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. 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. 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 $K(q) = /2.$ 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. SRN – The Bayesian Lasso by Park and Casella ‘The Bayesian Lasso’ paper by Park and Casella has been on my desk for a while. Thanks to the Sunday Reading Notes (SRN) series, I finally sit down to read and think about this paper. Lasso (Least absolute shrinkage and selection operator) is a regression method where comparing to ordinary least squares some estimators shrinks to zero while others are set to zero. Lasso is able to do both variable selection (done with step-wise regression before Lasso) and shrinkage to avoid overfitting (for example with ridge regression) at the same time. The objective function for Lasso is $\min_{\beta} (y-X\beta)^T(y-X\beta) + \lambda \sum_{j=1}^p |\beta_j|.$ In the 1996 paper, Tibshirani points out that Lasso estimates can be interpreted as posterior mode estimates (MAP) when we put i.i.d. Lapalace priors ($\pi(\beta) \propto \prod_{j=1}^p \exp(-\lambda |\beta_j|).$ When I took the Bayesian Data Analysis class (STAT 220 at Harvard), our textbook BDA3 contains an exercise on Lasso regularization and it claims that ‘a fully Bayesian lasso is the same model as lasso but assigning a hyperprior to $\lambda.$‘ But ‘The Bayesian Lasso’ paper sets up the hierarchy with a different route: The full conditional distributions on $\beta, \sigma^2$ and $\vec{\tau^2}$ have closed-form solutions and are provided in Section 2 of the paper. In Section 3, the authors discuss ways of choosing the Lasso parameter: 1) Empirical Bayes by marginal maximum likelihood: iteratively update $\lambda^{(k)} = \sqrt{\frac{2p}{\sum_{j=1}^p \mathbb{E}_{\lambda^{(k-1)}} [\tau_j^2|y] }}$; 2) hyper-priors on $\lambda^2$. The examples in this paper come from the diabetes dataset from the Tibshirani 1996 paper and can be found in the ‘lars‘ R package. On this dataset the results from a Bayesian lasso are very similar to a regular Lasso, as seen in Figure 1 & 2 from the paper or the plot below where I borrowed code from the Rdocumentation of the ‘monomvn‘ R package. In the plot above, we can see the shrinkage effect and the variable selection happening. Compared to ordinary least squares (green), the both the lasso estimates and samples from Bayesian lasso shrink towards zero (except for b.3) and the lasso estimate fixes some parameters (b.1,2,5,6,8,10) at zero. Comparing to Lasso where we do not have uncertainty estimates unless with bootstrap, the Bayesian lasso is providing these uncertainty estimates. References: Hastie, T., & Efron, B. (2013). lars: Least angle regression, lasso and forward stagewise, 2013. Gramacy, R. B., & Gramacy, M. R. B. (2013). Package ‘monomvn’. Park, T., & Casella, G. (2008). The bayesian lasso. Journal of the American Statistical Association103(482), 681-686. Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological), 267-288. SRN – The Hamming Ball Sampler by Titsias and Yau My Sunday Reading Notes (SRN) this semester will mostly be about Bayesian Computations. This week’s post is on The Hamming Ball Sampler proposed by Titsias and Yau.The hamming ball sampler is a MCMC algorithm for high-dimensional discrete-valued vectors or matrices. While reading this paper, I also found a blog post about it from Xi’an’s OG, which provided some high-level intuitions and background knowledge. The paper considers a state space model with discrete hidden space $X$ with parameters $\theta$ and observations $y.$ Factorial Hidden Markov Model (fHMM) is an example of such a model. In state space models, the complete data likelihood can be factorized with $p(y,X,\theta) = p(X,\theta) \prod_{i=1}^N p(y_i|X,\theta).$ Given some prior, we want to sample from the posterior distribution $X,\theta | y.$ When the dimension of $X$ is large, we would suffer from ‘the curse of dimensionality’. Using a Gibbs sampler, we can iteratively sample $\theta \sim \cdot | X,y$ and$\theta X \sim \cdot | \theta,y$. Because the dimension of$X$is high, we should also consider blocked Gibbs sampling on $X$ by for example updating one row (or column) of $X$ at a time. While this is conceptually straightforward and potentially also easy to implement, as the authors pointed out: Conditional sampling may lead to an inability to escape from local modes in the posterior distribution particularly if the elements of $X$ exhibit strong correlations with each other and together with $\theta$. The Hamming Ball Sampler (HBS) introduces an auxiliary variable $U$ that has the same dimension as the latent space $X$. The augmented joint probability can be factorized with as $p(y,X,\theta,U) = p(U|X) p(y,X,\theta).$ The conditional distribution $p(U|X)$ is chosen to be uniform over a neighborhood set $\mathcal{H}_m(X).$ This set $\mathcal{H}_m(X)$ is a Hamming Ball and it basically says that if $U,X$ are $K \times N$ matrices, then $U$ and $X$ can be different on at most $m$ positions among the $K$ rows. With the auxiliary variable $U$, the Hamming Ball Sampler alternate between the steps $U \gets p(U|X)$ and $(\theta, X) \gets p(\theta,X|U,y).$ The Hamming Ball Sampler is like slice-sampling in discrete spaces, and each Hamming Ball $\mathcal{H}_m(X)$ is a slice. Introducing the slice introduces random exploration, and makes it easier to escape from local modes. For the simplest example where $X$ is a $K \times N$ matrix and the hamming distance is defined the the number of different elements each column, if we set $m = K/2$ then we can potentially change all elements of $X$ in one update. But when $m$ is large, the algorithm complexity also increases. In this paper the authors provided several numerical examples comparing the Hamming Ball Sampler with block Gibbs Samplers. In the fHMM examples (Figure 4 in the paper) we can see that HBS with $m = 2$ or $3$ achieves having joint posterior density much faster than the block Gibbs Samplers. They also conclude that HB-2 is best balances computational time and sampling efficiency. Reference: Titsias, M. K., & Yau, C. (2017). The Hamming ball sampler. Journal of the American Statistical Association112(520), 1598-1611. Ghahramani, Z., & Jordan, M. I. (1996). Factorial hidden Markov models. In Advances in Neural Information Processing Systems (pp. 472-478). SRN – Should I follow the crowd? by Canamares and Castells from SIGIR’18 It’s time for Sunday Reading Notes again. This past week I have been reading the paper ‘Should I Follow the Crowd? A Probabilistic Analysis of the Effectiveness of Popularity in Recommender Systems‘ by Rocio Canamares and Pablo Castells. It won the best paper award at SIGIR 2018! Current state of the art recommender systems tend to recommend popular items to their users. As the paper points out in its abstract: The fundamental questions remains open though whether popularity is really a bias we should avoid or not, whether it could be a useful and reliable signal in recommendation, or it may be unfairly rewarded by the experimentation biases. This paper is concerned with un-personalized recommendations evaluated in off-line experiments. The dataset consists of rated items only and is randomly split into training and test. The ratings takes binary values (positive rating / negative rating). For every item, popularity is defined in terms of number of positive ratings, which is different from average rating. In order to make recommendations, we want to measure ‘relevance‘, which is a binary random variable and $res(i) = 1$ if the user likes item $i.$ From this set-up, we have $pop(i) \sim p(rel,rate|i)$ and $avg(i) \sim p(rel|rate,i).$ But we cannot directly measure $p(rel|i).$ The author uses the metric ‘expected precision’ as the metric for recommendations and emphasizes that true precision (cannot be measured from offline experiments) is different from observed precision (measured with rating from the test set). The authors find the optimal ranking function for both true precision and observed precision. Using relevance optimizes the true precision : $f(i) = p(rel|i)\frac{1-\rho p(rated|rel,i)}{1-\rho p(rated|i)}$ is the optimal ranking function for $\mathbb{E}[P@1|f]$, and as a result of experimental design the expected observed precision $\mathbb{E}[\hat{p}@1|f]$ is optimized by $\hat{f}(i) \sim p(rel|i)\frac{p(rated|rel,i)}{1-\rho p(rated|i)}.$ As we can see, popularity has advantage in optimizing the expected observed precision. Because customers have a tendency to rate items that they like, the rating information is not missing at random. To understand rated given relevance and item, the authors gives two simple conditions: conditional item independence and conditional relevance independence. Under both conditions, using $pop(i)$ optimizes the expected observed precision and using $avg(i)$ optimized the expected true precision. Because we can only measure the expected observed precision with offline experiments and because popularity has advantage optimizing $\mathbb{E}(\hat{P}@1)$, the current recommender systems that are based on offline evaluations favors popularity. Although not all recommender systems are designed the same way as described in this paper, I believe this elegant framework provides intuitions because the ‘popularity of popularity’. Acknowledging that a user can rate an item only if he or she has bought the item, the authors further consider ‘discovery bias’, because rating depends on all of relevance, discovery and item. This dependencies are characterized in Figure 2 from the paper. In a realistic senario, our data should fall into category 4 – ‘no assumption’. In this case the expected precision can be approximated with Monte Carlo integration. The key results and experiment are summaries in Table 1 and Figure 5 in the paper. Besides the probability based framework, what I really like about this paper is that the authors designed and collected a crowdsourced dataset that makes real data experiments of relevance-independent discovery and item-independent discovery possible. After reading going through the math and the experiments, I feel quite convinced that average rating may be in fact a better, safer, more robust signal than the number of positive ratings in terms of true achieved accuracy in most general situations. Although the metric $\mathbb{E}(P@1|R)$ seems too simple because essentially it is like we can only recommend one item to a user, it is a good and tractable measure to start with. The authors suggest that it empirically consistently generalize other metric such as nDCG@k. However I am not sure how much I would agree with this point, because nDCG@k cares much beyond the top ranked item. Overall I really like this paper and I think it touches many fundamental problems in popularity bias and provides enough mathematical clairty. I wonder if this paper suggests we should do more online experiments for recommender systems because true accuracy cannot be measured with offline experiments. I am also eager to see what the author has to say about temporal data splitting. Lastly I hope the authors talk about what we should do with the very common ‘5-star’ rating system. Reference: 1. Rocío Cañamares and Pablo Castells. 2018. Should I Follow the Crowd? A Prob- abilistic Analysis of the Effectiveness of Popularity in Recommender Systems. In Proceedings of ACM SIGIR ’18, July 8–12, 2018, Ann Arbor, MI, USA. ACM, NY, NY, USA, 10 pages. https://doi.org/10.1145/3209978.3210014 SRN – Objective Bayesian Two Sample Hypothesis Testing for Online Controlled Experiments by Alex Deng The Sunday Reading Notes (SRN) series is back! This Sunday’s topic is Bayesian A/B tests. Last month, I wrote a post on Bayesian continuous monitoring, commenting on the DSAA’16 paper Continuous Monitoring of A/B Tests without Pain by Deng et al. In that paper, the main take away is that peeking is not a problem in Bayesian A/B tests if we use genuine priors and prior odds, and Bayesian A/B tests controls False Discovery Rate (FDR). In this paper, the authors mention that we can learn an objective prior from empirical data, instead of using non-infomration priors in Bayesian A/B tests. Practical prior specification for Bayesian statistics is a big topic. Yesterday I met a machine learning engineer at a BBQ and told him that I have been thinking about objective prior learning and earning stopping for Bayesian A/B tests. His reaction was: don’t you know conjugate priors? don’t you know that priors will be overwhelmed by data very soon? My response was: yes you can use conjugate priors because it’s convenient. But I am not sure if we still want to assume we are in the asymptotic regime when we want to do an early stopping in A/B tests. While I convinced by him and myself with this argument, I am not sure whether this is the only reason prior specification matters in A/B tests. I’d be super happy if there can be some discussions in the comments! Going back to Deng’s paper, to use priors learned from past experiments for a new test, the only assumption we need to make is that we assume the prior behinds these experiments are the same. This would happen because the variants are designed by the same group of engineers and the treatment effort would have the same distribution. We assume that observations in control come i.i.d. from some distribution with mean $\tau_C$ and observations in treatment come i.i.d. from some distribution with mean $\tau_T.$ The null hypothesis is $H_0: \tau_C = \tau_T$ and alternative $H_1: \tau_T \not=\tau_C.$ Let $\sigma^2$ be the pooled variance, then the average treatment effect scaled by $\sigma$ is $\mu := \mathbb{E}(\delta) = \frac{\tau_T - \tau_C}{\sigma}.$ (For details of the t-test, see section 3.) We need to learn the prior odds $\frac{p}{1-p}$ and the prior density $\mu \sim \pi.$. The algorithm used to learn the parameters is Expectation-Maximization (EM). The M-step for $p$ is a straightforward one, and the M-step for $V$ is a generalized M-step using moment matching. (Details in Algorithm 1 in Section 3.2). Once the model is set-up, the procedure is straight-forward. What I find most interesting in this paper is Section 3.4 where the author discuss problems of the Beta-Bernoulli model, which is what people usually use for conversion rates or click-through-rates experiments. The first practical problem comes from a miss match between the experiment randomizing on the user-level while the model assumes page-views being iid. The second problem, which I honestly do not quite understand, is that the conjugate Beta prior cannot be a genuine prior. I wish the author had elaborated more on this point because conversion rate is tested so often in online experiments. When the author concludes the paper, he talks about what we should do if there are not thousands experiments from which we can learn an objective prior. He talks about the trade-off between the sample size of using all the experiments and the precision from using only relevant experiments. To this end, he suggests setting up a hierarchical model. I really like this paper and I have read it several times. Every time I read it, I learn a little more about Bayesian A/B tests. I like how it is a good blend of technical derivations, practical considerations and philosophical discussions.While reading the paper, I kind of feel that the author needs more space than what the page limit had given him. Because there are so many places where I hope he had elaborates on or given more details about. The DSAA’16 paper is a follow up on optional stopping for Bayesian A/B tests. I am personally very intrigued by the Beta-Bernoulli discussions and I also want to learn more about what the author has to say about multiple-testing! SRN – Winner’s Curse by Lee and Shen from Airbnb Online controlled experiments (A/B tests) has been my reading theme for this summer. This weekend I decide to read • Minyong R. Lee and Milan Shen. 2018. Winner’s Curse: Bias Estimation for Total E ects of Features in Online Controlled Experiments. In KDD ’18: The 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, August 19–23, 2018, London, United Kingdom. (link) This is the conference submission following the illuminating medium post • Milan Shen and Mining Lee. Selection Bias in Online Experimentation:Thinking through a method for the Winner’s Curse in A/B testing. (link) Today websites are testing many incremental changes to the webpage at the same time on large-scale experimentation platforms. The average treatment effect of each incremental change is unbiased estimated. Experiments with statistically significant improvements to some metric will be launched to the users. But the total effect of aggregating these incremental changes is over-estimated, if we simply add up the individual estimates. This is the winner’s curse the authors describe, and they quantified the bias in estimating the total effects in this paper. Suppose after running $N$ experiments, the set of chosen experiments is $A$ (which is a random set!) with true total effect $T_A$ and estimated total effect $S_A$. Then it can be shown that $\beta = \mathbb{E}\left[S_A - T_A\right] > 0$. The authors provides closed form solution for the bias in the simple case where each estimate follows a normal distribution with known standard deviation $X_i \sim \mathcal{N}(a_i,\sigma^2_i)$. Let’s say $b_i$ is some value choose by analysts running the experiments to select the set $A$. The authors show that $\beta = \sum_{i=1}^n \sigma_i \phi\left(\frac{\sigma_i b_i - a_i}{\sigma_i}\right)$. The point of this equation is that all the epxeriments contribute to the bias, not just those selected through experimentation, because the sum is over all the experiments! As the Medium post pointed out: If the set of experiments being aggregated A is fixed from the beginning, then no bias would be introduced. In the end, the bias exists because of the process of selecting the successful ones among many experiments we run, and we do this every day in a large scale experimentation platform. The authors also provide a bootstrap confidence interval for the bias corrected total effect estimator. I did not have time to go-through Section 5 – applications at Airbnb from the paper this time and I’d like to revisit it at another chance. I really like the authors’ way of formulating the solving this selective inference problem. Their de-biasing method requires very little assumptions and is straightforward to calculate. I have not given must thought to it but I am wondering how would a Bayesian approach this problem. Would that be setting up a hierarchical model by pulling all the incremental changes together? I would certainly meditate on this question during running or yoga! SRN – A near-optimal exploration-exploitation approach for assortment selection by Agrawal et al. I am very interested in exploration-exploitation trade-off. In a previous Sunday Reading Notes post, I discussed bayesian optimization for learning optimal hyper-parameters for machine learning algorithms, as an example for this trade-off. Today I study another exploration-exploitation algorithm for learning the best assortment: A Near-Optimal Exploration-Exploitation Approach for Assortment Selection by Agrawal et al. It has applications in online advertisement display and recommendation systems for e-commerce. If we take the recommendation system for e-commerce websites as an example. Suppose this website has $N$ products in total, and we can only recommend a subset $S$ to customers and this subset is at most size $K$. This subset $S$ is an assortment and $K$ is a display constraint. When an assortment $S$ is offered to a customer, the probability of the customer purchasing the $i$-th item in the assortment is $p_i(S) \propto \nu_i$ where $i=0,...,|S|$ and $p_o(S)$ is the probability of not purchasing any item from $S$. Further more if the price of each item in this assortment is $r_i$ then the expected revenue from assortment $S$ is $R(S) = \sum_{i\in S}$. If the probabilities are known, then we will have a static assortment optimization problem of $S^\star = \max_{s \in S} R(S)$. But if the preferences are not known or if they change over time, then we would want to learn user preference among assortments (explore) and recommend the optimal assortment to them (exploitation), and this is called a (dynamic) assortment optimization problem. In the optimization, we want to minimize the cumulative regret until time $T$ which is $reg(T) = \sum_{t=1}^T R(S^\star) - \mathbb{E}\left[R(S_t) \right]$. In this paper, the user preferences which are described through probabilities $p_i(S)$ are modeled with Multinomial Logistic model (MNL). With the MNL model, we have to estimate the model parameter $\nu_i$ with some multi-armed bandit algorithm to balance exploration and exploitation. Therefore the authors refer to this problem as bandit-MNL. The main contribution of this paper is Theorem 4.1, in which they give a regret bound of $O(\sqrt{NT} \log T + N \log^3T)$ with the proposed Algorithm 1. Section 4 gives the proof of the main theorem and it is in my opinion very well-written. Proof outline is given in Section 4.1 and the authors marked the 3 steps of the proof, which are detailed later in Sections 4.2 – 4.4. The exploration-exportation algorithm for bandit-MNL algorithm in Algorithm I iterations between learning the current best possible recommendation with using static assortment optimization methods. After recommending the calculated assortment to a customer, they observe purchasing decisions of customers until a no-purchase happens. Using information form user purchases, we can learn upper confidence bounds of model parameters$later \nu_{i}.\$ The UCB estimates are used to find the assortment for the next iteration. This process of going back between offering assortments and observing purchase decisions continuous until $T$ steps set before starting the experiment. In  Section 4.2, the authors should that the UCBs converge to the true value with high probability. (Be careful this is not a convergence in probability!).

In my opinion, this paper lies on the theoretical end of the spectrum and their is no simulation results or real-data examples accompanying the theoretical results. It would be nice to see some visualization of the cumulative regret with this algorithm and how it compare to the theoretical lower bound presented in Section 5.

I have two idea of how this problems can be extended to accommodate for complex applied problems. For examples, 1)  item prices are assumed to be known for now. It would be interesting extension for this paper to see if we can treat $r_i$‘s as random variables or something we can learn to optimize. 2) In this algorithm we have to assume customers are homogenous. Personalized assortment optimization would be an interesting direction to explore.

References:

• Agrawal, Shipra, et al. “A near-optimal exploration-exploitation approach for assortment selection.” Proceedings of the 2016 ACM Conference on Economics and Computation. ACM, 2016.