Using Empirical Bayes to approximate posteriors for large “black box” estimators



Many machine studying purposes have some type of regression at their core, so understanding large-scale regression techniques is necessary. However doing this may be laborious, for causes not usually encountered in issues with smaller or much less vital regression techniques. On this put up, we describe the challenges posed by one downside — tips on how to get approximate posteriors — and an method that we now have discovered helpful.

Suppose we need to estimate the variety of instances an advert will probably be clicked, or whether or not a consumer is on the lookout for photographs, or the time a consumer will spend watching a video. All these issues may be phrased as large-scale regressions. Now we have a set of things with covariates (i.e. predictive options) and responses (i.e. noticed labels), and for every merchandise, we need to estimate a parameter that governs the response. This downside is often solved by coaching an enormous regression system, like a penalized GLM, neural web, or random forest.

We regularly use massive regressions to make automated choices. Within the examples above, we would use our estimates to decide on advertisements, resolve whether or not to indicate a consumer photographs, or work out which movies to advocate. These choices are sometimes business-critical, so it’s important for information scientists to grasp and enhance the regressions that inform them.

The scale and significance of those techniques makes this tough. First, techniques may be theoretically intractable. Even techniques based mostly on well-understood strategies often have customized tweaks to scale or match the issue higher. Second, necessary techniques evolve shortly, since persons are continuously attempting to enhance them. Which means any understanding of the system can grow to be old-fashioned.

Thus, the information scientist’s job is to work with an enormous black field that may change at any time. This appears not possible. We’ve had some success, nonetheless, by ignoring techniques’ inside construction and analyzing their predictions. Under, we talk about a option to get approximate posteriors that’s based mostly on this method.

Needed: approximate posteriors

Suppose we now have $I$ objects, every with response $y_i$ and covariates $x_i$ and unknown parameter $theta_i$, and the responses come from a identified parametric household $f_theta$. We need to estimate the $theta$s. A black field regression offers us a degree estimate of $theta_i$ for every merchandise, $t_i = t(x_i)$, that could be a operate of the covariates. Nonetheless, we frequently need a posterior for $theta_i$, not only a level estimate.
Posteriors are helpful to grasp the system, measure accuracy, and make higher choices. However commonest machine studying strategies don’t give posteriors, and plenty of don’t have express chance fashions. Strategies just like the Poisson bootstrap may also help us measure the variability of $t$, however don’t give us posteriors both, notably since good high-dimensional estimators aren’t unbiased.

Actual posteriors are laborious to get, however we will get approximate ones by extending calibration, an ordinary option to post-process regression predictions. There are just a few completely different strategies for calibration, however all are based mostly on the identical thought: as a substitute of utilizing $t$, estimate and use $E(theta | t)$. Calibration fixes mixture bias, which lets us use strategies which can be environment friendly however biased. Calibration additionally scales simply and doesn’t rely upon the small print of the system producing $t$. Which means it might deal with the big, altering techniques we now have to take care of.

Calibration estimates $E(theta | t)$. We suggest estimating the distribution of $theta | t$ and utilizing this to approximate the distribution of $theta | x$. Determine 1 illustrates the concept by plotting $theta$ vs. $t$. Calibration adjusts our estimate, as a operate of $t$, by shifting from the $y = t$ line to the conditional imply curve $E(theta | t)$.

Fig 1: Extraordinary and second order calibration. Every panel plots $theta$ on the y-axis in opposition to $t$ on the x-axis. Extraordinary calibration, left, adjusts our estimate away from $t$ (the blue line, $y = t$) to $E(theta|t)$ (the purple line). Second order calibration, proper, makes use of $y$ to estimate the distribution of $theta | t$ (represented by the pink shade strip).

The proposed methodology, which we name second order calibration, goes additional and estimates the distribution of $theta$ round that curve. We don’t observe the true $theta$s, so we will’t estimate the distribution of $theta | t$ straight. As an alternative, we use an Empirical Bayes method — take the noticed $y | t$ distribution and “deconvolve” utilizing the identified household $f_theta$ to deduce the distribution of $theta | t$.

Extra exactly, our mannequin is that $theta$ is drawn from a previous that is determined by $t$, then $y$ comes from some identified parametric household $f_theta$. In our mannequin, $theta$ doesn’t rely straight on $x$ — all the data in $x$ is captured in $t$.
theta | t sim G_t
y | theta, t sim f_theta

We estimate $G_t$ by normal Empirical Bayes strategies, and use it to seek out fascinating posterior portions similar to $mathrm{var}(theta | t)$, $E(theta | t, y)$ and $mathrm{var}(theta | t, y)$.

Empirical Bayes posteriors in 4 simple steps

We’ll clarify our methodology in additional element by making use of it to advert click-through-rate (CTR) estimation. Right here, our objects are query-ad pairs. For instance, one merchandise is perhaps a specific advert on the question [flowers]. We observe how usually every pair happens (impressions $N_i$) and is clicked (clicks $y_i$). Our mannequin is that every pair has a real CTR $theta_i$, and that given impressions and CTR, clicks are Poisson: $y_i|theta_i sim mathrm{Pois}(N_i theta_i)$ (we will doubtlessly get extra clicks than impressions). A machine studying system produces an estimated CTR $t_i$ for every query-ad pair. We talk about this instance intimately in our paper [1] on which this put up is predicated. For extra on advert CTR estimation, confer with [2].

Our methodology has 4 steps:

  1. Bin by $t$.
  2. Estimate the prior distribution of $theta | t$ in every bin utilizing parametric Empirical Bayes. 
  3. Clean throughout bins and examine suits. 
  4. Calculate posterior portions of curiosity. 

Step 1: Binning

We finally need to estimate the $theta | t$ distributions for every $t$. Since $t$ is steady, essentially the most pure method can be to make use of a steady mannequin. Nonetheless, we’ve discovered it simpler, and no much less efficient, to easily bin our query-ad pairs by $t$, in order that $t$ is roughly fixed in every bin. This implies the following becoming steps in every bin can ignore $t$ and may be accomplished in parallel. The computational financial savings from this method make the issue rather more tractable — we will sort out tens of billions of query-ad pairs in just a few hours with a modest variety of machines.

We may additionally bin by extra low-dimensional variables which can be notably necessary. For instance, we would bin by $t$ and nation. Extra curiously, we will bin by $t$ and a few estimate of $t$’s variability. That is one thing we will get from many machine studying techniques, both utilizing an inside estimate like a Hessian, or a black field estimate as supplied by the Poisson bootstrap.

Step 2: Estimation

We now work inside a bin and assume t is successfully fixed. Our mannequin thus reduces to
theta sim G
y | theta sim mathrm{Pois}(N theta)

Fig 2: An instance of a histogram of $y/N$ and its fitted Gamma prior $G$ (purple). The histogram and prior are usually not alleged to agree: the histogram is the purple distribution,
plus Poisson noise.

Empirical Bayes strategies attempt to estimate $G$ utilizing a set of responses from a mannequin just like the one above. Determine 2 illustrates the concept. The histogram of empirical CTRs ($y_i / N_i$) is unfold out with a spike at 0. However a lot of that dispersion and spikiness is Poisson noise, so the true CTR distribution could look completely different. Empirical Bayes strategies discover a prior such that once we add Poisson noise, we match the distribution of our noticed information. In Determine 2, the purple line reveals a Gamma prior that results in match. For an introduction to Empirical Bayes, see the paper [3] by Brad Efron (with extra in his ebook [4]).

How precisely ought to we mannequin $G$? It seems that if we have an interest within the first few posterior moments $theta | t$ and $theta | t, y$, the small print of the $G$ mannequin don’t actually matter. If $G$ suits the marginal distribution of $y$ effectively and has cheap tail conduct, our second estimates will probably be cheap (Part Four of [1] has extra particulars on this). We modeled $G$ utilizing a Gamma distribution. In our utility, this easy mannequin was fast to suit utilizing most probability, and labored in addition to a extra versatile method based mostly on mixtures of Gamma distributions (gridded for stability).

Step 3: Smoothing and Diagnostics

The subsequent step is to easy and examine our fitted priors. The draw back of becoming every bin individually is our fitted priors $G_t$ aren’t constrained to differ properly with $t$. Typically this isn’t an issue — with sufficient information in every bin, the $G_t$ should behave effectively (unusual conduct may additionally level to issues upstream). To verify, and to enhance our priors utilizing info from close by bins, we will easy. Determine 2 reveals how we smoothed the prior imply and variance.

Fig 3: Unsmoothed (dots) and smoothed (purple traces) $E G(theta | t)$ and $mathrm{var} G(theta | t)$ (prime and backside, respectively). The previous is fairly easy, however the latter advantages from smoothing.

We additionally want to ensure our mannequin suits the information. A easy, needed examine is that the estimated priors result in marginal distributions that match the information. This isn’t ample, because it doesn’t examine our assumed mannequin for the response, $f_theta$. For instance, we is perhaps mistaken to imagine clicks are Poisson, however a Poisson mannequin may nonetheless match our information with sufficient freedom within the prior.

One option to examine $f_theta$ is to assemble take a look at information and examine whether or not the mannequin suits the connection between coaching and take a look at information. This assessments the mannequin’s skill to differentiate what’s widespread for every merchandise between the 2 information units (the underlying $theta$) and what’s completely different (the draw from $f_theta$).

Determine Four reveals the outcomes of such a take a look at. We used our mannequin and coaching information to assemble predictive distributions for take a look at information, and checked the place the take a look at information fell in these distributions. If our mannequin had been excellent, the take a look at quantiles can be uniform (we jittered the quantiles to account for the discreteness of the information). Our precise quantiles had been near, however not precisely, uniform, indicating however not excellent match.

Fig 4: Histograms for the predictive distribution quantiles inside every bin, coloured by bin. The densities are principally flat, indicating that our mannequin suits effectively, however there may be some misfit on the suitable for increased bins. The spikes are artifactual — they seem as a result of we plot the histograms at discrete factors, so we see the total vary of the curves at these factors and never in between.

Step 4: Posterior portions

Armed with our validated priors, we will lastly calculate fascinating posterior portions. We are able to’t belief our estimates of each amount we is perhaps keen on. Normal concept on Empirical Bayes and deconvolution says that we will’t estimate “spiky” capabilities like $P_G(theta=0|t,y)$ with out extra assumptions. However we will belief our estimates of the primary few posterior moments, similar to $mathrm{var}(theta | t)$, $E(theta | t, y)$ and $mathrm{var}(theta | t, y)$. Every is doubtlessly helpful.

$mathrm{var}(theta | t)$ is a option to assess accuracy. By evaluating $mathrm{var}(theta | t)$ to $mathrm{var}(theta)$, we will see what fraction of the true variation in $theta$ is captured by $t$.

$E(theta | t, y)$ balances memorization and generalization to get an improved estimate of $theta$. For instance, suppose we now have a query-ad pair with a single impression ($N = 1$). The empirical CTR for that merchandise will probably be both Zero or 1, and can inform us virtually nothing concerning the true CTR of that advert. In that case, the very best we will do is depend on international info and use $E(theta | t)$ as our estimate. Conversely, suppose we now have a query-ad pair with thousands and thousands of impressions. The empirical CTR tells us rather more concerning the advert’s true CTR than t does, so we must always memorize and use $y/N$ as our estimate. The posterior imply easily balances memorization and generalization, relying on the quantity of data from every supply. Doing this in our calibration as a substitute of simply counting on $t$ may be extra correct, and might result in system simplifications. For instance, we may use a comparatively coarse generalization mannequin for $t$ and depend on calibration to memorize item-specific info.

$mathrm{var}(theta | t, y)$ estimates the accuracy of $E(theta | t, y)$ — this tells us how a lot we find out about every merchandise. These estimates may be helpful to make risk-adjusted choices and explore-exploit trade-offs, or to seek out conditions the place the underlying regression methodology is especially good or unhealthy.


Second order calibration, like abnormal calibration, is meant to be simple and helpful, not complete or optimum, and it shares a few of abnormal calibration’s limitations. Each strategies may be mistaken for slices of the information whereas being appropriate on common, since they solely use the covariate info via $t$.

Second order calibration additionally has necessary extra limitations. We have to know $f_theta$ to work backwards from the distribution of $y | t$ to that of $theta | t$. It’s laborious to check this assumption straight — a helpful oblique examine is to ensure the predictive distributions are correct.

Extra subtly, second order calibration depends on repeated measurement, in a method that abnormal calibration doesn’t. We should outline a unit of statement, and this may be surprisingly difficult. In our CTR instance, we outlined the unit of statement to be the query-ad pair. However CTR is determined by different components, just like the consumer, and the UI the advert is proven in, and the time of day. It’s not possible to refine our unit of statement by all these potential components and nonetheless have repeated measurements for any particular person unit. Extraordinary calibration and the underlying regression system should not have to resolve this downside, since they’ll simply give predictions on the finest-grained stage.

We don’t have a totally satisfying reply for this challenge. In observe, we now have gotten good outcomes by normalizing responses for the first-order results of things ignored by our unit definition. Extra principled options could possibly be to account for dispersion inside a unit, or to suit some type of multi-way random results mannequin.


Second order calibration is a pleasant instance of how coping with massive, advanced, altering regression techniques requires a special method. As an alternative of attempting to give you a principled, appropriate reply for a specific system, we tried to seek out one thing approximate that might work fairly effectively with out figuring out a lot concerning the underlying system. The ensuing methodology has clear limitations, however is scalable, maintainable, and correct sufficient to be helpful.


[1] Omkar Muralidharan, Amir Najmi “Second Order Calibration: A Simple Way To Get Approximate Posteriors”,  Technical Report, Google, 2015.

[2] H. Brendan McMahan et al, “Ad Click Prediction: a View from the Trenches”, Proceedings of the 19th ACM SIGKDD Worldwide Convention on Data Discovery and Information Mining (KDD), 2013.

[3] Bradley Efron, “Robbins, Empirical Bayes, and Microarrays”, Technical Report, 2003.

[4] Bradley Efron, “Large-Scale Inference:  Empirical Bayes Methods for Estimation, Testing, and Prediction”, Cambridge College Press, 2013.


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