $$\tilde{w}^i_k$$ at any given time provide an estimate of the empirical CDF for the posterior.

• We draw a uniform $$u$$ on $$[0,1]$$ and find the first index $$i$$ for which the total sum of all the weights below index $$i$$ fall below the realized value $$u$$.

• We select this index and create a particle replicate of the associated $$\pmb{x}_k^i$$.
• This is repeated until we have $$N_e$$ total ensemble members once again, all given equal weights to restart the algorithm.

• Notice that assigning the map of $$u$$ to the associated particle $$\pmb{x}^{i}_k$$ is precisely the inverse empirical CDF map.

• In particular, under generic convergence conditions, and the limit in the sample size $$N_e \rightarrow \infty$$,

• the ensemble will generate statistics that follow the appropriate theoretical CDF in expectation, giving weak convergence as before.

### Systematic resampling algorithm

• More commonly, the standard technique for the bootstrap particle filter is the systematic resampling algorithm.

• Rather than using multiple draws of the uniform, this uses a single draw and make a stratified sample of the empirical CDF.
• Firstly, we draw $$u^1$$ uniform on $$[0, N_e^{-1}$$, i.e., on the restricted range up to one over the ensemble size.

• Then, we make a replicate of the first ensemble member for which the cumulative weight is greater than $$u^1$$.
• The first draw creates exactly one “representative” replicate corresponding to all particles with combined weight in the range $$[0,N_e^{-1}]$$.

• From this point, a new $$u^j$$ is defined as $$u^j= u^1 + N_e^{-1}(j-1)$$, where the same replication strategy follows:

• the first ensemble member for which the cumulative weight is greater than $$u_j$$ is selected as $$j$$ ranges up to $$N_e$$.
• With this strategy, we are guaranteed to draw exactly one “representative” replicate among particles for which the empirical CDF, $$c^i$$ falls in the range $$[(j-1)/N_e, j/N_e]$$.

• Particularly, $$u^j$$ is uniform on $$[(j-1)/N, j/N]$$, and this decides which particular particle will be replicated in this weight interval.