SRN – the magical 0.574

Hello 2021! This week, I keep reading about optimal scaling of Markov chain Monte Carlo algorithms and I move on to a new class of algorithms: MALA. This time, the magical number is 0.574. 

Hello 2021! Here is the first post in the Sunday Reading Notes series in the year of 2021. This week, I keep reading about optimal scaling of Markov chain Monte Carlo algorithms and I move on to a new class of algorithms: MALA.

Roberts and Rosenthal (1998, JRSSB) concerns the optimal scaling of Metropolis-Adjusted Langevin algorithm (MALA) for target distributions of the form {\pi^d(x) = \prod_{i=1}^d f(x_i)}. Unlike random walk Metropolis algorithm, in MALA, the proposal distribution uses the gradient of {\pi^d} at {x_t}:

\displaystyle x^{\star} = x_t + \sigma_n Z_{t+1} + \frac{\sigma^2}{2} \nabla \log (\pi^d(x_t)).

Here {\sigma_n} is the step variance and {Z_{t+1} \sim \mathcal{N}(0,I_d)} is a d-dimensional standard normal. Following this proposal, we perform an accept / reject step, so that the resulting Markov chain has the desired stationary distribution:
\displaystyle \alpha_d(x_t,x^{\star}) = \min \left\{ 1, \frac{\pi^d(x^{\star})}{\pi^d(x_t)} \frac{q_d(x^{\star},x_t)}{q_d(x_t,y_{t+1})}\right\}.

There are some smoothness assumptions on the log-density {g = \log f}: absolute value of the first 8 order derivatives are bounded, {g'} is Lipschitz, and all moments exist for {f}.

Theorem 3 (diffusion limit of first component)

Consider the chain {\{X\}} starting from stationarity and following the MALA algorithm with variance {\sigma_d^2 = l^2 n^{-1/3}} for some {l > 0}. Let {\{J\}} be a Poisson process with rate {n^{1/3}}, {\Gamma^n_t = X_{J_t}} and {U^d} is the first component of {\Gamma^n}, i.e. {U^d_t = X^d_{J_t,1}}. We must have, as {d \to \infty}, the process {U^d \to U}, where {U} is a Langevin diffusion defined by
\displaystyle dU_t = \sqrt{h(l)} dB_t + \frac{1}{2} h(l) \frac{d}{dx} \log (\pi_1(U_t))dt.

Here {h(l) = 2 l^2 \Phi(-Kl^3/2)} is the speed measure and {K = \sqrt{\mathbb{E} \left[\frac{5g'''(X)^3 - 3g''(X)}{48}\right]} > 0}.
The skeleton of the proof is similar to that of proving diffusion limit of random walk Metropolis. We will find sets {F_d^* \subseteq \mathbb{R}^d} such that {\mathbb{P}\left(\Gamma_s^d \in F_d^{\star} \ \forall \ 0 \le s \le t\right) \to 1} and on these sets the generators converges uniformly.

Theorem 4 (convergence of acceptance rate)
\displaystyle \lim_{d \to \infty} \alpha_d(l) = \alpha(l) where {a(l) = 2 \Phi(-Kl^3/2).}
As a result, when {h(l)} is maximized, the acceptance rate {\alpha(l) \approx 0.574}. Also the step variance should be tuned to be {\sigma_d^2 = (\hat{l}_d)^2 n^{-1/3}.}

MALA enjoys a faster convergence rate of order {n^{1/3}} compared to order {n} of RWM. But let’s also keep in mind that MALA requires an extra step to compute the gradients {\log \pi(x) = \sum_{i=1}^n g'(x)}, which takes {n} steps to compute.

Reference:

Roberts, G. O., & Rosenthal, J. S. (1998). Optimal scaling of discrete approximations to Langevin diffusions. Journal of the Royal Statistical Society: Series B (Statistical Methodology)60(1), 255-268.

SRN – about the magical 0.234 acceptance rate

Sunday Reading Notes series is back : Let’s understand the magical rule of ‘tuning your MH algorithm so that the acceptance rate is roughly 25%’ together!

‘Tune your MH algorithm so that the acceptance rate is roughly 25%’ has been general advice given to students in Bayesian statistics classes. It has been almost 4 years since I first read about it from the book Bayesian Data Analysis, but I never read the original paper where this result first appeared. This Christmas, I decided to read the paper ‘Weak Convergence and Optimal Scaling of Random Walk Metropolis Algorithms’ by Roberts, Gelman and Gilks and to resume by Sunday Reading Notes series with a short exposition of this paper.

In Roberts, Gelman and Gilk (1997), the authors obtain a weak convergence result for the sequence of algorithms targeting the sequence of distributions {\pi_d(x^d) = \prod_{i=1}^{d} f(x_i^d)} converging to a Langevin diffusion. The asymptotic optimal scaling problem becomes a matter optimizing the speed of the Langevin diffusion, and it is related to the asymptotic acceptance rate of proposed moves.

A one-sentence summary of the paper would be

if you have a d-dimensional target that is independent in each coordinate, then choose the step size of random walk kernel to be 2.38 / sqrt(d) or tune your acceptance rate to be around 1/4.

Unfortunately, in practice the ‘if’ condition is often overlooked and people are tuning the acceptance rate to be 0.25 as long as the proposal is random walk, no matter what the target distribution is. It has been 20 years since the publication of the 0.234 result and we are witnessing the use of MCMC algorithms on more complicated target distributions, for example parameter inference for state-space models. I feel that this is good time that we revisit and appreciate the classical results while re-educating ourselves on their limitations.

Reference:

Roberts, G. O., Gelman, A., & Gilks, W. R. (1997). Weak convergence and optimal scaling of random walk Metropolis algorithms. The annals of applied probability7(1), 110-120.

——–TECHNICAL EXPOSITION——-

Assumption 1 The marginal density of each component f is such that {f'/f} is Lipschitz continuous and

\displaystyle \mathbb{E}_f\left[\left(\frac{f'(X)}{f(X)}\right)^8\right] = M < \infty, \ \ \ \ \ (1)

\displaystyle \mathbb{E}_f\left[\left(\frac{f''(X)}{f(X)}\right)^4\right] < \infty. \ \ \ \ \ (2)

Roberts et al. (1997) considers random walk proposal {y^d - x^d \sim \mathcal{N}(0,\sigma_d I_d)} where {\sigma_d^2 = l^2 / (d-1).} We use {X^d = (X_0^d,X_1^d,\ldots)} to denote the Markov chain and define another Markov process {(Z^d)} with {Z_t^d = X_{[dt]}^d}, which is the speed-up version of {X^d}. Let {[a]} denote the floor of {a \in \mathbb{R}}. Define {U^d_t= X^d_{[dt],1}}, the first component of {X_{[dt]}^d = Z^d_t}.

Theorem 1 (diffusion limit of first component) Suppose {f} is positive and in {\mathbb{C}^2} and that (1)-(2) hold. Let {X_0^{\infty} = (X^1_{0,1},X^{2}_{0,2},\ldots)} be such that all components are distributed according to {f} and assume {X^{i}_{0,j} = X^{j}_{0,j}} for all {i \le j}. Then as {d \to \infty},
\displaystyle U^d \to U.

The {U_0 \sim f} and {U} satisfies the Langevin SDE
\displaystyle dU_t = (h(l))^{1/2}dB_t + h(l)\frac{f'(U_t)}{2f(U_t)}dt \ \ \ \ \ (3)

and
\displaystyle h(l) = 2 l^2 \Phi(-l\sqrt{I}/2)

with {\Phi} being the standard normal cdf and
\displaystyle I = \mathbb{E}_f\left[\left(f'(X)/ f(X)\right)^2\right].

Here {h(l)} is the speed measure of the diffusion process and the `most efficient’ asymptotic diffusion has the largest speed measure. {I} measures the `roughness’ of {f}.

Example 1 If {f} is normal, then {f(x) = (2\pi\sigma^2_f)^{-1/2}\exp(-x^2/(2\sigma_f^2)).}
\displaystyle I = \mathbb{E}_f\left[\left(f'(x) / f(x) \right)^2\right] = (\sigma_f)^{-4}\mathbb{E}_f\left[x^2\right] = 1/\sigma^2_f.

So when the target density {f} is normal, then the optimal value of {l} is scaled by {1 / \sqrt{I}}, which coincides with the standard deviation of {f}.

Proof: (of Theorem 1.1) This is a proof sketch. The strategy is to prove that the generator of {Z^n}, defined by

\displaystyle G_n V(x^n) = n \mathbb{E}\left[\left(V(Y^n) - V(x^n)\right) \left( 1 \wedge \frac{\pi_n(Y^n)}{\pi_n(x^n)}\right)\right].

converges to the generator of the limiting Langevin diffusion, defined by
\displaystyle GV(x) = h(l) \left[\frac{1}{2} V''(x) + \frac{1}{2} \frac{d}{dx}(\log f)(x) V'(x)\right].

Here the function {V} is a function of the first component only.
First define a set

\displaystyle F_d = \{|R_d(x_2,\ldots,x_d) - I| < d^{-1/8}\} \cap \{|S_d(x_2,\ldots,x_d) - I| < d^{-1/8}\},

where
\displaystyle R_d(x_2,\ldots,x_d) = (d-1)^{-1} \sum_{i=2}^d \left[(\log f(x_i))'\right]^2

and
\displaystyle S_d(x_2,\ldots,x_d) = - (d-1)^{-1} \sum_{i=2}^d \left[(\log f(x_i))''\right].

For fixed {t}, one can show that {\mathbb{P}\left(Z^d_s \in F_d , 0 \le s \le t\right)} goes to 1 as {d \to \infty}. On these sets {\{F_d\}}, we have
\displaystyle \sup_{x^d \in F_d} |G_d V(x^d) - G V(x_1)| \to 0 \quad \text{as } d \to \infty ,

which essentially says {G_d \to G}, because we have uniform convergence for vectors contained in a set of limiting probability 1.
\Box

Corollary 2 (heuristics for RWMH) Let
\displaystyle a_d(l) = \int \int \pi_d(x^d)\alpha(x^d,y^d)q_d(x^d,y^d)dx^d dy^d

be the average acceptance rate of the random walk MH in {d} dimensions.
We must have {lim_{d\to\infty} a_d(l) \to a(l)} where {a(l) = 2 \Phi(-l\sqrt{I}/2)}.
{h(l)} is maximized by {l = \hat{l} = 2.38 / \sqrt{I}} and {a(\hat{l}) = 0.23} and {h(\hat{l}) = 1.3 / I.}
The authors consider two extensions of the target density {\pi_d}, where the convergence and optimal scaling properties will still hold. The first extension concerns the case where {f_i}‘s are different, but there is an law of large numbers on these density functions. Another extension concerns the case {\pi_d(x^d) = f_1(x_1) \prod_{i=2}^{d} P(x_{i-1}, x_{i})}, with some conditions on {P}.

SRN – “Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets” by Cornish et al.

When faced with large datasets, standard MCMC methods have a cost of O(n) per step due to evaluating the log-likelihood for n data points. Can we make it faster?

My COVID-19 self-isolation plan includes trying to finish as many to-read articles as possible and to blog my progress on this blog as much as possible. This weekend, I read the paper “Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets” by Robert Cornish, Paul Vanetti, Alexandre Bouchard-Cȏté, George Deligiannidis, and Arnaud Doucet. This is an ICML 2019 paper and it is a dense one. It attempts to solve one problem. When faced with large datasets,  standard MCMC methods have a cost of O(n) per step due to evaluating the log-likelihood for n data points. Can we make it faster? In a typical MH step (with symmetric proposals), we accept the proposal with probability \alpha(\theta,\theta') = \min(1, \frac{\pi(\theta')}{\pi(\theta)}). This is typically implemented with

log_alpha <- log_post(theta_proposal) - log_post(theta_current);
if (log(runif(1)) < log_alpha ){
    theta_current <- theta_proposal;
}

The authors realize that it is wasteful having to evaluate the log-likelihood at all data points before rejecting the proposal. Their solution, Salable Metropolis-Hastings (SMH), is based on a combination of three ideas: (1) factorization of posterior density (trivial factorization is 1 + n factors), (2) fast simulation of rejection using thinning of a Poisson point process, and (3) posterior concentration around mode when n is large. 

I first get excited about this paper seeing the fast simulation of Bernoulli random variables (in Algorithm 1) because there is a very similar problem on simulating a Poisson process with inhomogenous rates in Stat 210, for which I was a Teaching Fellow (TF). I am very happy to see this result being useful in cutting-edge research. The fast simulation relies on access to a lower bound for each factor in the posterior distribution and the tighter this lower bound the more efficient this implementation. Suppose the target distribution has m factors, then once we pay a once-off O(m) cost, each acceptance step can be implemented in O(upper bound of negative-log-posterior) number of steps on average. 

Another highlight of the paper is the smart application of the Bernstein-von Mises (BvM) phenomenon, that which states that posterior distribution concentrates around the mode with rate 1/\sqrt{n} as n goes to infinity. BvM was an important topic in Stat 213, which is another class that I TFed. I have seen BvM used (1) to select step-size in random-walk MH, (2) to initialize an MH chain, and (3) to motivate Laplace approximation of the posterior distribution. In this paper, it motivates a Taylor expansion of the potential function near posterior mode, which subsequently leads to the factorization behind the SMH kernel. The SMH kernel leaves the target distribution invariant, and, if not mistaken by me, this property explains the use of “exact” in the title. More importantly, since the acceptance probability in SMH is factorizable, we can implement the acceptance step with the aforementioned fast simulation, provided that log-density can be differentiated enough times. Intuitively, the higher the differentiability, the closer the Taylor expansion, the tighter the bound for acceptance rate and the fewer the number of evaluations we need. In fact, as later justified with Theorem 3.1, the average computational cost decays geometrically with respect to the order of Taylor expansion. With a second-order Taylor expansion, which requires 3rd-order differentiability, the expected computational cost behaves like O(1/\sqrt{n}). Hooray! I am always a big fan of “the bigger the problem, the easier the solution”. 

I learn a lot from this paper, especially from the opening paragraph of Section 3 and Section 3.2. But there is a question that I keep asking myself: if n is very large and we have an estimate of posterior mode that is O(1/\sqrt{n}) distance from the true mode, how long does it take for SMH to be better than that declaring the posterior is Gaussian by invoking BvM and sleeping on the result?

SRN – Further and stronger analogy between sampling and optimization by Arnak Dalalyan

Lately, I have been pondering the connection between Monte Carlo based sampling methods and optimation methods. It brings me to read a paper that I have been wanting to read in detail long ago. In this week’s Sunday Reading Notes series on Phyllis with Data, I discuss the paper “Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent” by Arnak Dalalyan.

Lately, I have been pondering the connection between Monte Carlo based sampling methods and optimation methods. It brings me to read a paper that I have been wanting to read in detail long ago. In this week’s Sunday Reading Notes series on Phyllis with Data, I discuss the paper “Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent” by Arnak Dalalyan. 

This paper focuses on the Langevin Monte Carlo algorithm, which I believe is also called the Unadjusted Langevin algorithm (ULA) in contrast to the Metropolis adjusted Langevin algorithm (MALA). Using the Wasserstein distance as the metric, this paper establishes an upper bound of the Wasserstein distance, more precisely it shows that one needs O(\frac{d}{\epsilon^2}\log(\frac{d}{\epsilon})) number of iterations to reach the precision level . Considering the close connection between ULA and gradient descent algorithm for optimization, the author proves an upper bound for the L2 distance for gradient descent by cleverly utilizing a tempering argument (details in section 3 of the paper). The well-known result for optimization is that it takes O(\log(\frac{1}{\epsilon})) iterations to reach precision in gradient descent. Intuitively sampling should be a more difficult task than optimization because it requires an exploration of the whole parameter space, which can be of high-dimension. The bounds in this paper confirm this intuition. More importantly, it points out how these two problems are “continuously” connected (as the temperature converges to 0) and that this connection naturally leads to the deduction of the optimization bound from the sampling bound. 

Admittedly this is not a new paper as it was first arxived in 2016, there have been many papers tackling similar problems in sampling, for example using the KL-divergence (Cheng and Bartleet 2019) or extending the results to MALA (Dwivedi et al. 2019). When I first picked up this paper two years ago as a second-year Ph.D. student, I was intimidated by the proof, which in fact was only 6-pages long. This time, as I sat down and went through the proof line by line, I found it quite accessible and elegant. I have been equipped with the mathematical knowledge required to understand the proof in college. This time, I was defeated by my own fear. For me, this paper is not only a delightful read, but also a wake-up call to trust myself. 

SRN – A Framework for Adaptive MCMC Targeting Multimodal Distribution by Pompe et al.

After a very long break, I come back to the Sunday Reading Notes and write about the paper “A Frame for Adaptive MCMC Targeting Multimodal Distributions” by Emilia Pompe et al. This paper proposes a Monte Carlo method to sample from multimodal distributions.

After a very long break, I come back to the Sunday Reading Notes and write about the paper “A Frame for Adaptive MCMC Targeting Multimodal Distributions” by Emilia Pompe et al. This paper proposes a Monte Carlo method to sample from multimodal distributions. The “divide and conquer” idea separates the sampling task into (1) identifying the modes, (2) sample efficiently within modes and (3) moving between the modes. While the paper emphasizes more heavily on addressing task (2) and (3) and establishing ergodicity , it provides a framework that unites all three tasks.

The main algorithm in this paper is Jumping Adaptive Multimodal Sampler (JAMS). It has a burn-in algorithm that uses optimization-based procedure to identify modes, to merge them and to estimate covariance matrices around the modes. After task (1) is solved by the burn-in algorithm, JAMS alternates between the later two tasks with local moves that samples around a mode and jump moves that facilitate crossing the low density regions. The local moves are adaptive in the sense that it relies on local kernels with parameters that are learned on the fly. The jump moves propose a new mode and choose a new point around the new mode. 

This paper proves ergodicity of JAMS by considering it as a case of a general class of Auxiliary Variable Adaptive MCMC. It also provides extensive simulation studies on multivariate Gaussian (d = 200) , mixtures of t-distributions and bananas and a Bayesian model for sensor network localisation. The results are compared against Adaptive Parallel Tempering (APT). In all the cases tested, JAMS out performs APT in terms of root mean squared error scaled by dimension of the problem. JAMS is also more robust to initialization in term of recovering all the modes. 

JAMS identities all the unknown sensor locations. But posterior samples using APT misses the location around (0.5,0.9) in the bottom-left panel. (Figure 6 from Pompe et al.)

I usually think of Monte Carlo integration and optimization as very similar tasks and even neglect the area of optimization. This paper is a wake up call for me, as I believe the current JAMS algorithm relies on a successful mode identifying stage. The discussions in Section 6 also suggest mode finding as the bottle neck. 

SRN – Bayesian Calibration of Microsimulation Models by Rutter et al.

The Sunday Reading Notes paper for this week is ‘Bayesian Calibration of Microsimulation Models’ by Carolyn Rutter, Diana Miglioretti and James Savarino. This is a 2009 JASA Applications and Case Studies paper.

According to a 2012 Review paper by Rutter et al,

Microsimulation models (MSMs) for health outcomes simulate individual event histories associated with key components of a disease process; these simulated life histories can be aggregated to estimate population-level effects of treatment on disease outcomes and the comparative effectiveness of treatments. Although MSMs are used to address a wide range of research questions, methodological improvements in MSM approaches have been slowed by the lack of communication among modelers. In addition, there are few resources to guide individuals who may wish to use MSM projections to inform decisions.

In this paper, the authors propose a Bayesian method to calibrate microsimulations models, using Markov Chain Monte Carlo. The case study in this paper is the history of colorectal cancer (CRC). In this paper, the authors assume all CRCs arise from an adenoma and the history of CRC consists of four components: 1) adenoma risk, 2) adenoma growth, 3) transition from adenoma to preclinical cancer and 4) transition from preclinical cancel to clinical cancer. These four components are not observed directly and the calibration data consists of prevalence of adenomas and preclinical cancers and the size and/or number of adenomas from many (independent) studies from different years and about different subpopulations and using different colonoscopy methods.

The authors use \theta to denote MSM parameters and these parameters can be separated from K components with independent priors: \pi(\theta) = \prod_{i=1}^k \pi_i(\theta_i). The calibration data come from N independent sources and each follows some distribution  y_j|\theta \sim f_j(g_j(\theta)) parametrized by some unknown functions of the MSM parameters. The likelihood is therefore f(Y | \theta) = f(Y|g(\theta) )= \prod_{j=1}^n f_j(y_j|g_j(\theta)).

What makes the calibration problem hard is the unknown function g(\theta). Suppose we want to simulate from the posterior distribution of \theta|Y using a Metropolis-Hastings(MH) algorithm, we need to know r(\theta,\theta^*) = \frac{\pi(\theta)f(y|g(\theta)}{\pi(\theta^*)f(y|g(\theta^*)}. But because g is unknown, the MH step cannot be performed.

So the authors propose an approximate MH algorithm that includes a step to estimate $g(\theta)$ for both the current value \theta and the proposed value \theta^*. To me this feels like an ‘EM’ step: simulate m copies of \tilde{y}_{ji} from f_j(y_j| g_j(\theta)) and calculate the MLE \hat{g}_j(\theta). With this approximation, the resulting transition probability function \hat{\alpha}(\theta,\theta^*)for the Metropolis-within-Gibbs step on \theta_i is based on \frac{\pi(\theta_i)}{\pi(\theta_i^*)}\cdot \frac{\prod_{j=1}^N f_j(y_j|\hat{g}_j(\theta^*_i, \theta_{(-i)})}{\prod_{j=1}^n f_j(y_j|\hat{g}_j(\theta))}. In the Appendix, the authors prove that this approximation satisfies the detailed-balanced condition in the limit of m goes to infinity.

I think this paper provides an interesting example of how to incorporate data from multiple sources. As the authors point out,

how closely the model should calibrate to observed data is unclear, especially when calibration data are variable and many provide conflicting interest. […] It depends on how modelers trade-off concerns about possibly overparameterizing and overfitting calibration data relative to the importance of exactly replicating observed or expected results.

 

References:

  • Rutter, C. M., Miglioretti, D. L., & Savarino, J. E. (2009). Bayesian calibration of microsimulation models. Journal of the American Statistical Association104(488), 1338-1350.
  • Rutter, C. M., Zaslavsky, A. M., & Feuer, E. J. (2011). Dynamic microsimulation models for health outcomes: a review. Medical Decision Making31(1), 10-18.

SRN – A Geometric Interpretation of the Metropolis-Hastings Algorithm by Billera and Diaconis

Coming back to the Sunday Reading Notes, this week I discuss the paper ‘A Geometric Interpretation of the Metropolis-Hastings Algorithm’ by Louis J. Billera and Persi Diaconis from Statistical Science. This paper is suggested to me by Joe Blitzstein.

In Section 4 of ‘Informed proposals for local MCMC in discrete spaces’ by Giacomo Zanella (see my SRN Part I and II), Zanella mentions that the Metropolis-Hasting acceptance probability function(APF) \min\left(1,\frac{\pi(y)p(x,y)}{\pi(x)p(y,x)}\right) is not the only APF that makes the resulting kernel \pi-reversible as long as detailed-balance is satisfied. This comes first as a ‘surprise’ to me as I have never seen another APF in practice. But very quickly I realize that this fact was mentioned in both Stat 213 & Stat 220 at Harvard and I have read about it from Section 5.3 – ‘Why Does the Metropolis Algorithm Work?‘ of ‘Monte Carlo Strategies in Scientific Computing‘ by Jun S. Liu. Unfortunately, I did not pay enough attention. Joe suggested this article to me after I posted on Facebook about being upset with not knowing such a basic fact.

In this Billera and Diaconis paper, the authors focus on the finite state space case X and considers the MH kernel as the projection of stochastic matrices (row sums are all 1 and all entries are non-negative, denoted by\mathcal{s}(X)) onto the set of \pi-reversible Markov chains (stochastic matrices that satisfy detailed balance \pi(x)M(x,y) = \pi(y)M(y,x), denoted by R(\pi)). If we introduce a metric on the stochastic matrices: d(K,K') = \sum_{x} \sum_{x\not=y} \pi(x) |K(x,y)-K'(x,y)|.

The key result in this paper is Theorem 1. The authors prove that the Metropolis maps M := M(K)(x,y) = \min\left( K(x,y), \frac{\pi(y}{\pi(x)}K(y,x)\right) minimizes the distance d from the proposal kernel K to R(\pi). This means that M(K) is the unique closest element in R(\pi) that is coordinate-wise smaller than K on its off-diagonal entries. So M is in a sense the closest reversible kernel to the original kernel K.

I think this geometric interpretation offers great intuition about how the MH algorithm works: we start with a kernel K and change it to another kernel with stationary distribution \pi. And the change must occur as follows:

from x, choose y from K(x,y) and decide to accept x or stay at y; this last choice may be stochastic with acceptance probabilty F(x,y) \in [0,1]. This gives the new chain with transition probabilities: K(x,y) F(x,y), x \not =y$. The diagonal entries are changed so that each row sums to 1.

Indeed the above procedure describes how the MH algorithm works. If we insist on \pi-reversibility, we must have 0 \leq 0 \leq \min(1,R(x,y) where R(x,y) =  \frac{\pi(y)K(y,x)}{\pi(x)K(x,y)}. So the MH choice of APF is one that maximizes the chance of moving from x to y. The resulting MH kernel M has the largest spectral gap (1 – second largest eigenvalue) and by Peksun’s theorem must have the minimum asymptotic variance estimating additive functionals.

In Remark 3.2, the authors point out if we consider only APF that are functions of R(x,y), then the function must satisfy g(x) = x g(1/x) which is the characteristic of balancing functions in Zanella’s ‘informed proposals’ paper.

This paper allows me to study Metropolis-Hastings algorithm from another angle and review facts I have neglected in my coursework.

References:

  • Billera, L. J., & Diaconis, P. (2001). A geometric interpretation of the Metropolis-Hastings algorithm. Statistical Science, 335-339.
  • Zanella, G. (2017). Informed proposals for local MCMC in discrete spaces. arXiv preprint arXiv:1711.07424.
  • Liu, J. S. (2008). Monte Carlo strategies in scientific computing. Springer Science & Business Media.

SRN – Informed proposals for local MCMC in discrete spaces by Zanella (Part II)

This is a super delayed Sunday Reading Notes post and today I come back to finish discussing the informed proposals paper by Giacomo Zanella. In Part I of my discussions, I reviewed point-wise informed proposals and locally-informed criterion. In Part II, I hope to focus on asymptotical optimality of locally informed proposals and their simulation studies.

The author uses Peskun ordering to compare efficiency of MH schemes. He deduces that ‘locally-balanced proposals are asymptotically optimal in terms of Peskun ordering’ as dimensionality of the underlying state space increases. Conditions for asymptotic optimality are indeed mild (Proposition 1) for the three illustrative examples: 1) independent binary components, 2) weighted permutation and 3) Ising model.

In Section 4, Zanella points out a connection between the balancing function and acceptance probability function (APF) for MH algorithms, which I find very interesting. He also shows that the optimal proposal for independent binary variables is Barker choice g(t) = \frac{t}{1+t}. The proof goes by finding the limiting continuous-time process of the MH chain and finding the optimal g for the limiting process.

The simulation studies use the illustrative examples: weighted permutations and Ising models. The comparisons are in terms of 1) acceptance rate, 2) number of successful flips per computation time and 3) mixing of some summary statistics. The second criterion concerns the trade-off between computational cost (of calculating the informed proposal) and statistical accuracy (by producing efficient moves). For simple target distributions (such as Uniform), using locally-balanced proposals does not bring much benefits but it achieves a much higher number of flips per unit time for more complicated and ‘rough’ targets. See second row of Figure 1 of the paper.

Screen Shot 2018-10-14 at 5.37.28 PM

I find it interesting that globally-balanced proposals (aka ‘naively-informed’ or g(t) = t) are extremely sensitive to initialization. Looking at the effective sample size (ESS) per time, the chains from GB has much more stable behavior if initialized from stationarity. See GB v.s. GB( station.) in Figure 2. But in Figure 5, this phenomenon does not show up for Ising models: initializing from stationarity does not yield ESS performance comparable to that of LBs.

Screen Shot 2018-10-14 at 5.44.23 PM.png

In the simulation studies section, the author emphasizes the cost-vs-efficiency trade-off, which I find very important. I feel I have ignored this aspect of designing MCMC algorithms and should think more about it in my future studies. The author indicates that ‘computations required to sample from locally-balanced proposals are trivially parallelizable’. This is also something very interesting to me and I hope to learn more about multi-core computations during my PhD.

In his discussions, the author makes reference to Multiple-Try Metropolis. The connection between LB proposals and MT proposals is not entirely obvious to me, but I intuitively agree with the author’s comment that the weight function in MT serves a very similar purpose as the balancing function term g\left(\frac{\pi(y)}{\pi(x)}\right) in LB.

Side note:
From my downloaded version of the paper, the transition kernel of the first equation on Page 1 is wrong, because the Q(x,dy) terms needs to be multiplied by its acceptance probability a(x,y).

References:

  •  Zanella, G. (2017). Informed proposals for local MCMC in discrete spaces. arXiv preprint arXiv:1711.07424.
  • Liu, J. S., Liang, F., & Wong, W. H. (2000). The multiple-try method and local optimization in Metropolis sampling. Journal of the American Statistical Association95(449), 121-134.

SRN – Informed proposals for local MCMC in discrete spaces by Zanella (Part I)

This week I am reading ‘Informed proposals for local MCMC in discrete spaces‘ by Giacomo Zanella. This paper is about designing MCMC algorithms for discrete-values high-dimensional parameters, and the goal is similar to the papers discussed in previous posts (Hamming ball sampler & auxiliary-variable HMC). I decide to split the Sunday Reading Notes on this paper into two parts, because I find many interesting ideas in this paper.

In this paper, Zanella come up with locally-balanced proposals. Suppose \pi(x) is the target density and K_{\sigma}(x,dy) is an uninformed proposal. We assume that as \sigma \to 0 the kernel K_{\sigma}(x,dy) converges to the delta measure. Zanella seeks to modify this uninformed proposal so that it incorporates information about the target \pi and is biased towards areas with higher density. An example of locally-balanced proposals is Q_{\sqrt{\pi}} (x,dy) = \frac{\sqrt{\pi(y) }K_{\sigma}(x,dy)}{(\sqrt{\pi} * K_{\sigma})(x)}. This kernel is reversible with respect to \sqrt{\pi(x)}(\sqrt{\pi} * K_{\sigma})(x), which converges to \pi(x)dx as x \to 0. [Note the normalizing constatn is the convolution \sqrt{\pi(x)}* K_{\sigma} = \int \sqrt{\pi(y)} K_{\sigma}(x,dy)].]

More generally, Zanella considers a class of pointwise informed proposals that has the structure Q_{g,\sigma} = \frac{1}{Z_{g}}\cdot g\left(\frac{\pi(y)}{\pi(x)}\right) K_{\sigma}(x,dy). It is suggested that the function g satisfy g(t) = t g(1/t).

I will save the discussion on locally-balanced proposals and Peskun optimality to Part II. In this part, I want to discuss Section 5: Connection to MALA and gradient-based MCMC. In continuous space, the point-wise informed proposal Q_{g,\sigma} would be infeasible to sample from because of the term g\left(\frac{\pi(y)}{\pi(x)}\right) . If we take a first-order Taylor expansion, we would have Q_{g,\sigma}^{(1)} \propto g \left( \exp ( \nabla  \log \pi(x) (y-x)) \right) K_{\sigma}(x,dy). If we choose g(t) = \sqrt{t} and K_{\sigma}(x,\cdot) =N(x,\sigma^2), this is the MALA proposal.

I find this connection very interesting, although I do not have a good intuition about where this connection comes from. One way to explain it is that gradient-based MCMC in continuous space is using local information to design informed proposals. In the conclusions, the author mentions that this connection should improve robustness of gradient-based MCMC schemes and help with parameter tuning.

References:(x)

  •  Zanella, G. (2017). Informed proposals for local MCMC in discrete spaces. arXiv preprint arXiv:1711.07424.

SRN – Auxiliary-variable Exact HMC samplers for Binary Distributions by Pakman and Paninski

It’s time for Sunday Reading Notes again! This week I am discussing another computational statistics paper: ‘Auxiliary-variable Exact Hamiltonian Monte Carlo Samplers for Binary Distributions’ by Ari Pakman and Liam Paninski from Columbia University. This paper is published at NIPS 2013.

In the previous Hamming Ball sampler SRN post, the algorithm uses data augmentation to sample from discrete distributions. In this week’s paper, the goal is to sample from generic binary distributions with data augmentation into continuous variables.

Let’s say we want to sample from the distribution p(s) defined over s \in \{\pm 1 \}^d given an un-normalized density f(s). The authors propose augmenting with a continuous variable y \in \mathbb{R}^d with joint density p(s,y) = p(s)p(y|s) where p(y|s) is a density we can design but it must satisfy s_i = \mathrm{sgn}(y_i) for all i = 1,\cdots,d. The marginal distribution of $y$ is $p(y) = p(s)p(y|s)$ as a result of this constraint. It turns out that at this point we transformed a d-dimentional binary  problem on s into a d-dimensional continuous problem on y.

To sample from y, the authors suggest using Hamiltonian Monte Carlo, the potential energy is U(y) = - \log p(y) = -\log p(y|s) - log f(s) and the kinetic energy terms is K(q) = <q,q>/2. The HMC sampler involves simulating a trajectory of y that preserves the Hamiltonian H(y,q) = U(y) + K(q) and typically leap-frog simulation is used. With the constraint in p(y|s), the potential function is defined only piece-wise and we need to be careful when the trajectory crosses regions. o this end, the authors insist we choose p(y|s) such that \log p(y|s) is quadratic, so that the trajectory is deterministic and approximations methods are not necessary.

Because U(y) has a jump at y_i = 0, the value of momentum q_i should change when we cross boundaries. This is, in my opinion, the most interesting part of the paper. Suppose at time t_j we have y_j = 0, then a change in trajectory must happen and let’s say the momentum just before and after y_j = 0 are q_j(t_j^-) and q_j(t_j^+). Conservation of energy says we must have q_j^2(t_j^+)/2+ U(y_j = 0, s_j = -1) = q_j^2(t_j^-)/ 2+ U(y_j = 0, s_j = +1) if y_j <0 before the y_j = 0. From this equation, if q_j^2(t_j^+)>0 then we continue the trajectory with q_j(t_j^+) =q_j(t_j^-); however, if q_j^2(t_j^+)<0  then the particle is reflected from a wall at y_j = 0 and the trajectory gets reflected with q_j(t_j^+) = - q_j(t_j^-).

The two augmentations mentioned in this paper are Gaussian augmentation and exponential augmentation. Both results in quadratic log likelihood. The Gaussian augmentation is very interesting because there is a fixed order that each coordinate y_j reaches zero and the successive hits occur at t_j + n \pi. The authors makes an observation that:

Interestingly, the rate at which wall y_j = 0 is crossed coincides with the acceptance rate in a Metropolis algorithm that samples uniformly a value for i and makes a proposal of flipping the binary variable s_i.

To me this is a sanity check rather than a surprise because each coordinate hits the boundary the same number of times and making a decision to continue or to bounce back in y_j is the same as deciding whether we should flip the sign of s_i. But I think the authors give a very help comment pointing out that although the acceptance probability is the same,  the method proposed is still different from Metropolis because

in HMC the order in which the walls are hit is fixed given the initial velocity, and the values of q_i^2 at successive hits of y_i = 0 within the same iteration are not independent.

What’s interesting for the exponential augmentation method is that

particles moves away faster from areas of lower probability.

This is certainly a nice feature to have so that the sample mixes well.

In the simulation examples, the authors compared Gaussian HMC and Metropolis on 1d and 2d ising models and showed that:

  1. ‘the HMC sampler explores faster the samples space once chain has reached equilibrium distribution.’
  2. ‘the HMC sampler is faster in reaching the equilibrium distribution.’

I think the take away from this paper is the continuous data augmentation to sample discrete variables and their dealing with piece-wise defined potential function.

 

Reference:

  • Pakman, A., & Paninski, L. (2013). Auxiliary-variable exact Hamiltonian Monte Carlo samplers for binary distributions. In Advances in neural information processing systems (pp. 2490-2498).
  •  Neal, R. M. (2011). MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo2(11), 2.