31:T87d,# Data Externalities The goal of this writeup is to summarize the model from Acemoglu et al. (2022) in a way that makes it relevant to the class: relating it to the topics we've discussed thus far and then using it to usher in the unit on **Policy Evaluation**! Note two possibly confusing changes we've made here relative to the paper: > ### *Notation Changes* > > *(You can skip this if you haven't read the paper! This is just for ease of "converting" between their notation and ours)* > > * Here we use ${tex`y_i`} in place of the paper's ${tex`x_i`}, since in an ML setting we typically use ${tex`y_i`} to represent the ${tex`i`}th **label** that we're hoping to predict > * Here we use ${tex`x_i`} in place of the paper's ${tex`s_i`}, to represent **feature values** (which also follows the "standard" ML setup) > > The model remains the same, however! These changes just modify the _terminology_, to make what we're talking about feel a bit more natural within a Machine Learning setup. In other words, the above changes in mathematical notation really just correspond to these terminological changes: > > * What economists call a **"type"** we'll also call a **"label"** in the ML sense (either way, it's something the platform hopes to predict) > * What economists call a **"signal"** we'll also call a **"feature"** in the ML sense (either way, it's some piece of information that the platform can use to predict the type/label) So, to start off, we consider a scenario with **${tex`n`} users** indexed by ${tex`\mathcal{V} = \{1, 2, \ldots, n\}`}, who are all on some **platform** (say, a social media platform) that has an interest in collecting user data. Each of these users ${tex`i`} has a **type** value ${tex`y_i`} that they might be concerned about a company learning, since the company can derive additional revenue if they learn this value. This notion of a hidden ["type"](https://www.youtube.com/watch?v=wzMrK-aGCug) comes up again and again in policy evaluation (and general econometric) frameworks, so it's worth pausing to consider what this really means.32:T1045,## What is a "Type" in an Economic Modeling Framework? As an example that will help for interpreting this model, you can imagine ${tex`y_i`} as a ${tex`z`}-score (below we'll specify that the ${tex`y_i`} values are standard-normally distributed) representing ${tex`i`}'s **systolic** (immediately after a heartbeat) or **diastolic** (immediately before a heartbeat) blood pressure. > ### *Running Example: Prediction of Hypertension* > > In the following chart from [Baylor Medical Center](https://www.bcm.edu/bodycomplab/BPappZjs/BPvAgeAPPz.html), for example, we can see how ${tex`i`}'s placement relative to others on this scale determines whether they are diagnostically classified into "normal", "elevated", or "hypertension" categories: > > ![Baylor Medical Center blood pressure chart.](${await FileAttachment("image.png").url()}) > > Although the 2014 Affordable Care Act in theory prevents health insurance companies from denying coverage to individuals with "pre-existing conditions", in practice (as with many other examples of law-versus-reality we've looked at in class) these companies easily find ways around this provision, e.g. through claim denial, with the usual kinds of disproportionate effects on the poor and on people of color that we've come to expect in the US (in this case, for example, lower-income patients are 43% more likely to have claims denied). Notice, however, that a given user ${tex`i`}'s concern **isn't necessarily a concern about a harm!** For example, patients who **do** have blood pressure levels near the median will likely view revelation of their data as a **benefit**, since it reveals them to be "healthy" and thus may enable them to secure **lower** health insurance premiums. One key goal of the model here is to capture **both** the case where the user is concerned to **give** their data to the company (as an even simpler example than the blood pressure case, for example, revelation of a user's movie preferences may lead to better movie recommendations from Netflix) — **and** the case where the user is concerned to **hide** the data — for example, since it may lead their healthcare claims to be denied. Therefore, as a general modeling strategy you'll see again and again as we enter this **Policy Evaluation** unit, information that may be relevant to evaluation of particular policies in particular cases (e.g., how a policy might "balance" the harms to those with high blood pressure against the benefits to those with low blood pressure, after these harms/benefits have been identified, operationalized, and then measured in some way) is not "built into" the model itself but is instead "encoded" in each user's **utility function** ${tex`u_i(\cdot)`}. This is what makes "utility" such an important concept throughout this unit! Since most policies are not universally "good" or "bad"^[1], but are instead "good" for some stakeholders and "bad" for others, we need to be able to **incorporate** this **variety** of effects **across** different individuals into our models, which is exactly what utility functions are for! In the lecture on the effects of GDPR, for example, our discussion could have been framed like: * If the **utility function representing the preferences of individual customers** is decreasing in the number of trackers monitoring them across different websites (all else held constant), then the implementation of GDPR probably **increased** the utility of these agents, whereas * If the **utility function representing the preferences of startup entrepreneurs** is decreasing in the cost required to comply with legal privacy regulations, then the implementation of GDPR probably **decreased** the utility of these agents. The task of policy evaluation therefore, at least for the sake of this class, is to **model** these utility functions and their interrelationships in order to assess the **differential impact** that **changes** to a policy like GDPR might have on different stakeholders affected by these changes.33:Tb4c,## The Distribution of User *Labels* We let ${tex`Y_i`} be the random variable whose realizations are the ${tex`y_i`} values, and model the **distribution of labels** among users via the assumption that ${tex.block` \mathbf{Y} = (Y_1, \ldots, Y_n) \sim \mathcal{N}(0, \Sigma) `} which is the standard *multivariate* normal distribution you know and love from e.g. DSAN 5100. The fact that **all** of the labels are generated from a **single** distribution is important here: it is exactly what enables us to model **correlations** between the attributes of different users, and thus capture the core dynamic of data externalities: the fact that **learning information about others besides a user ${tex`i`}** can **reveal information about ${tex`i`}**. The role played by the covariance matrix ${tex`\Sigma`} in this model is thus that: * The greater a particular entry, the covariance value ${tex`\sigma_{ij}`}, * The more that learning information about ${tex`j`} also reveals information about ${tex`i`}. The following plot visualizes this distribution for **two users** (whose labels are jointly distributed as a two-dimensional Normal distribution), where the **correlation coefficient ${tex`\rho`}** determines whether **learning one user's label lets us make a better "guess" for other user's label**. The fact that ${tex`y_1`} and ${tex`y_2`} are jointly Normally distributed means that the vertical "slice" in the plot on the left corresponds exactly to the distribution of ${tex`[Y_2 \mid Y_1 = y_1]`} plotted in the bottom right. [This alternative visualization from distill.pub lets you do the opposite: drag the distribution and see the resulting covariance matrix!] * Click the **"Set to Zero"** button to set ${tex`\rho = 0`}. In this case, there is **no correlation** between the two users' labels, which means that **learning the value of ${tex`y_1`} provides no information about ${tex`y_2`}**. You can see the result by moving the slider in the right panel: no matter what particular value of ${tex`y_1`} we observe, the distribution of ${tex`Y_2`} remains the same. * Next click the **"Set Positive"** button to set ${tex`\rho = 0.75`}. Unlike the previous case, there is now a **positive** correlation between the two users' labels, which means that **learning the value of ${tex`y_1`} provides information about ${tex`y_2`}**! You can see this by moving the slider on the right panel: learning that ${tex`y_1`} is **positive**, for example, tells us that ${tex`Y_2`} is also more likely to be a positive number. * Finally, click the **"Set Negative"** button to set ${tex`\rho = -0.75`}. This case is just the "mirror image" of the previous case: here learning that ${tex`y_1`} is **positive** tells us that ${tex`Y_2`} is more likely to be **negative**, and vice-versa.34:T10bc,## The Distribution of *Features* We're "halfway there", if you think about what elements go into a Machine Learning task. We outlined how the **labels** work in the previous section, so what's left is to describe how the **features** are set up. Since the core goal of this model is to capture the dynamics of **cross-individual prediction of labels** by the platform (predicting ${tex`y_i`} even if user ${tex`i`} refuses to share their information), the features are a bit of an afterthought in a sense: the idea is just that we'd like to model features as "some pieces of information that help the platform predict labels". Thus, the feature setup is simple: Each user has a **single** feature value ${tex`x_i`}, that is modeled as a **"noisy version"** of the label, where the noise has a standard normal distribution: ${tex.block` X_i \sim \mathcal{N}(Y_i, 1) `} If you recall from DSAN 5100, this means that e.g. about 68% of the realized feature values ${tex`x_i`} will be within 1 unit of the label ${tex`y_i`}, 95% will be within 2 units of that label value, and 99.7% will be within 3 units of that label value. This relationship between the feature values and the labels is one reason why the platform wants to obtain these feature values, and thus may offer money to users to obtain it. This leads to the user's decision around whether to provide or withhold this feature from the platform, described in the next section. Importantly, though, the value of this data for the platform **extends beyond just the single user ${tex`i`}!** While the above expression for the distribution of ${tex`X_i`} makes clear the "first-order" relationship between ${tex`X_i`} and ${tex`Y_i`}, there is **also** a **"second-order"** relationship between ${tex`X_i`} and ${tex`X_j`} for any other user ${tex`j \neq i`}. This boils down to the following two-step "chain" of inferences that the platform can make: * **First,** obtaining the value ${tex`x_i`} improves prediction of ${tex`Y_i`} specifically, the "direct" (or "first-order") effect of obtaining this information. * Since we measure prediction performance in terms of Mean Squared Error (as described below), this corresponds to a decrease in the ${tex`i`}th term within the MSE equation: ${tex`(\hat{y}_i - y_i)^2`}. * Before acquiring ${tex`x_i`}, the platform's best guess for ${tex`y_i`} would just be the population average label of 0. * After acquiring ${tex`x_i`}, the platform can make a better guess (for example by using ${tex`x_i`} and other acquired features to estimate a regression model) * **Second,** obtaining the value ${tex`x_i`} also improves prediction of **all other labels** ${tex`Y_j`} for ${tex`j \neq i`}, which we'll call the "indirect" (or "second-order") effect of obtaining this information. * The intuition here is that we've not only learned a piece of information about the specific user ${tex`i`}, but also a piece of information about the **relationship between features and labels more broadly!** * Mathematically, this second step just follows from the fact that ${tex`Y_i`} and ${tex`Y_j`} are correlated for any users ${tex`i`} and ${tex`j`} given the setup in the previous section: if learning that ${tex`X_i = x_i`} improves prediction of ${tex`Y_i`}, then in turn we can leverage this improved "knowledge" of ${tex`y_i`} to further predict some other label ${tex`y_j`} better than we could before, **even if we never acquire ${tex`j`}'s feature ${tex`x_j`}!** Following the same logic as the explanation of ${tex`\rho`} in the previous section, for example, consider a platform starting with **no** knowledge of the two users' features or labels. The platform then pays user ${tex`i`} to acquire their feature value, and learns that > **User ${tex`i`}'s feature value is ${tex`X_i = 4`}** Given the distribution we defined above for the feature values, ${tex`X_i \sim \mathcal{N}(Y_i, 1)`}, this means it is **extremely unlikely** that user ${tex`i`}'s **label** is negative, for example: since values generated by a standard Normal distribution will lie within the interval ${tex`[-3,3]`} about 99.7% of the time, the likelihood that the label ${tex`y_i`} is 0 but the feature is a value that high is tiny (well below 0.3%). Thus, if the company also knows that 35:Ta59,## The Platform and User Objectives This will look familiar from... all your other data science courses! The platform's aim is to derive optimal **predictions** from all available user data. In the paper (and in the setup we use across most courses), since each user's type ${tex`x_i`} is a **continuous** variable, we have a **regression problem** where we can capture the accuracy of a learned prediction function via the **Mean Squared Error (MSE)**: ${tex.block` \pi_{\text{Platform}} = -\text{MSE}(\hat{y}, \mathbf{p}) = -\sum_{i = 1}^{n}\left( \mathbb{E}\left[(\hat{y}(\mathbf{X}) - Y_i)^2\right] - \sigma^2_i + p_i \right) `} The only important new thing here, relative to the ML models you've looked at in previous courses, is the ${tex`p_i`} term within the sum: this is the **amount of money** that the platform pays to user ${tex`i`} in order to acquire their **single feature value ${tex`x_i`}**. This **price** variable ${tex`p_i`} is a key "link" between the platform's objective and each user's objective, since: * The platform wants to strike a balance between **minimizing** the prices ${tex`p_i`} paid to users and **maximizing** the revenue they can derive from data they acquire, whereas * A user ${tex`i`} wants to strike a balance between **maximizing** the price ${tex`p_i`} they receive from the platform for their data and **minimizing** the amount of privacy they sacrifice through this revelation of their data to the platform. This "tension" between the two agents is thus captured by the following **utility function for a user ${tex`i`}**: ${tex.block` u_i(p_i) = v_i\underbrace{\left( \mathbb{E}\left[ \left(\hat{y}_i(\mathbf{X}) - Y_i\right)^2 \right] - \sigma^2_i \right)}_{\mathclap{\text{Increase in prediction accuracy over }\sigma^2_i}} + p_i `} The quantity in parentheses is important here! Since ${tex`v_i`} is the degree to which user ${tex`i`} values their own data privacy, everything in these two objective functions is "calibrated" (via the subtraction of ${tex`\sigma^2_i`} and the choice of unit for ${tex`p_i`}) such that: * Individuals who place a *high* value on the privacy of their feature value — individuals with ${tex`v_i > 1`} — experience **disutility** when their value can be predicted with greater accuracy than ${tex`\sigma^2_i`}, whereas * Individuals who place a *low* value on the privacy of their feature value — individuals with ${tex`v_i < 1`} — experience **utility** when their value can be predicted with greater accuracy than ${tex`\sigma^2_i`}. This is what allows the model to capture the **between-user tension** inherent in scenarios of data externalities!36:Tce0,## The Key Quantity: Information Leakage The key result here, it turns out, can be derived even in the ultra-simple case with **two users**! To save you the pain of having to manually compute conditional distributions and Mean Squared Errors, we'll use the following facts from [this textbook chapter](https://catalogimages.wiley.com/images/db/pdf/0471461261.excerpt.pdf): > ### *Predicting ${tex`X_i = \; ?`} from ${tex`X_j = x_j`} when ${tex`(X_i, X_j)`} are Jointly Normal* > > If ${tex`X_i`} and ${tex`X_j`} are jointly Normally-distributed Random Variables with mean ${tex`(\mu_i, \mu_j) = (0, 0)`} (the case we defined above), but we don't have any information about either of these Random Variables (besides their joint Normal distribution), the best guesses for both are ${tex`\hat{x}_i^* = \mu_i`} and ${tex`\hat{x}_j^* = \mu_j`}, so that in this case ${tex`\hat{x}^*_i = \hat{x}^*_j = 0`}. > > However, if we **observe** the value one of these RVs, say ${tex`X_j = x_j`}, and we want to use this observation to derive the **best possible (that is, MSE-minimizing) prediction** for the other user's value ${tex`X_i`}, this best-possible prediction turns out to be > > ${tex.block` \hat{x}_i^* = \argmin_{\hat{x}_i}\text{MSE} = \boxed{\frac{\sigma_{ij}}{\sigma_j^2}x_j}, \tag{1} `} > > which corresponds to the following **minimal possible MSE value** (here a "mean" squared error over just one prediction, hence the absence of a summation): > > ${tex.block` \text{MSE}^* = \min_{\hat{x}_i}\text{MSE} = \boxed{\sigma^2_i - \frac{\sigma^2_{ij}}{\sigma^2_j}} \tag{2} `} Why do these results help us? Because essentially all of the paper's results relate to a quantity they call **information leakage**, which they define such that the "amount" of leaked information about user ${tex`i`} is the **reduction in MSE** that results from all users' **information sharing decisions** ${tex`\mathbf{a} = (a_1, a_2)`}, where ${tex`a_i = 1`} if user ${tex`i`} decides to share their feature value with the platform, and is 0 otherwise. The expression they provide for this quantity looks as follows: ${tex.block` \mathcal{I}_i(\mathbf{a}) = \sigma^2_i - \min_{\hat{y}_i}\mathbb{E}\left[(Y_i - \hat{y}_i(\mathbf{X}_\mathbf{a}))^2\right] `} which would be scary on its own, since it would require us to solve an optimization problem (due to the ${tex`\min`} operator) every time we wanted to compute information leakage. Instead, the above results provide us with a **closed-form expression** for the MSE-minimizing value(s), thus allowing us to skip over this optimization entirely in this simple case This closed-form expression becomes much more complicated for more than two users, so in those cases it may make more sense to "switch back to" the non-closed-form optimization. For intuition around the "leaked information" value, we can think through what this value would be if the platform managed to learn the value of User 2's **label** with certainty (though this is not a possibility in the "data sharing game" below). In this case, the amount of leaked information would be the **maximum possible value**, ${tex`\mathcal{I}_2 = \sigma^2_2`}, since the platform can now "predict" ${tex`Y_i`} with certainty, resulting in an MSE value of 0.37:T9f4,## Leakage of "Imperfect" Information Within the model, however, users can only share their **feature values**, not their labels. So, if User 2 **does** decide to share their feature value ${tex`x_2`} with the platform, the platform doesn't have **perfect** information about ${tex`y_2`}, but information that is highly correlated with this value. How much would this leakage reduce the MSE of the platform's prediction for **User 2** herself? (In other words, we're not considering the *externalities* of this data leakage yet — only the "self-impact" of User 2's decision to share their own feature value) This is where the above closed-form optimal-predictor results come into play. If User 2 alone reveals ${tex`x_2`} to the platform, what is the platform's resulting **optimal estimator** for this user's label value? The above results require us to have the **covariance matrix** for the joint distribution of ${tex`(X_2, Y_2)`}. To avoid forcing you to compute this, here is the cell from my Mathematica notebook which symbolically computes this matrix: You don't need to worry about this computation, obviously, besides just noting that the covariance matrix in this case turns out to be: ${tex.block` \Sigma = \begin{pmatrix} \sigma_2^2 & \sigma_2^2 \\ \sigma_2^2 & \sigma_2^2 + 1 \end{pmatrix} `} which means that, looking at the results from the textbook chapter above, the optimal prediction of ${tex`Y_2`} using this value is ${tex.block` \hat{y}_2^* = \frac{\sigma_2^2}{\sigma_2^2 + 1}x_2, `} the new minimized MSE at this value is ${tex.block` \text{MSE}^* = \sigma_2^2 - \frac{\sigma_2^4}{\sigma_2^2 + 1} = \frac{\sigma_2^2}{\sigma_2^2 + 1}, `} and thus the information leakage value is ${tex.block` \mathcal{I}_2(\underbrace{(0,1)}_{\mathclap{\text{User 2 shares data}}}) = \sigma_2^2 - \left( \sigma_2^2 - \frac{\sigma_2^4}{\sigma_2^2 + 1} \right) = \frac{\sigma_2^4}{\sigma_2^2 + 1} `} That's all a bit abstract, but it's the foundation for the following demo which I hope makes it a bit more clear. It starts where the previous demo ends, using the two-dimensional Normal distribution to **generate** the label values ${tex`y_1`} and ${tex`y_2`} (and then the feature values via *one*-dimensional Normal distributions centered at ${tex`y_1`} and ${tex`y_2`}), and then measuring how much learning **one** user's feature value improves the prediction of **both** users' labels.38:T10ca,## The Key Result: Information Leakage → "Submodular" Equilibria → Prisoners' Dilemma By doing more math that I won't force you to do, you would obtain a similar result for the information leakage impact on User 2 if **only User *1*** revealed their feature value: in that case we'd compute the covariance matrix for the joint distribution of ${tex`(X_1, Y_2)`} and apply the same closed-form expressions we used above, to obtain ${tex.block` \mathcal{I}_2(\underbrace{(1,0)}_{\mathclap{\text{User 1 shares data}}}) = \sigma_2^2 - \left( \sigma_2^2 - \frac{\sigma_{12}^2}{\sigma_2^2 + 1} \right) = \frac{\sigma_{12}^2}{\sigma_2^2 + 1} `} This is very similar to the result from the previous section, just with the numerator changed from ${tex`\sigma_2^4`} to ${tex`\sigma_{12}^2`}. And, with these two cases of **"unilateral"** data sharing in hand (i.e., looking at the effects if *only* User 1 or *only* User 2 shares data), we have a proof of the **strong monotonicity of information leakage**: > ### *Theorem 1: Strong Monotonicity* > > As long as there is **any correlation at all** between the labels of User 1 and User 2, revelation of one user's feature value will **always leak some nonzero amount of information** about the other user's label. This may be somewhat obvious/contrived, especially after you've gotten this far into the writeup. There's one more **even more important result** here, however, that you could arrive at by computing the information leakage when **both** users share data. By comparing this information leakage value with the values in the other three cases (no data sharing, User 1 only, and User 2 only), you would have a proof of the property mentioned in the title of this section, the **strong submodularity of information leakage**: > ### *Theorem 2: Strong Submodularity* > > As long as there is **any correlation at all** between the labels of User 1 and User 2, the amount of information leakage from revelation of one user's feature value is "more than half" of the total platform-discoverable information, meaning that leakage from one user inherently makes the other user's data **less valuable** to the platform. This second theorem is the "data-economy" version of what economists call **"diminishing returns"**, and it may make more sense if you imagine a **classification** task with ${tex`n`} users rather than just the two we analyzed above. Starting from a "baseline" where the platform has yet to collect **any** data from users, the submodularity property tells us that the following sequence of data-acquisition decisions would occur: * The platform will be willing to pay a **lot** to obtain a **first** user's data, since this may "boost" their prediction accuracy from 1/2 (no better than random guessing of labels) to 2/3... * After this, the platform will value a **second** user's data **less** than the first user's data, since this second datapoint may only "boost" their prediction accuracy from 2/3 to 3/4... * The platform values a **third** user's data even less than this, since it may only "boost" prediction accuracy from 3/4 to 4/5... * And so on, to the point that eventually it is **precisely those who have the most to lose from sharing their data** who are the **most harmed** in the resulting equilibrium outcome, since: * Their private data has become predictable with arbitrarily-high accuracy, * **And** (unlike those who valued privacy less and sold their data earlier) they didn't even receive a "consolation prize" in the form of some non-zero **monetary compensation** for this revelation of their data! It is this last point that makes this scenario structurally equivalent to the **Prisoner's Dilemma 😨**: no matter how much a user values their own privacy, they can carry out in their heads what economists call **"backwards induction"**, to realize that no matter what any other user does their **best response** is to always share their data (just as the Prisoners in the Dilemma realize that they should always Confess/Defect) — even for a tiny fraction of a cent! — since their only options are to exchange this data for the fraction-of-a-cent or be "left out in the cold", having their label predicted regardless and ending up without even this tiny amount of remuneration.39:T104f,## Escaping the Prisoners' Dilemma "Race to the Bottom" via Policy Intervention But don't worry, there is still hope for us Data-Ethical Policy Proposers! As we discussed at the beginning of the Policy Evaluation unit, the data-policymaker's "task" is to explore how/whether a given Prisoner's Dilemma scenario such as this can be **"converted"** into an Assurance Game via policy interventions. We'll discuss this more next week, but there are a few general (albeit very vague) principles regarding when and how these "conversions" may be possible: If wearker, **"one-time" policy interventions** are possible (that is, interventions that doesn't affect the core structure of interactions between agents, but may "inject" resources into the system to "bump" these agents out of worst-cases)... * Prisoners' Dilemmas can sometimes be "converted" into Assurance Games, and * Agents who are stuck within the "bad" equilibrium in an Assurance Game can also sometimes be collectively "pushed" into the "good" equilibrium ...Both of which are already promising reasons to not lose hope! For example, the so-called **"Folk Theorem"** for repeated games states that, if policymakers are able to induce even a single round where agents **cooperate**, this cooperation can often be sustained **on its own** from then onwards (as in, without requiring any additional policy interventions) by way of **reputational dynamics**: agents see the benefits from cooperation in this special "intervention" round, then continue cooperating to avoid being ostracized from the group and thus losing the continued stream of benefits-from-cooperation. If stronger, **"continuous" and/or "structural" policy interventions** are possible (for example, if permanent and effective monitoring systems can be established, or if large firms can be broken up or small firms merged together), then the possible outcomes are even more promising ***(in theory...)***: * Both Prisoners' Dilemmas and Assurance Games can be "converted" into Invisible Hand Games, where agents' **individual-utility maximizing** behavior is nonetheless **calibrated** (via incentives and interaction structures which then induce particular "rules of the game") to give rise to **social-welfare maximizing** outcomes. I say ***(in theory...)*** up there because... well... uncritical belief in this ability to just "fix" things in society via technocratic solutions is (I would argue) exactly the kind of **hubris** that has been at the root of countless atrocities committed by politicians, economists, and others over the past few centuries. Instead of ranting about this, though, I'll link you to PDFs of two books I've referenced a bunch in class (when yapping about the "Nobel" Prize in Economics, for example) — Bradley R. Simpson's *Economists with Guns* (2008) and Juan Gabriel Valdés' *Pinochet's Economists* — and then leave you with a poem by Bertolt Brecht (it's the one I was trying to think of in class at some point during Week 2), *Die Lösung*, written in the aftermath of the East German government's 1953 suppression of a workers' uprising in East Berlin: > *After the uprising of the 17th of June*
> *The Secretary of the Writers' Union*
> *Had leaflets distributed in the Stalinallee*
> *Stating that the people*
> *Had forfeited the confidence of the government*
> *And could only win it back*
> *By redoubled efforts.*
> *Would it not in that case be simpler*
> *For the government*
> *To dissolve the people*
> *And elect another?*3a:T49d,viewof labelGenPlot = Plot.plot({ // Big thanks to https://observablehq.com/@mattiasvillani/multivariate-normal-distribution width: 460, aspectRatio: 1, x: { label: "y₁", domain: [-4,4], ticks: 9, }, y: { label: "y₂", domain: [-4,4], ticks: 9, }, color: { scheme: "Oranges", legend: true, type: "sequential", label: "Probability Density", domain: [0, maxpdfFixed] }, marks: [ Plot.contour( mvnormal_on_grid_fixed, { filter: (d) => d.pdf > Number.EPSILON, x: "y1", y: "y2", fill: "pdf", stroke: "black", blur: 5, // levels: [0.01, 0.02, 0.3] }), Plot.dot([[0,0]], { fill: 'black', stroke: 'black', }), Plot.ruleX([simData.y1], { strokeDasharray: [3], }), Plot.ruleY([simData.y2], { strokeDasharray: [3], }), Plot.dot([[simData.y1, simData.y2]], { r: 4, fill: 'black', stroke: 'white', strokeWidth: 1, }), Plot.text([`Generated Value of (Y₁,Y₂): (${simData.y1}, ${simData.y2})`], { frameAnchor: 'top', dy: -17, fontSize: 14, }), ] })3b:T6ec,viewof labelGenPanel = Inputs.form({ r1: md` Generated **label** values: ${tex`y_1 = ${simData.y1}`}, ${tex`y_2 = ${simData.y2}`}
`, r2: md` Generated **feature** values: ${tex`x_1 = ${simData.x1}`}, ${tex`x_2 = ${simData.x2}`}
`, r3: md` Before collecting data, best guess is ${tex`\hat{y}^*_1 = \hat{y}^*_2 = 0`}
`, r31: md`${tex`\implies \text{MSE}_{\text{pre}} = \frac{1}{2} \left[ \overset{\text{SE}^{\text{pre}}_1}{(\underset{\hat{y}^*_1}{0} - \underset{y_1}{${simData.y1}})^2} + \overset{\text{SE}^{\text{pre}}_2}{(\underset{\hat{y}^*_2}{0} - \underset{y_2}{${simData.y2}})^2} \right] = ${msePre}`}
`, r4: viewof xChecks, r5: md`${tex`\implies \hat{y}_1^* ${optCondition} = \vphantom{\frac{\sigma_{12}}{\sigma_2^2}} ${optimalY1.formula} = ${optimalY1.value}, \; \; \hat{y}_2^* ${optCondition} = ${optimalY2.formula} = ${optimalY2.label}`}
`, r6: md`${tex`\implies \text{MSE} ${optCondition} = \frac{1}{2} \left[ \overset{\text{SE}_1 ${optCondition}}{(\underset{\hat{y}^*_1}{${optimalY1.label}} - \underset{y_1}{${simData.y1}})^2} + \overset{\text{SE}_2 ${optCondition}}{(\underset{\hat{y}^*_2}{${optimalY2.label}} - \underset{y_2}{${simData.y2}})^2} \right] = ${+msePost.toFixed(3)}`}
`, r7: md` Information Leakage ${tex`\triangleq`} Decrease in MSE:`, r71: md`${tex.block` \begin{aligned} \mathcal{I}_1 &= \text{SE}^{\text{pre}}_1 - \text{SE}_1 ${optCondition} = ${+leak1.toFixed(3)} \\ \mathcal{I}_2 &= \text{SE}^{\text{pre}}_2 - \text{SE}_2 ${optCondition} = ${+leak2.toFixed(3)} \\ \mathcal{I} &= \text{MSE}^{\text{pre}} - \text{MSE} ${optCondition} = ${+leakTotal.toFixed(3)} \end{aligned} `}`, })3c:T587,viewof typeDistPlot = Plot.plot({ // Big thanks to https://observablehq.com/@mattiasvillani/multivariate-normal-distribution width: 420, aspectRatio: 1, marginTop: 20, // clip: true, x: { label: "y₁", domain: [-4,4], ticks: 9, }, y: { label: "y₂", domain: [-4,4], ticks: 9, }, color: { scheme: "Oranges", // legend: true, type: "sequential", label: "Probability Density", domain: [0, maxpdf] }, marks: [ Plot.contour( mvnormal_on_grid, { filter: (d) => d.pdf > Number.EPSILON, x: "y1", y: "y2", fill: "pdf", stroke: "black", blur: 5, // levels: [0.01, 0.02, 0.3] }), Plot.dot([[mu1, mu2]], { r: 4, fill: 'black', }), // Marginal y1 dist Plot.areaY( normal_on_grid_y1, { x: "y1", y1: -4, y2: d => 2*d["density"] - 4, fill: '#e69f00', stroke: 'black', clip: true, } ), // Marginal y2 dist Plot.areaX( normal_on_grid_y2, { y: "y2", x1: -4, x2: d => 2*d["density"] - 4, fill: '#e69f00', stroke: 'black', clip: true, } ), Plot.ruleX([y1obs], { strokeDasharray: [3], }), Plot.text(['Joint Distribution of (Y₁,Y₂)'], { frameAnchor: 'top', dy: -17, fontSize: 14, }), ] })3d:[{"id":"3c051b09103b30caef7ec8c6a661bbf6eb18b804c3a1cd407262b991a9f2f96bcd103df8fa27d719a0ac89655f3aa943161683caf212aee72038756cefe8255b","href":"https://static.observableusercontent.com/files/3c051b09103b30caef7ec8c6a661bbf6eb18b804c3a1cd407262b991a9f2f96bcd103df8fa27d719a0ac89655f3aa943161683caf212aee72038756cefe8255b","name":"image.png","type":"image/png","lastModified":1774684148354.054,"size":28988}] 2f:{"title":"Data Externalities","cells":[{"id":0,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":5,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":189,"value":"$33","pinned":false,"hidden":false,"mode":"md"},{"id":1714,"value":"md`### *Demo 1: Uncertainty About ${tex`Y_2`} Given ${tex`Y_1`}*\n\n${viewof modelViz}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":918,"value":"$34","pinned":false,"hidden":false,"mode":"md"},{"id":256,"value":"As a final step in describing the model, we now need to add in the **objective functions** of both the platform and the user, which will determine the outcome of the model in terms of which users opt to sell their data to the platform and which do not.","pinned":false,"hidden":false,"mode":"md"},{"id":207,"value":"$35","pinned":false,"hidden":false,"mode":"md"},{"id":912,"value":"$36","pinned":false,"hidden":false,"mode":"md"},{"id":1070,"value":"$37","pinned":false,"hidden":false,"mode":"md"},{"id":1724,"value":"infoLeakageViz = md`\n### *Demo 2: Information Leakage from Data Acquisition*\n\n${viewof simViz}\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":1025,"value":"$38","pinned":false,"hidden":false,"mode":"md"},{"id":1044,"value":"$39","pinned":false,"hidden":false,"mode":"md"},{"id":155,"value":"## Footnotes\n\n1. In fact, in Week 11 when we opt to \"zoom in\" specifically on **policies lying on the Pareto Frontier**, this assumption is made explicit: by definition, policies on the Pareto frontier are \"as good as possible\" in terms of **overall** (aggregate) utility, which in turn means that the **choice of a particular policy along this frontier** is (again by definition) a specific **tradeoff** between one agent's utility and the utility of others.","pinned":false,"hidden":false,"mode":"md"},{"id":906,"value":"

","pinned":false,"hidden":false,"mode":"md"},{"id":485,"value":"## JS Parameter Values","pinned":false,"hidden":false,"mode":"md"},{"id":401,"value":"viewof mu1 = genSlider(tex`\\Large\\mu_1`, [-1,1], defaults.stepSize, defaults.mu1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":405,"value":"viewof mu2 = genSlider(tex`\\Large\\mu_2`, [-1,1], defaults.stepSize, defaults.mu2)","pinned":false,"hidden":false,"mode":"ojs"},{"id":408,"value":"viewof sigma1 = genSlider(tex`\\Large\\sigma_1`, [0.1,2], defaults.stepSize, defaults.sigma1)","pinned":false,"hidden":false,"mode":"ojs"},{"id":411,"value":"viewof sigma2 = genSlider(tex`\\Large\\sigma_2`, [0.1,2], defaults.stepSize, defaults.sigma2)","pinned":false,"hidden":false,"mode":"ojs"},{"id":414,"value":"viewof rho = genSlider(tex`\\Large\\rho`, [-0.9,0.9], defaults.stepSize, defaults.rho)","pinned":false,"hidden":false,"mode":"ojs"},{"id":289,"value":"viewof params = Inputs.form({\n r1: viewof mu1,\n r2: viewof mu2,\n // r3: viewof sigma1,\n // r4: viewof sigma2,\n r5: viewof rho,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":614,"value":"viewof y1obs = genSlider(md`Observed ${tex`\\large y_1`}: `, [-4, 4], 0.5, 1, 100)","pinned":false,"hidden":false,"mode":"ojs"},{"id":562,"value":"## Calculations","pinned":false,"hidden":false,"mode":"md"},{"id":1684,"value":"se1Pre = (simData.y1 - 0)**2","pinned":false,"hidden":false,"mode":"ojs"},{"id":1691,"value":"se2Pre = (simData.y2 - 0)**2","pinned":false,"hidden":false,"mode":"ojs"},{"id":1202,"value":"msePre = (1/2) * (se1Pre + se2Pre)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1693,"value":"se1Post = (optimalY1.value - simData.y1)**2","pinned":false,"hidden":false,"mode":"ojs"},{"id":1698,"value":"se2Post = (optimalY2.value - simData.y2)**2","pinned":false,"hidden":false,"mode":"ojs"},{"id":1280,"value":"msePost = (1/2) * (se1Post + se2Post)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1701,"value":"leak1 = se1Pre - se1Post","pinned":false,"hidden":false,"mode":"ojs"},{"id":1703,"value":"leak2 = se2Pre - se2Post","pinned":false,"hidden":false,"mode":"ojs"},{"id":1705,"value":"leakTotal = msePre - msePost","pinned":false,"hidden":false,"mode":"ojs"},{"id":1112,"value":"simData = ({\n y1: 0.5,\n y2: 1.0,\n x1: 0.1,\n x2: 1.5,\n rho: 0.75,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1269,"value":"optimalY1 = {\n if (!isLearned.x1 && !isLearned.x2) {\n return {\n formula: \"\\\\mu\",\n value: 0,\n label: \"0\",\n }\n } else if (isLearned.x1) {\n return {\n formula: \"x_1\",\n value: simData.x1,\n label: +simData.x1.toFixed(3),\n }\n } else {\n // We know x2, use it to infer x1\n return {\n formula: \"\\\\frac{\\\\sigma_{12}}{\\\\sigma_1^2}\",\n value: simData.rho * simData.x2,\n label: +(simData.rho * simData.x2).toFixed(3),\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1248,"value":"// optimalY2 = (0.75 / 1) * simData.x1\noptimalY2 = {\n if (!isLearned.x1 && !isLearned.x2) {\n return {\n formula: \"\\\\mu\", \n value: 0,\n label: \"0\",\n };\n } else if (isLearned.x2) {\n return {\n formula: \"x_2\",\n value: simData.x2,\n label: +simData.x2.toFixed(3),\n }\n } else {\n // We know x1, use it to infer x2\n return {\n formula: \"\\\\frac{\\\\sigma_{12}}{\\\\sigma_2^2}\",\n value: simData.rho * simData.x1,\n label: +(simData.rho * simData.x1).toFixed(3),\n }\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":314,"value":"meanVec = [mu1, mu2]","pinned":false,"hidden":false,"mode":"ojs"},{"id":317,"value":"covarMatrix = [\n [sigma1**2, sigma1 * sigma2 * rho],\n [sigma1 * sigma2 * rho, sigma2**2],\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":1152,"value":"covarMatrixFixed = [\n [1, 0.75],\n [0.75, 1],\n]","pinned":false,"hidden":false,"mode":"ojs"},{"id":633,"value":"conditionalDist = ({\n mu: mu2 + rho * (1/(sigma1**2)) * (y1obs - mu1),\n sigma: (sigma2**2) - rho * (1/(sigma1**2)) * rho,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":564,"value":"function mvnpdf(x, mu, Sigma, log = false){\n const p = math.matrix(mu).size()[0];\n if (log) {\n return -(p/2)*math.log(2*math.pi) -0.5*math.log(math.det(Sigma)) \n -0.5*math.multiply( math.multiply( math.transpose(math.subtract(x,mu)), math.inv(Sigma)), math.subtract(x,mu))\n } else{\n return (2*math.pi)**(-p/2)*math.det(Sigma)**(-0.5)*\n math.exp(-0.5*math.multiply( math.multiply( math.transpose(math.subtract(x,mu)), math.inv(Sigma)), math.subtract(x,mu)))\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":566,"value":"normpdf = (x, mu, sigma) => {\n let left_chunk = 1 / (Math.sqrt(2 * Math.PI * sigma**2))\n let right_top = -((x - mu)**2)\n let right_bottom = 2 * (sigma**2)\n return left_chunk * Math.exp(right_top / right_bottom)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":571,"value":"mvnormal_on_grid = {\n const ngridx1 = 50;\n const ngridx2 = 50;\n var griddata = [];\n for (let y1 of d3.range(-5, 5, 10/ngridx1)){\n for (let y2 of d3.range(-5, 5, 10/ngridx2)){\n griddata.push({\n y1: y1,\n y2: y2,\n pdf: mvnpdf(\n [y1, y2],\n meanVec,\n covarMatrix) \n })\n }\n }\n return griddata\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1150,"value":"mvnormal_on_grid_fixed = {\n const ngridx1 = 50;\n const ngridx2 = 50;\n var griddata = [];\n for (let y1 of d3.range(-5, 5, 10/ngridx1)){\n for (let y2 of d3.range(-5, 5, 10/ngridx2)){\n griddata.push({\n y1: y1,\n y2: y2,\n pdf: mvnpdf(\n [y1, y2],\n meanVec,\n covarMatrixFixed) \n })\n }\n }\n return griddata\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1756,"value":"normal_on_grid_y1 = {\n const ngrid = 100;\n var griddata = [];\n for (let y1 of d3.range(-5, 5, 10/ngrid)) {\n griddata.push({\n y1: y1,\n density: normpdf(y1, mu1, sigma1),\n })\n }\n return griddata\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":573,"value":"normal_on_grid_y2 = {\n const ngrid = 100;\n var griddata = [];\n for (let y2 of d3.range(-5, 5, 10/ngrid)) {\n griddata.push({\n y2: y2,\n density: normpdf(y2, mu2, sigma2),\n })\n }\n return griddata\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":638,"value":"cond_normal_on_grid = {\n const ngrid = 100;\n var griddata = [];\n for (let y2 of d3.range(-5, 5, 10/ngrid)) {\n griddata.push({\n y2: y2,\n density: normpdf(y2, conditionalDist.mu, conditionalDist.sigma),\n })\n }\n return griddata\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":70,"value":"## UI Pieces","pinned":false,"hidden":false,"mode":"md"},{"id":1126,"value":"viewof simViz = inputsGroup(\n [[viewof labelGenPlot, viewof labelGenPanel]],\n [['label-gen-plot', 'label-gen-panel']],\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1284,"value":"viewof xChecks = genCheckboxes([\"x₁\",\"x₂\"], md` Platform learns...`)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1443,"value":"learnedVars = xChecks.map(d => d.replace(\"₁\",\"_1\").replace(\"₂\",\"_2\"))","pinned":false,"hidden":false,"mode":"ojs"},{"id":1290,"value":"isLearned = ({\n x1: xChecks.includes(\"x₁\"),\n x2: xChecks.includes(\"x₂\"),\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1437,"value":"optCondition = (\n ((isLearned.x1 || isLearned.x2) ? \"\\\\mid \" : \"\") +\n learnedVars.join(\", \")\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1108,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs"},{"id":552,"value":"viewof modelViz = inputsGroup(\n [[viewof distPanel, viewof conditionalPanel]],\n [['dist-panel','conditional-panel']],\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1129,"value":"$3b","pinned":true,"hidden":false,"mode":"ojs"},{"id":1288,"value":"xChecks","pinned":false,"hidden":false,"mode":"ojs"},{"id":556,"value":"viewof conditionalPanel = Inputs.form({\n r0: md`Distribution of ${tex`Y_2`} **before** observing ${tex`Y_1`}: ${tex`\\mathcal{N}(${+mu2.toFixed(3)},${+sigma2.toFixed(3)})`}`,\n r1: viewof y2DistPre,\n r2: viewof y1obs,\n r3: md`Distribution of ${tex`Y_2`} **after** observing ${tex`Y_1 = ${y1obs}`}: ${tex`\\mathcal{N}(${+conditionalDist.mu.toFixed(3)},${+conditionalDist.sigma.toFixed(3)})`}`,\n r4: viewof y2DistPost,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":390,"value":"viewof paramsPanel = Inputs.form({\n r1: viewof params,\n r2: viewof resetButtons,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":392,"value":"viewof resetButtons = {\n const buttonElt = Inputs.button([\n [\"Set Negative\", resetParamsNegative],\n [\"Set to Zero\", resetParamsZero],\n [\"Set Positive\", resetParamsPositive],\n ]);\n buttonElt.classList.add('reset-buttons');\n return buttonElt;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":363,"value":"viewof distPanel = Inputs.form({\n r1: viewof paramsPanel,\n r2: viewof typeDistPlot,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":258,"value":"$3c","pinned":false,"hidden":false,"mode":"ojs"},{"id":558,"value":"viewof y2DistPre = Plot.plot({\n width: 400,\n height: 150,\n color: {legend: true},\n marks: [\n Plot.ruleY([0]),\n Plot.areaY(\n normal_on_grid_y2, {\n x: \"y2\",\n y2: \"density\",\n fill: '#e69f00',\n stroke: 'black',\n }\n ),\n ],\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":602,"value":"viewof y2DistPost = Plot.plot({\n width: 400,\n height: 150,\n x: {\n ticks: 9,\n },\n color: {legend: true},\n marks: [\n Plot.ruleY([0]),\n Plot.lineY(cond_normal_on_grid, {\n x: \"y2\", y: \"density\",\n stroke: 'black',\n }),\n Plot.areaY(\n cond_normal_on_grid, {\n x: \"y2\",\n y2: \"density\",\n fill: '#e69f00',\n }\n ),\n Plot.ruleX([conditionalDist.mu], {\n strokeDasharray: [3],\n }),\n ],\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":1225,"value":"genCheckboxes = (boxLabels, fieldLabel) => {\n const checkboxElt = Inputs.checkbox(boxLabels, {\n label: fieldLabel,\n });\n checkboxElt.classList.add('demo-checkbox-form');\n var styleStr = `--label-width: auto`;\n Object.assign(checkboxElt, {style: styleStr});\n checkboxElt.querySelectorAll('label').forEach((x) => x.classList.add('checkbox-label'));\n return checkboxElt;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":93,"value":"function genSlider(varLabel, varRange, varStep, defaultVal, labelWidth = 40) {\n var sliderElt = Inputs.range(varRange, {\n value: defaultVal,\n label: varLabel,\n width: 20,\n step: varStep,\n });\n var styleStr = `--label-width: ${labelWidth}px`;\n Object.assign(sliderElt, {style: styleStr});\n return(sliderElt);\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1599,"value":"calloutViz = null;\n// This would place the visualizations in a callout box\n// md`\n// > ### Uncertainty about ${tex`Y_1 \\mid Y_2`}\n// > \n// > ${viewof modelViz}\n// `","pinned":false,"hidden":false,"mode":"ojs"},{"id":678,"value":"## JS Pieces / Helpers","pinned":false,"hidden":false,"mode":"md"},{"id":51,"value":"defaults = ({\n mu1: 0,\n mu2: 0,\n sigma1: 1,\n sigma2: 1,\n rho: 0.75,\n simRho: 0.75,\n stepSize: 0.05,\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":264,"value":"maxpdf = mvnpdf(\n meanVec,\n meanVec,\n covarMatrix,\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":1579,"value":"maxpdfFixed = mvnpdf(\n meanVec,\n meanVec,\n covarMatrixFixed,\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":366,"value":"simpleTwoCol = (c1, c2) => {\n return html`

${c1}

${c2}

`\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":374,"value":"// https://observablehq.com/@kin-fung-negawatt/inputs-in-grid\nfunction inputsGroup(views,names, minWidth = 300, margin = 10){\n const form = html`

${\n views.map(row => html`

${\n row.map(input => html`

${input}

`)\n }

`;\n \n form.oninput = () => {\n form.value = views.map(row => row.map(input => input.value))\n if(names){\n names.forEach((row,i) => row.forEach((c,j) => form.value[i][j] && (form.value[c]=form.value[i][j])))\n }\n };\n form.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":442,"value":"resetParamsNegative = () => {\n set(viewof rho, -defaults.rho)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":447,"value":"resetParamsZero = () => {\n set(viewof rho, 0)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":449,"value":"resetParamsPositive = () => {\n set(viewof rho, defaults.rho)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":387,"value":"function set(input, value) {\n input.value = value;\n input.dispatchEvent(new Event(\"input\"));\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":548,"value":"function centerCell(orig_cell, full_width=false) {\n let fw_text = full_width ? \"width: 100%;\" : \"\";\n return html`

${orig_cell}

`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1221,"value":"## Bootstrap Icons Import","pinned":false,"hidden":false,"mode":"md"},{"id":1207,"value":"bootstrapIcons = {\n const src = cdn;\n const elem = html``;\n const _html = elem.outerHTML;\n\n const exists = [...document.querySelector(\"head\").querySelectorAll('link')].map(elem=>elem.href).filter(href=>href === src).length > 0;\n \n const api = {\n attach: () => document.querySelector(\"head\").prepend(elem),\n detach: () => elem.parentNode && elem.parentNode.removeChild(elem),\n elem, html: _html, src, exists\n };\n api.attach();\n invalidation.then(api.detach);\n\n api.exists = [...document.querySelector(\"head\").querySelectorAll('link')].map(elem=>elem.href).filter(href=>href === src).length > 0;\n \n return api;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":1215,"value":"cdn = `https://cdn.jsdelivr.net/npm/bootstrap-icons@1.13.1/font/bootstrap-icons.css`;","pinned":false,"hidden":false,"mode":"ojs"},{"id":1359,"value":"## Misc Imports","pinned":false,"hidden":false,"mode":"md"},{"id":268,"value":"math = require(\"mathjs\");","pinned":false,"hidden":false,"mode":"ojs"},{"id":1077,"value":"","pinned":false,"hidden":false,"mode":"html"}],"files":"$W3d","stdlib":"1","titling":"auto"}