Sequential Monte Carlo in Statistics

# Sequential Monte Carlo in Statistics
## July 2019
### Anthony Lee University of Bristol & Alan Turing Institute With many thanks to the Statistical Society of Australia and ACEMS

---

# Outline

- Introduction: Classical Monte Carlo & Importance Sampling

- Sequential Monte Carlo & hidden Markov models

- Arbitrary target distributions

- Twisted flows

- Variance estimation

For more details, and references, please see the recent survey

Doucet & Lee. Sequential Monte Carlo methods. In Handbook of Graphical Models, 2018.

A preprint is linked to from my website.

---

# Introduction

## Classical Monte Carlo & Importance Sampling

---

# Classical Monte Carlo: integral

Main idea: approximate sums/integrals with random variables.

Notation: measurable space `$(\mathsf{X}, \mathcal{X})$`, `$f:\mathsf{X}\rightarrow\mathbb{R}$` and `$\mu$` a measure on `$\mathcal{X}$`,
`$$\mu(f):=\int_{\mathsf{X}}f(x)\mu({\rm d}x).$$`
--

For example, consider
 
 - `$\mathsf{X} = \{1,\ldots,s\}$` so `$\mu(f)$` is a sum,
 - `$\mathsf{X} = \mathbb{N}$` so `$\mu(f)$` is an infinite sum,
 - `$\mu$` having a density w.r.t. Lebesgue measure on `$\mathbb{R}^d$`.

If `$\mu$` is a probability measure / distribution, `$\mu(1)=1$` and
`$$\mu(f)=\mathbb{E}\left[f(X)\right],\qquad X\sim\mu.$$`

---

# Classical Monte Carlo: approximation

Assume `$\mu$` is a probability measure / distribution.

Monte Carlo (particle) approximation of `$\mu$` is a discrete probability measure

`$$\mu^{N} := \frac{1}{N} \sum_{i=1}^N \delta_{\zeta_i},\qquad\zeta_{i}\overset{\rm iid}{\sim}\mu$$`
--

So the Monte Carlo approximation of `$\mu(f)$` is the integral of `$f$` w.r.t. `$\mu^N$`

`$$\mu^{N}(f):=\frac{1}{N}\sum_{i=1}^{N}f(\zeta_{i}).$$`

**Law of large numbers**: `$\mu^{N}(f)\overset{\rm a.s.}{\rightarrow}\mu(f)$` as `$N\rightarrow\infty$`.

Fundamental justification, but no handle on accuracy for finite `$N$`.

---

# Classical Monte Carlo: accuracy

Assume `$\mu(f^2) < \infty$`, i.e `$f \in L_2(\mathsf{X}, \mu)$`.

The variance of `$\mu^N(f)$` is
`$${\rm var}(\mu^N(f)) = \frac{1}{N} {\rm var}(f(X))  = \frac{1}{N}\mu(\bar{f}^2),$$`
where `$\bar{f} := f - \mu(f)$`.

**Central Limit Theorem**: as `$N \to \infty$` we have asymptotic normality

`$$N^{1/2} ( \mu^N(f) - \mu(f) ) \overset{L}{\to} N(0, \mu(\bar{f}^2)).$$`

Good news: `$|\mu^N(f) - \mu(f)|$` is `$\mathcal{O}_P(N^{-1/2})$`.

Bad news: `$\mu(\bar{f}^2)$` may be very large, e.g. when `$\mathsf{X}$` is "large".

---

# Importance sampling

Say we want to approximate `$\pi(f)$` but we can only simulate according to `$\mu$`?

If `$\mu$` dominates `$\pi$`, i.e. `$\pi(x) > 0 \Rightarrow \mu(x) > 0$`, then we can write

`$$\pi(f) = \mu(w \cdot f) = \int f(x) w(x) \mu({\rm d}x),$$`

where `$w = {\rm d} \pi / {\rm d} \mu$` is the ratio of densities of `$\pi$` and `$\mu$`.

So we can approximate `$\pi(f)$` by `$\mu^N(w \cdot f)$`.

This is an unbiased approximation with variance

`$${\rm var}(\mu^N(w \cdot f)) = \frac{\mu(w^2 \cdot f^2) - \pi(f)^2}{N} = \frac{\pi(w \cdot f^2) - \pi(f)^2}{N}.$$`

---

# Self-normalized importance sampling

We can also "self-normalize" the approximation, i.e. compute

`$$\pi^N_{\rm SN}(f) = \frac{\mu^N(w \cdot f)}{\mu^N(w)} = \frac{\sum_{i=1}^N w(X_i)f(X_i)}{\sum_{i=1}^N w(X_i)} = \frac{\mu^N \cdot w}{\mu^N(w)} (f).$$`

Useful when `$w$` can be computed only up to an unknown normalizing constant, e.g. when `$\pi$` is a posterior distribution.

This is biased but strongly consistent and asymptotically normal as `$N \to \infty$`.

The asymptotic variance is

`$$\lim_{N \to \infty} N ~ {\rm var}(\pi^N_{\rm SN}(f) ) = \pi(w \cdot \bar{f}^2),$$`
where `$\bar{f} = f - \pi(f)$`.

The asymptotic variance can be smaller than the asymptotic variance for IS.

---

# Sequential Monte Carlo

## Motivated by Hidden Markov Models

---

# Hidden Markov model

Let `$(X_1,\ldots,X_n)$` be a Markov chain with transition kernels `$M_2,\ldots,M_n$` and initial distribution `$\mu$`. So `$X_1 \sim \mu$` and `$X_p \mid X_{p-1} \sim M_p(X_{p-1}, \cdot)$`.

Let `$Y_p \mid (X_1,\ldots,X_n) \sim g(X_p,\cdot)$`.

In an HMM, the observations are `$Y_1,\ldots,Y_n$`.

<center>

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</center>

---

# HMM examples

- Economics: asset prices driven by a latent process

- Chemistry: reactions driven by concentrations of chemicals

- Physics: imperfect measurements of a dynamical system

- Robotics: localization in space through noisy observations

- Ecology: sparse observations of animals in space and time

- Environment: noisy/partial observations of climate

- Essentially any discretely and partially observed Markov process.

---

# Objects of interest

For `$p \in \{1,\ldots,n\}$`, we are interested in,

1. `$\eta_p$` : the distribution of `$X_p$` given `$Y_1,\ldots,Y_{p-1}$`. [ Note `$\eta_1 \equiv \mu$` ].

1. `$\hat{\eta}_p$` : the distribution of `$X_p$` given `$Y_1,\ldots,Y_{p}$`.

1. `$Z_p$` : the marginal likelihood associated with `$y_1,\ldots,y_p$`.

For example, we might have densities in common notation

`$$\begin{align}
\eta_p(x_p) &= f(x_p \mid y_1,\ldots,y_{p-1}) \\
\hat{\eta}_p(x_p) &= f(x_p \mid y_1,\ldots,y_p) \\
Z_p &= f(y_1,\ldots,y_p)
\end{align}$$`

To simplify notation in what follows, we define

`$$G_p(x) := g(x, y_p),$$`

rendering the dependence on `$y_1,\ldots,y_n$` implicit.

---

# Recursive updates

With densities, we have `$\eta_1(x_1) = \mu(x_1) = f(x_1)$`, and

`$$\hat{\eta}_1(x_1) = f(x_1 \mid y_1) = \frac{f(x_1) f(y_1 \mid x_1)}{f(y_1) } = \frac{\eta_1 \cdot G_1}{\eta_1(G_1)}(x_1).$$`

Similarly, for `$p \in \{2,\ldots,n\}$`,

`$$\eta_p(x_p) = f(x_p \mid y_{1:p-1}) = \int f(x_{p-1} \mid y_{1:p-1}) f(x_p \mid x_{p-1}) {\rm d}x_{p-1} = \hat{\eta}_{p-1}M_p(x_p),$$`

and

`$$\hat{\eta}_p(x_p) = \frac{f(x_p \mid y_{1:p-1}) f(y_p \mid x_p)}{\int f(y_p \mid x_p')f(x_p' \mid y_{1:p-1}) {\rm d}x_p' } = \frac{\eta_p \cdot G_p}{\eta_p(G_p)}(x_p).$$`

We also have
`$$Z_p = f(y_{1:p}) = f(y_{1:p-1}) f(y_p \mid y_{1:p-1})  = Z_{p-1} \eta_p(G_p).$$`

---

# Forward algorithm (finite state space)

**Algorithm**:

1. Set
`$$\eta_1 = \mu, \qquad Z_1 = \eta_1(G_1), \qquad \text{ and } \qquad \hat{\eta}_1 = \frac{\eta_1 \cdot G_1}{\eta_1(G_1)}.$$`
2. For `$p = 2,\ldots,n$`: set
`$$\eta_p = \hat{\eta}_{p-1} M_p, \qquad Z_p = Z_{p-1} \eta_p(G_p), \qquad \text{ and } \qquad \hat{\eta}_p = \frac{\eta_p \cdot G_p}{\eta_p(G_p)}.$$`
--

In SMC we replace intractable objects with particle approximations.

---

# SMC (general state space)

Sample `$\zeta_1^i \overset{\rm iid}{\sim} \mu = \eta_1$` for `$i \in \{1,\ldots,N\}$`. Set
`$$\eta_1^N = \frac{1}{N} \sum_{i=1}^N \delta_{\zeta_1^i}, \qquad Z_1^N = \eta_1^N(G_1), \qquad \hat{\eta}^N_1 = \frac{\eta_1^N \cdot G_1}{\eta_1^N(G_1)}.$$`

For `$p = 2,\ldots,n$`: sample
`$$\zeta_p^i \overset{\rm iid}{\sim} \hat{\eta}^N_{p-1}M_p = \frac{\sum_{j=1}^N G_{p-1}(\zeta_{p-1}^j)M_p(\zeta_{p-1}^j, \cdot)}{\sum_{j=1}^N G_{p-1}(\zeta_{p-1}^j)}, \qquad i \in \{1,\ldots,N\}.$$`

and then set

`$$\eta_p^N = \frac{1}{N} \sum_{i=1}^N \delta_{\zeta_p^i}, \qquad Z_p^N = Z_{p-1}^N \eta_p^N(G_p), \qquad \hat{\eta}^N_p = \frac{\eta_p^N \cdot G_p}{\eta_p^N(G_p)}.$$`
---

# Evolutionary algorithm interpretation

Sampling
`$$\zeta_p^i \overset{\rm iid}{\sim} \hat{\eta}^N_{p-1}M_p = \frac{\sum_{j=1}^N G_{p-1}(\zeta_{p-1}^j)M_p(\zeta_{p-1}^j, \cdot)}{\sum_{j=1}^N G_{p-1}(\zeta_{p-1}^j)}, \qquad i \in \{1,\ldots,N\},$$`
can be viewed as a two-stage procedure.

For each `$i \in \{1,\ldots,N\}$`,

1. Sample `$A_{p-1}^i \sim {\rm Categorical}(G_{p-1}(\zeta_{p-1}^1),\ldots,G_{p-1}(\zeta_{p-1}^N))$`.
2. Sample `$\zeta_p^i \sim M_p(\zeta_{p-1}^{A_{p-1}^i}, \cdot)$`.

---

1. Selection: `$G_{p-1}$` is like a fitness function.

2. Mutation: `$M_p$` is a mutation kernel.

---

# R code (univariate states)

```r
smc <- function(mu, M, G, n, N) {
 zetas <- matrix(0,n,N)
 as <- matrix(0,n-1,N)
 gs <- matrix(0,n,N)
 log.Zs <- rep(0,n)
 
 zetas[1,] <- mu(N)
 gs[1,] <- G(1, zetas[1,])
 log.Zs[1] <- log(mean(gs[1,]))
 
 for (p in 2:n) {
 # simulate ancestor indices, then particles
 as[p-1,] <- sample(N,N,replace=TRUE,prob=gs[p-1,]/sum(gs[p-1,]))
 zetas[p,] <- M(p, zetas[p-1,as[p-1,]])
 gs[p,] <- G(p, zetas[p,])
 log.Zs[p] <- log.Zs[p-1] + log(mean(gs[p,]))
 }
 
 return(list(zetas=zetas,gs=gs,as=as,log.Zs=log.Zs))
}
```

---

# Evolution of the particle system

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---

# A little bit of theory

For simplicity, assume `$G_1,\ldots,G_n$` and `$f$` are bounded.

**Lack of bias of `$Z_n^N$`**: `$\mathbb{E}[Z_n^N] = Z_n$`.

**Law of large numbers**: `$Z_n^N \overset{\rm a.s.}{\to} Z_n$` and `$\eta_n^N(f) \overset{\rm a.s.}{\to} \eta_n(f)$` as `$N \to \infty$`.

**Asymptotic normality**: `$N^{1/2}(Z_n^N - Z_n)$` and `$N^{1/2}(\eta_n^N(f) - \eta_n(f))$`  converge in distribution to (different) normal random variables.

**Time uniform convergence**: Under special assumptions one has
`$$\sup_{n \geq 1} \mathbb{E}\left [ | \eta_n^N(f) - \eta_n(f) |^2 \right ]^{1/2} \leq \frac{E(f)}{\sqrt{N}}.$$`

**Stability of `$Z_n^N$` with `$N \propto n$`**: Under similar assumptions,
`$${\rm var}\left (\frac{Z_n^N}{Z_n} \right ) \leq (1+C/N)^n.$$`

---

# Extensions to paths

One can also define the measure

`$$\bar{\gamma}_n({\rm d}x_1,\ldots,{\rm d}x_n) = \mu({\rm d}x_1)\prod_{p=2}^n G_{p-1}(x_{p-1})M_p(x_{p-1}, {\rm d}x_p).$$`

In an HMM: this has density `$f(x_{1:n}, y_{1:n-1}) \propto f(x_{1:n} \mid y_{1:n-1})$`.

By tracing an ancestral line from each of the time `$n$` particles, one can define a particle (path) approximation of `$\bar{\gamma}_n$`.

`$$\bar{\gamma}_n^N(f) = Z_{n-1}^N \frac{1}{N} \sum_{i=1}^N f(\zeta_{1:n}^{\text{path }i}),$$`
where the `$i$`th path arises by tracing the ancestors of `$\zeta_n^i$`.

These approximations are unbiased and consistent. A corresponding `$\bar{\eta}_n^N(f)$` is obtained by omitting the `$Z_{n-1}^N$` term, which is biased but consistent.

---

# Ancestral lineages, `$N=16$` particles

![](smc-tutorial_files/figure-html/unnamed-chunk-4-1.png)

---

# Ancestral lineages, `$N=256$` particles

![](smc-tutorial_files/figure-html/unnamed-chunk-5-1.png)

---

# Arbitrary target distributions

---

# Flow of distributions

The algorithm does not necessarily require a HMM interpretation.

One defines `$\eta_1 = \mu$` and for `$p=2,\ldots,n$`
`$$\eta_p = \frac{\eta_{p-1}\cdot G_{p-1}}{\eta_{p-1}(G_{p-1})} M_p = \Phi_p(\eta_{p-1}).$$`

In order to use SMC for approximating `$\pi(f)$` for an arbitrary `$\pi$` we just need to construct an appropriate flow from `$\mu$` to `$\pi$`, where `$\mu$` is easy to sample from.

---

# An SMC sampler

Define `$\mu  = \eta_1 \gg \ldots \gg \eta_n = \pi$` directly. Then let

`$$G_{p-1} \propto \frac{{\rm d}\eta_p}{{\rm d}\eta_{p-1}},$$`

and each `$M_p$` be a `$\eta_p$`-invariant Markov kernel, i.e. `$\eta_p M_p = \eta_p$`.

We can then verify that for `$p=2,\ldots,n$`,

`$$\frac{\eta_{p-1} \cdot G_{p-1}}{\eta_{p-1}(G_{p-1})}M_p = \eta_p M_p = \eta_p,$$`
so the flow is valid.

In practice, one might choose some `$\mu \gg \pi$` and let

`$$\eta_p(x) \propto \mu(x)^{1-\beta_p} \pi(x)^{\beta_p},$$`

for some `$0=\beta_1 < \ldots < \beta_n = 1$`.

---

# Twisted flows

---

# Twisting flows

A "good" flow has small associated variances.

Once you have a flow, you can also *twist* it.

A given `$\mu$`, `$M_1,\ldots,M_n$`, `$G_1,\ldots,G_n$`, can be twisted.

Notation: `$M(f)(x) = \int f(x')M(x,{\rm d}x')$`.

For some positive functions `$\psi_1,\ldots,\psi_{n}$` define

`$$\mu^\psi = \frac{\mu \cdot \psi_1}{\mu(\psi_1)}, \qquad G_1^\psi(x) := \frac{G_1(x)M_2(\psi_2)(x)}{\psi_1(x)} \mu(\psi_1),$$`

and for `$p \in \{2,\ldots,n\}$`

`$$M_p^\psi(x, {\rm d}x'):=\frac{M_p(x, {\rm d}x') \cdot \psi_p(x')}{M_p(\psi_p)(x)},\qquad G_p^\psi(x) = \frac{G_p(x)M_{p+1}(\psi_{p+1})(x)}{\psi_p(x)}.$$`

Calculations give `$\eta_p^\psi \propto \eta_p \cdot \psi_p$` for each `$p$`.

---

# Optimal twisting functions

To obtain a zero variance approximation of `$Z_n$`, one should use

`$$\psi_p(x) = G_p(x) M_{p+1}(\psi_{p+1})(x).$$`

In the HMM context, this corresponds to

`$$\psi_p(x_p) = f(y_{p:n} \mid x_p).$$`

Intuition: particles are weighted by how they explain the "future" as well as the present.

The optimal `$\psi$` functions are typically intractable. But one can use any reasonable approximation.

Example: for `$\psi_p = G_p$`, one obtains the fully-adapted auxiliary particle filter.

Caution: the intermediate distributions are changed by twisting.

---

# Variance estimation

---

# How accurate are our SMC approximations?

There is substantial theory on how variable SMC approximations are.

Especially for the unbiased, "unnormalized" approximations.

In practice this is not *that* helpful for assessing approximation quality.

A simple estimate of the variance can be phrased in terms of the time `$n$` particles and the *Eve* indices of those particles: i.e. the index of their time `$1$` ancestors, `$E_n^1,\ldots,E_n^N$`.

![](smc-tutorial_files/figure-html/unnamed-chunk-6-1.png)

---

# Estimate of variance of `$Z_{n-1}^N$`

Let `$E_n^j$` denote the index of the ancestor of `$\zeta_n^j$`.

Then an estimate of `${\rm var}(Z_{n-1}^N/Z_{n-1})$` is
`$$V_n^N = 1 - \left (\frac{N}{N-1} \right)^n \left( 1 - \frac{1}{N^2} \sum_{i=1}^N | \{j : E_n^j = i \} |^2 \right)$$`

If for some `$k$`, `$E_n^1 = \cdots = E_n^N = k$` then `$V_n^N = 1$`.

We have lack-of-bias in the sense that

`$$\mathbb{E}[(Z_{n-1}^N)^2 V_n^N] = {\rm var}(Z_{n-1}^N).$$`

We have consistency in the sense that

`$$N V_n^N \overset{P}{\to} \lim_{N \to \infty} N {\rm var}(Z_{n-1}^N/Z_{n-1}).$$`

---

# R code

```r
VnN <- function(as) {
 n <- dim(as)[1] + 1
 N <- dim(as)[2]
 eves <- 1:N
 # recursively update the Eve indices
 for (p in 1:(n-1)) {
 eves <- eves[as[p,]]
 }
 1 - (N/(N-1))^n*(1 - sum(table(eves)^2)/N^2)
}
```

The estimate depends only on the Eve indices, which depend only on the ancestors.

Approximating the variance of `$\eta_n^N(f)$` or `$Z^N_{n-1}\eta_n^N(f)$` is only a little bit more complicate, and does depend on particle values.

---

```r
lZ <- -12.4395996645203368302645685616880655
trials <- 1e3
lZs <- rep(0, trials)
Vs <- rep(0, trials)
for (i in 1:trials) {
 out <- smc(mu, M, G, 10, 128) 
 lZs[i] <- out$log.Zs[9]
 Vs[i] <- VnN(out$as)
}
mean(exp(lZs-lZ)) # lack-of-bias: should be close to 1
```

```
## [1] 0.9941002
```

```r
var(exp(lZs-lZ)) # sample relative variance of Zs
```

```
## [1] 0.02712202
```

```r
mean(exp(2*(lZs-lZ))*Vs) # unbiased estimate of relative variance
```

```
## [1] 0.02755751
```

```r
mean(Vs) # biased estimate of relative variance
```

```
## [1] 0.02746865
```

---

![](smc-tutorial_files/figure-html/unnamed-chunk-9-1.png)

```
## [1] "VnN(out$as) = 0.032616368188082"
```

---

# Remarks

---

# Remarks

SMC is useful for approximating integrals with a particular structure.

It can be very computationally efficient in suitable scenarios, e.g. time-uniform convergence.

One can also transform complex, high-dimensional integrals into integrals with this structure.

Not covered:

- Particle MCMC: use of SMC within MCMC, e.g. to infer parameters.
 
 - Specific smoothing algorithms.
 
 - Refined theoretical results.

---

# Thanks!