The use of a single pseudo-sample in approximate Bayesian computation

Abstract

We analyze the computational efficiency of approximate Bayesian computation (ABC), which approximates a likelihood function by drawing pseudo-samples from the associated model. For the rejection sampling version of ABC, it is known that multiple pseudo-samples cannot substantially increase (and can substantially decrease) the efficiency of the algorithm as compared to employing a high-variance estimate based on a single pseudo-sample. We show that this conclusion also holds for a Markov chain Monte Carlo version of ABC, implying that it is unnecessary to tune the number of pseudo-samples used in ABC-MCMC. This conclusion is in contrast to particle MCMC methods, for which increasing the number of particles can provide large gains in computational efficiency.

Appendices

Proofs

Proof of Theorem 3

Denote by $H_{2,\alpha }$ the transition kernel of the pseudo-marginal algorithm with proposal kernel q, target marginal distribution $\mu$, and estimator $T_{2,x,\alpha }$ of the unnormalized target. If we denote by $\{ (X_{t}^{(1)}, T_{t}^{(1)}) \}_{t \in \mathbb {N}}$ and $\{ (X_{t}^{(2)}, T_{t}^{(2)}) \}_{t \in \mathbb {N}}$ the Markov chains driven by the kernels $H_{2,\alpha }$ and $\alpha \mathrm {I} + (1-\alpha ) H_2$ respectively, then $\{ X_{t}^{(1)} \}_{t \in \mathbb {N}}$ and $\{ X_{t}^{(2)} \}_{t \in \mathbb {N}}$ have the same distribution.

If $T_{1,x} \le _{cx} T_{2,x,\alpha }$ then by Theorem 3 of Andrieu and Vihola (2014),

$$\begin{aligned} v(f,H_{1})&\le v(f, H_{2,\alpha }) \end{aligned}$$

(6)

for any $f \in L^2(\mu )$. We also have

$$\begin{aligned} v(f, H_{2,\alpha })\le & {} \frac{1}{1-\alpha } v(f,H_{2}) + \frac{\alpha }{1 - \alpha } v(f) \nonumber \\\le & {} \frac{1+ \alpha }{1 - \alpha } v(f, H_{2}), \end{aligned}$$

(7)

where the first inequality follows from Corollary 1 of Latuszyński and Roberts (2013) and the second follows from the fact that $H_{2}$ has nonnegative spectrum. Combining this with (6) yields the desired result.$\square$

Proof of Proposition 4

For any $M\ge 1$, let $T_{M,\varvec{\theta }}$ be the estimator $\hat{\pi }_{K,M}(\varvec{\theta }|\varvec{y}_{\mathrm {obs}})$ of the target $\pi _K$, so that $T_{M,\varvec{\theta },\alpha }$ is $T_{M,\varvec{\theta }}$ handicapped by $\alpha$ as defined in (4) of the main document. To obtain (5) of the main document, by Theorem 3 it is sufficient to take $\alpha = 1-\frac{1}{M}$ and show that $T_{1,\varvec{\theta }} \le _{cx} T_{M,\varvec{\theta },\alpha }$. By Proposition 2.2 of Leskelä and Vihola (2014), it is furthermore sufficient to show that, for all $c \in \mathbb {R}$,

$$\begin{aligned} \mathbb {E}[\vert T_{1,\varvec{\theta }} - c \vert ] \le \mathbb {E}[\vert T_{M,\varvec{\theta },\alpha } - c \vert ]. \end{aligned}$$

(8)

Let $\mathrm {Bin}(n,\psi )$ denote the binomial distribution with n trials and success probability $\psi$. For a given point $\varvec{\theta }\in \Theta$, let $\tau = \tau (\varvec{\theta }) \equiv \mathbb {P}[T_{1,\varvec{\theta }} \ne 0] = \int \mathbf 1 _{\{\Vert \eta (\varvec{y}_{\mathrm {obs}}) - \eta (\varvec{y})\Vert < \epsilon \}} p(\varvec{y}|\varvec{\theta }) d \varvec{y}$. Noting that $\frac{T_{1,\varvec{\theta }}}{\pi (\varvec{\theta })} \in \{0, 1 \}$, we may then write $T_{1,\varvec{\theta }}$ and $T_{M,\varvec{\theta },\alpha }$ as the following mixtures

$$\begin{aligned}&\frac{T_{1,\varvec{\theta }}}{\pi (\varvec{\theta })} \mathop {=}\limits ^{D} \mathrm {Bin}(1,\tau ),\nonumber \\&\frac{T_{M,\varvec{\theta },\alpha }}{ \pi (\varvec{\theta })} \mathop {=}\limits ^{D} \frac{M-1}{M} \delta _{0} + \frac{1}{M} \mathrm {Bin}(M,\tau ), \end{aligned}$$

where $\delta _0$ is the unit point mass at zero. Denote $T_{1,\varvec{\theta }}' = \frac{T_{1,\varvec{\theta }}}{\pi (\varvec{\theta })}$ and $T_{M,\varvec{\theta },\alpha }'=\frac{T_{M,\varvec{\theta },\alpha }}{\pi (\varvec{\theta })}$. We will check condition (8) for $T_{1,\varvec{\theta }}', T_{M,\varvec{\theta },\alpha }'$ and $0 \le c \le 1$, then separately for $c < 0$ and $c > 1$. For $0 \le c \le 1$, we compute:

$$\begin{aligned}&\mathbb {E}\left[ \vert T_{M,\varvec{\theta },\alpha }' - c \vert \right] \nonumber \\&\quad =\left( 1 - \frac{1}{M} \right) c + \frac{1}{M} (1 - \tau )^{M} c \nonumber \\&\qquad \; +\,\frac{1}{M} \left( \sum _{j=1}^{M} \frac{M!}{j! (M-j)!} \tau ^{j} (1-\tau )^{M-j} \left( j - c \right) \right) \nonumber \\&\quad = \tau + c \left( 1 - \frac{2}{M}\left( 1 - (1-\tau )^{M} \right) \right) \ge \tau + c \left( 1 - 2\tau \right) \nonumber \\&\quad = \mathbb {E}\left[ \vert T_{1,\varvec{\theta }}' - c \vert \right] . \end{aligned}$$

(9)

For $c < 0$, we have

$$\begin{aligned} \mathbb {E}\left[ \vert T_{M,\varvec{\theta },\alpha }' - c \vert \right]= & {} \mathbb {E}\left[ T_{M,\varvec{\theta },\alpha }'\right] - c = \mathbb {E}\left[ T_{1,\varvec{\theta }}'\right] - c \\= & {} \mathbb {E}\left[ \vert T_{1,\varvec{\theta }}' - c \vert \right] , \end{aligned}$$

and the analogous calculation gives the same conclusion for $c \ge M$. Finally, For $1< c < M$, note

$$\begin{aligned}&\mathbb {E}\left[ \vert T_{1,\varvec{\theta }}' - 1 \vert \right] \le \mathbb {E}\left[ \vert T_{M,\varvec{\theta },\alpha }' - 1 \vert \right] , \nonumber \\&\mathbb {E}\left[ \vert T_{1,\varvec{\theta }}' - M \vert \right] = \mathbb {E}\left[ \vert T_{M,\varvec{\theta },\alpha }' - M \vert \right] . \end{aligned}$$

(10)

Also, the functions $f_1(c) \equiv \mathbb {E}[\vert T_{1,\varvec{\theta }}' - c \vert ]$ and $f_2(c) \equiv \mathbb {E}[\vert T_{M,\varvec{\theta },\alpha }' - c \vert ]$ are continuous, convex and piecewise linear. For $c \ge 1$, they satisfy

$$\begin{aligned} \frac{d}{dc} f_{1}(c) = 1 \ge \frac{d}{dc} f_{2}(c) \end{aligned}$$

(11)

where the derivative of $f_{2}$ exists. Combining inequalities (10) and (11), we conclude that

$$\begin{aligned} \mathbb {E}[\vert T_{1,\varvec{\theta }}' - c \vert ] \le \mathbb {E}[\vert T_{M,\varvec{\theta },\alpha }' - c \vert ] \end{aligned}$$

for all $1 < c < M$. Thus we have verified (8) and the proposition follows.$\square$

Analysis of an alternative ABC-MCMC method

We give a result analogous to Corollary 2 for the version of ABC-MCMC proposed in Wilkinson (2013), given in Algorithm 3 below. The constant c can be any value satisfying $c \ge \sup _{\varvec{y}} K(\eta (\varvec{y}_{\mathrm {obs}}) - \eta (\varvec{y}))$.

Lemma 6 compares Algorithm 3 (call its transition kernel $\tilde{Q}$) to $Q_\infty$.

Lemma 6

For any $f\in L^2(\pi _K)$ we have $v(f, \tilde{Q}) \ge v( f, Q_{\infty })$.

Proof

Both $\tilde{Q}$ and $Q_\infty$ have stationary density $\pi _K$, so by Theorem 4 of Tierney (1998), it suffices to show that $Q_\infty (\varvec{\theta },A\backslash \{\varvec{\theta }\}) \ge \tilde{Q}(\varvec{\theta },A\backslash \{\varvec{\theta }\})$ for all $\varvec{\theta }\in \Theta$ and measurable $A \subset \Theta$. Since $\tilde{Q}$ and $Q_\infty$ use the same proposal density q, it is furthermore sufficient to show that for every $\varvec{\theta }^{(t-1)}$ and $\varvec{\theta }'$, the acceptance probability of $Q_\infty$ is at least as large as that of $\tilde{Q}$. Since $p(\cdot |\varvec{\theta })$ is a probability density,

$$\begin{aligned}&\int K(\eta (\varvec{y}_{\mathrm {obs}})-\eta (\varvec{y})) p(\varvec{y}|\varvec{\theta }) d\varvec{y}\nonumber \\ {}&\quad \le \sup _{\varvec{y}} K(\eta (\varvec{y}_{\mathrm {obs}})- \eta (\varvec{y})) \qquad \forall \varvec{\theta }\in \Theta . \end{aligned}$$

(12)

So the acceptance probability of $\tilde{Q}$, marginalizing over $\varvec{y}_{\varvec{\theta }'}$, is

$$\begin{aligned} a_{\mathrm {ABC}}= & {} \int r(\varvec{\theta }^{(t-1)}, \varvec{\theta }' | \varvec{y}) p(\varvec{y}| \varvec{\theta }') d \varvec{y}\nonumber \\\le & {} \frac{\int K(\eta (\varvec{y}_{\mathrm {obs}})-\eta (\varvec{y})) p(\varvec{y}| \varvec{\theta }') d\varvec{y}}{\sup _{\varvec{y}} K(\eta (\varvec{y}_{\mathrm {obs}})-\eta (\varvec{y}))}\nonumber \\&\times \min \left\{ 1,\frac{\pi (\varvec{\theta }') q(\varvec{\theta }^{(t-1)} \vert \varvec{\theta }')}{\pi (\varvec{\theta }^{(t-1)}) q(\varvec{\theta }' \vert \varvec{\theta }^{(t-1)})}\right\} \nonumber \\\le & {} \min \left\{ 1, \frac{\int K(\eta (\varvec{y}_{\mathrm {obs}})-\eta (\varvec{y})) p(\varvec{y}| \varvec{\theta }') d\varvec{y}}{\sup _{\varvec{y}} K(\eta (\varvec{y}_{\mathrm {obs}})-\eta (\varvec{y}))}\right. \nonumber \\&\left. \times \left( \frac{\pi (\varvec{\theta }') q(\varvec{\theta }^{(t-1)} \vert \varvec{\theta }')}{\pi (\varvec{\theta }^{(t-1)}) q(\varvec{\theta }' \vert \varvec{\theta }^{(t-1)})}\right) \right\} . \end{aligned}$$

(13)

The acceptance probability of the Metropolis-Hastings algorithm is

$$\begin{aligned} a_{\mathrm {MH}}= & {} \min \left\{ 1, \frac{\int K(\eta (\varvec{y}_{\mathrm {obs}})-\eta (\varvec{y})) p(\varvec{y}| \varvec{\theta }') d\varvec{y}}{\int K(\eta (\varvec{y}_{\mathrm {obs}})-\eta (\varvec{y})) p(\varvec{y}| \varvec{\theta }^{(t-1)}) d\varvec{y}}\right. \nonumber \\&\times \left. \left( \frac{\pi (\varvec{\theta }') q(\varvec{\theta }^{(t-1)} \vert \varvec{\theta }')}{\pi (\varvec{\theta }^{(t-1)}) q(\varvec{\theta }' \vert \varvec{\theta }^{(t-1)})}\right) \right\} . \end{aligned}$$

(14)

Using (12), (13), and (14), $a_{\mathrm {ABC}}/a_{\mathrm {MH}} \le 1$.$\square$