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# Find the trigonometric Fourier series (FS) and the exponential FS of the following: 2 *(1) 1.5...

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Find the trigonometric Fourier series (FS) and the exponential FS of the following: 2 *(1) 1.5 1 0.5 O -0.5 1 -1 2 -1 0 2 b) 2 3 4 6 exponential FS Cnejnwot f(t) = En=-00 Where Cn 7Se+ f(t)e-inwot dt trigonometric f(t)= a, +Ža, cos(n6,t)+b, sin(n0,1 ao 1 T. 2 to an S f(t)dt sº f(t)cos(n0,1)dt f(t)sin (no,t)dt To 2 pt b,

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text:
Letter

# Millions of online book co-purchases reveal partisan differences in the consumption of science

• Nature Human Behaviour 1, Article number: 0079 (2017)
• doi:10.1038/s41562-017-0079
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## Abstract

Passionate disagreements about climate change, stem cell research and evolution raise concerns that science has become a new battlefield in the culture wars. We used data derived from millions of online co-purchases as a behavioural indicator for whether shared interest in science bridges political differences or selective attention reinforces existing divisions. Findings reveal partisan preferences both within and across scientific disciplines. Across fields, customers for liberal or ‘blue’ political books prefer basic science (for example, physics, astronomy and zoology), whereas conservative or ‘red’ customers prefer applied and commercial science (for example, criminology, medicine and geophysics). Within disciplines, ‘red’ books tend to be co-purchased with a narrower subset of science books on the periphery of the discipline. We conclude that the political left and right share an interest in science in general, but not science in particular. This underscores the need for research into remedies that can attenuate selective exposure to ‘convenient truth’, renew the capacity for science to inform political debate and temper partisan passions.

In its quest for an objective understanding of the world1, modern science has practised two distinct forms of political neutrality: as an apolitical ‘separate sphere’ detached from ideological debates, and as a ‘public sphere’ relevant to political issues but with balanced political engagement that aids reasoned deliberation and deference to evidence2,3,4,5. Recent surveys support the view that science contributes not only to human knowledge but also to social integration, both as a voice of reason and also as a shared value. Joint surveys conducted by the American Association for the Advancement of Science (AAAS) and the Pew Research Center in 2009 and 2014 found that science remains near the top in public rankings of professions, well above that of clergy, despite the prevalence of liberals among scientists6,7,8. Although nearly two-thirds of respondents question evolution, even those who see conflict with issues of personal faith overwhelmingly support scientific contributions to public well-being (67%). In a highly polarized electorate, these responses invite reassurance that science continues to command political deference as a voice of reason and bridges the partisan divide. We may disagree on emotionally charged social issues, but at least we can agree on science.

Political and cultural polarization within the United States, however, raises questions about the validity of this interpretation9. A less comforting possibility is that verbal survey responses may simply echo an Enlightenment commitment to value-free scientific inquiry that masks underlying scepticism about science. In recent years, conservative politicians and pundits have challenged scientific positions on evolution, cosmology, climate change and the perceived liberal bias in policies advocated by social scientists. For example, the conservative-funded scientific counter-movement in climate change research suggests the possibility of politically driven scientific polarization10,11. When science becomes politicized, partisans tend to cast doubt on scientific consensus through questioning its inherent uncertainty12,13,14.This process is manifest not only in conservative resistance to climate change, but in historically liberal resistance to consensus over the positive benefits of genetically modified organisms, vaccination, nuclear power and the safe storage of nuclear waste15.

Survey data show little overall change in public confidence in science since 1970, but beneath the surface there is a marked shift: conservatives in the Vietnam era were more confident in science than liberals, but today that pattern has reversed16 (Supplementary Fig. 1). Does public exposure to science play an integrative role by encouraging and informing empirical validation? Or has selective attention instead reinforced the ‘Big Sort’ of American politics17,18,19— the tendency to cluster in like-minded communities?

Much previous research has used surveys to investigate political alignments of the producers of science (with a few exceptions20,21). We focus instead on the consumers of science, using online co-purchases of books on science and politics as a behavioural indication of preferences held by customers who ‘vote with their pocketbook’, in contrast to survey responses that are costless. Surveys measure what researchers think is important, not what respondents care about, whereas online consumers can register their preferences by purchasing books on any topic they choose. Retrospective self-reports are vulnerable to lapses of memory, whereas online sellers track every purchase. Survey responses are difficult to align across instruments that ask different questions and ask questions differently, whereas books from different stores can be classified using consistent typologies (for example the Library of Congress). Surveys are vulnerable to response bias from participants reluctant to reveal views regarded as politically incorrect; books purchased online arrive cloaked in cardboard. Finally, although surveys can use stratified random samples to generalize results to the underlying population, which is not possible with data from a convenience sample, rates of non-response are rising in landline-administered surveys, which raises concerns about their external validity22.

We addressed concerns about generalizability in two ways — by replicating our analysis using two independent samples of purchasing behaviour from two online merchants (Amazon and Barnes & Noble), and also by the size of these samples, collectively comprising hundreds of millions of online customers, including members of hidden populations (such as those without landlines) who may be undercounted in surveys based on at most a few thousand respondents.

Our approach also differs from survey methods in the unit of analysis. Individuals are the units in surveys, but online retailers do not provide access to individual customer behaviour. Instead, we use individual books as the unit of analysis in constructing a co-purchase network. Bipartite network analysis has been widely used in research on co-citation and co-author networks23,24,25,26,27 where individual information is aggregated to provide connections between scientific papers. What is lost in the absence of individual-level responses for a small sample is compensated for with massive numbers of observations we can use to detect population-level patterns generated by individual behaviour. Here we extend bipartite network analysis to the study of online co-purchase behaviour among a diverse population of tens of millions of consumers. These data provide an unprecedented opportunity to study the entire popular audience for science in ways that are not possible using traditional methods.

These data do not speak to the partisan alignment of scientists, the policy relevance of scientific research or the political polarization of science as an institution. Nor do we address the political preferences of science’s consumers. Rather, our attention is focused exclusively on the science preferences of those who purchase liberal and conservative political books, a group whose science preferences could differ from those who do not shop for books online or who shop for science books but not for politics. Within that constraint of available data, we ask to what extent purchasers of political books are also interested in science, and in what parts of science they are most interested. A shared interest in science might provide a bridge across partisan divisions, whereas selective attention to ‘convenient truths’ risks reinforcement of existing political identities.

To find out, we constructed two undirected co-purchase networks of books from the American domain of the world’s two largest online book stores, following an approach pioneered by Valdis Krebs28,29,30. Up to 100 unranked books are listed on each book page under the heading “Customers Who Bought This Item Also Bought”. According to previous work31, these recommendations are based on a disproportionate number of customers who bought both books and are designed exclusively to identify those books (up to 100) that a customer is most likely to be interested in also purchasing. We compared results derived from two different bookstores — Amazon, and Barnes & Noble — each with its own co-purchasing algorithm. No important differences were found (see Supplementary Information).

Differences in the relative popularity of two books A and B could cause B to be included in the top 100 list for A while A does not make it into the list for B. Co-purchase behaviour is inherently undirected at the level of an individual customer’s multiple purchases, however, as the purchase of book A cannot be said to cause an individual to purchase B any more than the converse. We therefore ignore direction and reciprocation and define an undirected co-purchase link between two books as the level of bi-directed co-purchasing required to trigger a co-purchase listing in either direction.

Beginning with several seed books, we collected data recursively by tracing co-purchase links, iterating the search until no new titles could be identified. In total, we collected 26,467,385 co-purchase links among 1,303,504 books from amazon.com after consolidating multiple editions. (For details, see Methods. These counts refer to the number of titles, not purchases; we use ‘books’ to refer to titles and never to the physical objects.) From this collection, we identified three groups: political books, science books, and non-science books. From 3,530 politically relevant titles, two independent coders (with a third as tie-breaker) identified 673 conservative (‘red’) books, 583 liberal (‘blue’) books, and discarded 2,274 indeterminate books. The ‘indeterminate’ books include some that are centrist, and these could be used to investigate different interests in science held by moderates and those with more extreme views (whether red or blue). That possibility falls outside the scope of the present study, which is limited to comparisons of red and blue co-purchases.

As an additional validation, we imputed red and blue codings based on the relative number of links to other red and blue books and compared these with the hand codings. Over 96% were in agreement. The network among blue and red books is visualized in Fig. 1a. The monochromatic clustering reveals the political ‘echo chambers’ in a highly polarized population. (See Methods for methodological details, red/blue book comparisons that show similarity in publication year and sales rank, and information about the handful of ‘misfits’ in each cluster.)

The political and science books identified in amazon.com were also collected from barnesandnoble.com, to test the consistency of co-purchasing patterns between the two bookstores. These co-purchase networks, composed of millions of distinct purchases, are different in important ways: only 9% of Amazon co-purchase links are found in the Barnes & Noble network, and only 21% of Barnes & Noble links are found in Amazon. Nevertheless, the number of political links with each book in Amazon and Barnes & Noble is highly correlated (0.60). The consistency of our findings in these two environments suggests robust patterns of co-purchase behaviour, in which the political preferences of science book consumers are very similar even though the expression of those preferences in the purchase of particular books differs across the two websites.

We identified 428,433 titles that appeared under science categories in the Library of Congress and Dewey Decimal classifications. We grouped these into 27 exclusive high-level topics, corresponding to broadly defined scientific disciplines (such as physics, chemistry, medicine or economics). These 27 disciplines fall under four main scientific ‘schools’ (humanities, physical sciences, life sciences and social sciences). An additional 494,278 non-science titles were grouped in four main topics — arts, sports, literature (fiction and poetry), and religion — as a baseline for assessing co-purchase links between science and politics (see Methods for detailed categories).

We used co-purchase links to measure the political relevance, alignment and polarization among online customers. Political relevance of a topic is measured by the fraction of links from political books (whether red or blue) among all co-purchase links to books in the topic. Given the measure’s uncertainty with small samples, we used a Bayesian probabilistic framework with a prior distribution induced by the configuration model32. Links between science, non-science and political books are randomly generated, given the network degree of each book. The higher the political relevance, the greater the likelihood that purchasers of political books will purchase books in the topic.

Political alignment of a topic is measured by the fraction of links from red books among co-purchase links from political books (red or blue). As with relevance, we account for statistical uncertainty by estimating this measure as the probability that books, in a particular topic will link with red books, conditioned on their links from political books. Alignment captures partisan interest in each scientific discipline on the red–blue spectrum, where purple (alignment = 0.5) could indicate the balanced political interest required for a public sphere of reasoned discourse.

Alternatively, a ‘purple’ discipline could also be internally polarized, with equal interest from left and right but in separate subsets of books. Political polarization is a function of the number of books within a discipline linked with both red and blue books, compared with a null model in which red and blue links are randomly assigned to books in the topic. Polarization equals 0 when disciplinary books are co-purchased with red and blue books uniformly at random, but increases as the sets of red- and blue-linked books diverge, indicating red and blue preferences for distinct books. (Note that we use the term ‘polarization’ to refer to the political segregation of the population into opposing groups of like-minded members who influence one another to adopt similar positions on other issues on which they did not initially agree33. Our usage should not be confused with an alternative definition of polarization that indicates the tendency of people who agree on a given issue to influence one another to adopt more extreme opinions on that same issue.)

In addition to these three metrics, we also measured characteristics of scientific books and fields to account for diverging red and blue scientific interests. We scored fields as basic or applied science based on the ratio of patent to article citations in each field. We also considered whether books are ‘academic’ or ‘popular science’, based on publication by an academic or popular press. Finally, we measured the relative scientific breadth of liberals versus conservatives as the difference between the average number of disciplinary titles linked with a blue book and those with a red book, and the network location of books with ties to red and blue relative to the core or periphery of the co-purchase network within the disciplines. (See Methods for detailed discussion of these measures.)

Our analysis proceeds in three steps. We first assess political relevance, alignment and polarization as measures of political interest in science compared with political interest in books and topics outside science; second, we report differences in these measures across scientific disciplines, broken down by the continuum of applied and basic science; third, we report the scientific breadth and location of red and blue books within disciplines.

First, compared with the number of co-purchase links expected by chance, people who buy liberal and conservative books are more likely to buy books on science than on important topics outside science, but this interest in science does not appear to be insulated from the ‘Big Sort’ of American politics. Figure 2 reports political relevance and polarization of science, religion, sports, arts and literature, with alignment indicated by colour. Results for political relevance show that political readers have greater interest in science relative to non-science topics, largely owing to books on social science. The physical and life sciences do not attract markedly greater interest among political readers compared with topics outside science, but this political interest in science is significantly more polarized than for arts and sports, indicating that liberals and conservatives are less likely to read the same science book.

Second, we found a significant positive correlation (r = 0.43, p = 0.002) between political alignment and the normalized number of citations by patents, our indicator of applied, commercial science (Fig. 3b). For example, organic chemistry is the most applied sub-discipline, as measured by patent versus article citations, and it aligns closely with red books (0.75 on an alignment scale from 0 to 1). This contrasts with a sub-discipline such as zoology, which is largely driven by curiosity and basic scientific concerns, and appeals more to those on the left (0.1). This pattern can also be observed in Fig. 3a, which reports co-purchase alignment within the four ‘schools’. Applied disciplines such as medicine and law attract readers at the red end compared with other disciplines in their respective schools, whereas anthropology and astronomy attract readers at the blue end. This mirrors the ideological differences among scientists employed in academia versus industry, as reported in AAAS/Pew surveys6,7 from 2009 and 2014. A possible interpretation is that scientific puzzles appeal more to the left, while problem-solving appeals more to the right.

A few disciplines, notably palaeontology, bridge political divisions and are politically purple because books in these disciplines attract equal interest from both left and right. However, most purple disciplines (with equal likelihood to have red and blue book links) do not bridge political divisions. Figure 4 illustrates the internal network structure of seven natural and social science disciplines. Monochromatic clusters, located in different regions of the network, indicate that red and blue political books are linked to different interconnected clusters of disciplinary books. Simply put, even when left and right are equally likely to read books in a discipline, they are rarely the same books or even from the same topical cluster.

We drilled down further to examine the location of science books linked with red and blue books within co-purchase networks at the disciplinary level. The scatter plot in Fig. 4 presents the relative scientific breadth of blue versus red (y axis). Note that higher values in the y axis indicate greater difference between the breadths of blue and red books. The figure reveals that blue books link to a larger number of scientific titles than red within most disciplines, and the difference increases for more polarized disciplines.

The disciplinary networks show the location of red and blue political books linked to disciplinary books within a field (coloured grey). All books are positioned such that pairwise proximities correspond only to co-purchase links with disciplinary books. The visualizations show that red books tend to cluster on the periphery of the co-purchase networks for climatology, environmental science, political science and biology, which indicates that conservatives tend to purchase books likely to be co-purchased with each other but not with other books in the discipline (Supplementary Fig. 4). In contrast, blue books are less clustered and linked to books closer to the core, which indicates that liberals tend to purchase a more diverse set of science books, including books frequently co-purchased with other books in the discipline. The greater centrality of blue-linked books does not appear to be a consequence of academic liberalism. When books by academic publishers are removed, the pattern remains, as do all the other patterns that we report (see Supplementary Fig. 5).

These results need to be qualified by inherent limitations in the use of co-purchase links to measure partisan interest in science. First, although half of the US population purchases books online, this is not a random sample, which limits the ability to generalize our results to the other half. We cannot be certain that those who purchase books online are similar to those who do not, or that those who purchase red and blue books have the same interests in science as liberals and conservatives who do not purchase political books.

Second, we do not have individual-level co-purchase data and therefore cannot pursue analytical strategies for which such data would be required, such as multivariate causal modelling. Co-purchasing patterns reveal the population-level distribution of political interests in science but not the underlying individual-level causes.

Third, co-purchase links can influence purchases that then reinforce new co-purchase links, thereby magnifying estimates of the underlying consumer interest in politics and science. We believe that this reinforcing dynamic does not invalidate the patterns that we find. Rather, this provides a possible explanation of the findings — an ‘echo chamber’ constructed not only by consumer preferences but also by the ‘filter bubble’ constructed by collaborative filtering technology that reinforces those preferences34. Future research using randomized trials is needed to test whether recommender systems can induce users to make choices orthogonal to or counter to their existing preferences.

These three sets of concerns are mitigated, however, by similarity in the results derived from distinct bookstores, with different purchase patterns and company-specific algorithms. We replicated the Amazon-based analyses using the Barnes & Noble network and found consistent results across datasets, with political polarization (r = 0.97, p < 0.001) and alignment (r = 0.76, p < 0.001) highly correlated across the two websites (Supplementary Fig. 6).

Keeping the limitations of these data in mind, online co-purchases reveal a similar level of interest in science among liberal and conservative readers, but not in the same topics. We found little support for the view that science is either an apolitical separate sphere, largely ignored by partisans, or a public sphere in which left and right share common scientific interests. Books on science are more likely than novels or books on religion, sports or the arts to be co-purchased with political books, but left and right rarely purchase the same books. Scientific preferences are polarized at the aggregate level, with liberals attracted to basic science, and conservatives attracted to applied, commercial science. Books in a few ‘purple’ disciplines such as palaeontology are co-purchased by both the left and right, but these ‘bridging’ disciplines also tend to be those with the lowest relevance to political readers. Disciplines with high political relevance, such as social science, law, history and biology, tend to attract politically aligned readers, and even when they attract readers from both left and right, it is not to the same books. Although our analysis cannot uncover the cognitive and social processes underlying these partisan differences in the consumption of science, it is consistent with research on ideologically motivated reasoning35, in which political differences seed different predilections toward politically consequential scientific positions or amplify path-dependent routes of political and cultural sorting19.

Our finding that red books tend to link to a narrow cluster of relatively peripheral books in the more polarized disciplines is consistent with recent research documenting growing scepticism among conservatives of the cultural authority held by professional scientists16. This may also reflect efforts by conservative political movements to offer politicized alternatives to consensus scientific positions. Such efforts have been observed in previous research on climate change10 and a range of other areas36. Science may not be on the front lines of the culture wars, but it is not above the battle, nor is it immune to the ‘echo chambers’ that have been widely observed in political discourse.

This conclusion underscores the need for research into possible remedies. Scholars have begun to explore counter-measures of scientific communication that include helping scientists to establish shared interests with audiences to enhance credibility37, adding public deliberation alongside scientific analysis to separately identify fact and value38,39, and communicating consensus when it exists to help the public set aside protective motivations14. It is hoped that these will counter selective exposure to ‘convenient truth’ and renew the promise of science to inform and elevate political debate.

## Data collection

We collected metadata for 1,449,525 books from the largest online book retailer, Amazon.com, in spring 2013. Starting from two seed books, one liberal (Barack Obama’s Dreams from My Father) and one conservative (Mitt Romney’s No Apology), we collected all information accessible from the webpage of each book, including descriptive information, reviews, and the list of up to 100 books under “Customers Who Bought This Item Also Bought”. We then collected identical data for each book on the recommended list, iterating that process until no new books could be identified. The resulting ‘snowball sample’ contained virtually all books in the largest strongly connected component in Amazon’s directed co-purchase network.

Because every title may have multiple editions or formats, for example paperback and hardcover, each of which is associated with a distinct ISBN (International Standard Book Number), we consolidated different editions and formats based on the unique ASIN (Amazon Standard Identification Number) provided in the source code of each book page. After consolidation we ended up with 1,303,504 unique titles in the dataset. In the main text and below, we use ‘books’ to refer to the distinct titles and never to the physical objects.

The political and science books identified in amazon.com were also collected from barnesandnoble.com, together with books from the Barnes & Noble co-purchase recommendation lists for those books. See below for how science and political books are identified. A brief summary of the two datasets is given in Supplementary Table 1.

## Political books

Our 3,714 candidate political books were chosen from three sources: 1,667 books listed in Amazon’s ‘Liberalism & Conservatism’ category; 1,812 additional books listed in 20 Amazon categories that share the most books with ‘Liberalism & Conservatism’; and 320 books written by prominent US political leaders, including members of the US Congress since 1993 and major party presidential candidates since 1992.

Each candidate political book was examined by at least two independent coders, with another independent coder as a tiebreaker. The coders classified each book as liberal, conservative or ideologically indeterminate, based on the information available on the book’s Amazon webpage. For a book to be coded as liberal or conservative, it had to meet two basic requirements. First, the book must have political content. That is, the book must express an ideological position on a partisan political or social issue, even if that issue is not necessarily the main topic of the book. For example, autobiographies are nominally about a person, but autobiographies of political figures often express ideological positions on contested political issues, in which case the book meets the requirement to have political content. Second, the ideology that the book espouses must be consistently liberal or conservative. If the book expressed contradictory ideological positions, or centrist positions that are shared by both liberals and conservatives or by neither, the book was coded as indeterminate.

Two independent coders reached agreement on 83.5% of the candidate books listed in Amazon’s ‘Liberalism & Conservatism’ category. Another two independent coders reached agreement on 70% of the books in the 20 categories that overlap with ‘Liberalism & Conservatism’. The lower agreement reflects the greater diversity among books that were not listed by Amazon under ‘Liberalism & Conservatism’. Conflicts were resolved by other independent coders and, in rare cases, by the authors. In total, we identified 677 conservative (‘red’) books, 587 liberal (‘blue’) books, and 2,545 indeterminate books. Details of the conservative and liberal books are given in Supplementary Tables 3 and 4.

We tested for possible bias introduced by the exclusion of political books that did not meet both our criteria for red–blue classification, by including 315 ‘indeterminate’ books written by US Congressmen coded red or blue based on the authors’ party memberships. We found that adding these ‘author-coded’ books to the ‘content-coded’ books did not weaken the reported results (see Supplementary Fig. 8).

Distributions of sales ranks for liberal and conservative books are shown in Supplementary Fig. 2, along with distributions of publication years for the two sets of books. The average logarithmic sales ranks for liberal and conservative books are 13.5 and 13.1, respectively, and the medians are 13.7 and 13.4. The average publication years of both sets of books are 1999, and the median publication year is 2007 for liberal books and 2009 for conservative books. A summary is provided in Supplementary Table 2.

## Science books and categories

We consolidated the science-related categories in the Library of Congress and Dewey Decimal classification systems and reorganized these categories into 27 exclusive high-level topics, corresponding to broadly defined scientific disciplines (such as physics, chemistry, medicine or economics). These 27 disciplines fall under four main scientific ‘schools’ (humanities, physical sciences, life sciences and social sciences). We then used the Library of Congress and Dewey Decimal codes of the books to sort them into disciplines.

An additional 494,278 non-science titles were grouped in four main topics — arts, sports, literature (fiction and poetry) and religion — as a baseline for assessing co-purchase links between science and politics. (Detailed organization of the Library of Congress and Dewey Decimal categories can be found in Supplementary Tables 5 and 6).

## Political relevance

Political relevance measures how likely it is that books from a given topic will be co-purchased with political books. It is a number between 0 and 1, defined as the probability θ that a co-purchase link from books in a given topic will link that topic with political books (red or blue). More formally, we assume that the number of undirected co-purchase links X between the topic (for example climatology) and political books has a binomial distribution, X binomial(K,θ), where K is the total number of links attached to the topic and indicates ‘distributed as’.

A straightforward estimator of this probability θ is the number of co-purchase links between the topic and political books divided by the total number of co-purchase links between the topic and all other topics, X/K. However, this estimator is not appropriate in this application for two reasons. First, a topic might have few links to other topics, which renders this measure of relevance unreliable. For example, if a topic has only one link to other topics and it links to political books, then we have θ = X/K = 1, implying that the topic is extremely relevant, which is dubious because there is only one co-purchase link in total. Second, although the uncertainty of this estimator resulting from too few observations could be captured by a confidence interval, we needed to compare relevance across topics while still taking this uncertainty into consideration. We needed to normalize the scores across topics. For these two reasons, we developed a Bayesian estimate for the probability θ. The procedure is as follows. First, we assume a null model in which all the co-purchase links between topics are randomly shuffled while preserving the number of links attached to each topic (compare with the configuration model32). This ‘best guess’ of the co-purchase network (without knowing the identities of the topics) represents a prior distribution on θ, θ beta (dkpolitical/m, d(1 – kpolitical/m), where kpolitical is the total number of links attached to political books and m is the total number of links between topics (including political books). kpolitical/m is the probability of linking to political books in our random null model, and d controls the strength of our prior. The values of θ will depend on d, but we are interested in the relative size of θ across topics, and thus our results are insensitive to the choice of d. We use the average number of links to political books over all disciplines as d. After observing the number of links between the topic and political books, we can update our estimate of θ using Bayes rule: $P(θ|X)=P(X|θ)P(θ)P(X)$

We can then derive a Bayesian estimate for the probability θ: $θ∼beta(dkpolitical/m,d(1−kpolitical/m)$ $X|θ∼binomial(K,θ)$ $θ|X∼beta(X+dkpolitical/m,K−X+d(1−kpolitical/m)$ where the posterior distribution θ|X combines our initial guess of θ (based on the randomized network) with the actual data X and weights the actual data by the number of co-purchase links attached to the topic.

Finally, the political relevance of the topic is defined as the mean of the posterior distribution of θ: $E[θ|X]=X+dkpolitical/mK+d$

This model further reflects how the actual observations (X/K) are combined with prior beliefs (kpolitical/m) to incorporate uncertainty resulting from too few observations on the point estimator of θ. If the number K of co-purchase links attached to a topic is small, we do not have enough evidence to estimate its political relevance, and this will be close to the level in the random null model; if a topic has many co-purchase links, we have greater trust in the estimate.

## Political alignment

Political alignment locates each discipline on the blue–red spectrum based on the interest among conservative readers relative to liberal. Analogous to political relevance, alignment is defined as the probability that a co-purchase link attached to a book in a given topic is to a red book, conditioned on the link being to a political book (red or blue). Note that we restrict our focus to links with political books rather than to all topics. Hence, it can be viewed as a measure of how relevant the topic is to readers of red books as opposed to blue books. For ease of notation, we also denote this probability θ. One way to estimate this probability θ is to divide the number of links (Xred) between the topic and red books by the total number of links (Kp) between this topic and political books, Xred/Kp. Similar issues arise as for political relevance, however, and hence we developed an analogous Bayesian model to estimate θ.

First, we assume a null model in which co-purchase links between a given topic and political books are randomly shuffled while preserving the number of links attached to each (compare with the configuration model32). Let kred and kblue be the total numbers of links attached to red and blue books, respectively. Then the probability of linking to red books in the random null model is kred/(kred + kblue), which suggests a prior distribution on θ, $θ∼beta(dkred/(kred+kblue),dkblue/(kred+kblue))$ where d controls the strength of our prior assumption. After observing the number of links (Xred) between the topic and red books, and the number of links (Xblue) between the topic and blue books, we update our knowledge on θ and obtain the posterior distribution of θ. In summary, the Bayesian model for estimating θ is as follows: $θ∼beta(dkred/(kred+kblue),dkblue/(kred+kblue))$ $Xred|θ∼binomial(Kp,θ)$ $θ|Xred∼beta(Xred+dkred/(kred+kblue),Kp−Xred+dkblue/(kred+kblue))$

Accordingly, political alignment is calculated as the mean of the posterior distribution: $E[θ|Xred]=Xred+dkred/(kred+kblue)Kp+d$

We note one extra step in presenting the alignment score. Naively, one might expect a topic to be right-leaning if the topic has a larger probability of linking with red books than blue books (that is, θ > 0.5), but this is not a fair comparison. Because red books have more total co-purchase links than blue books, red books would also possess more co-purchase links in the random null network. Therefore, the fair comparison is to compare θ with kred/(kred + kblue), which is the probability of linking to red books in the random null model. To make the presentation of results more accessible and intuitive, we scaled θ linearly so that when θ = kred/(kred + kblue) the rescaled θ would be 0.5. In the main text and in the Supplementary Information, θ is always rescaled and 0.5 is the ‘neutral’ point, which accords with intuition. The rescaling is straightforward: θ is rescaled to 0.5θ/[kred/(kred + kblue)] when θ < (kred + kblue), and to 0.5[θkred/(kred + kblue)]/[1 − kred/(kred + kblue)] + 0.5 otherwise.

Lastly, we checked the consistency of our measurements of political alignment by measuring the alignment of all political books and comparing the results with the ‘ground truth’ red and blue hand-codings. Specifically, for every political book that had been hand-coded as blue or red, we ignored the ‘ground truth’ and instead imputed the book’s alignment using the same procedure for measuring alignment of scientific topics introduced above. We then classified a book with alignment θ > 0.5 as red and otherwise blue. The imputed ideology agreed with human codings for over 96% of the political books. Inspection of the handful of anomalies revealed red books that appeal to liberals (for example conservative criticisms of religious political influence) and blue books that appeal to the Tea Party (for example Alinsky-inspired community organizing).

## Political polarization

Political polarization measures the extent to which interests from conservatives and liberals in a given topic diverge. In other words, even if conservatives and liberals are equally likely to buy books of a given topic, they might buy distinct, ideology-relevant books. Polarization identifies this possibility. For a given topic, we computed the number of books within the topic linked to both red and blue books, divided by the number of books linked to either (we call this quantity ‘overlap’, denoted by o). We then compared observed overlap with the expected value in a null model in which links from red and blue books are randomly assigned to books in the topic.

Specifically, for each link between books in a given topic and political books, we shuffle the book in the topic to a randomly chosen topical book, also linked to political books. This results in all politically relevant books from the topic being linked to political books uniformly at random. After all political links have been randomized, the fraction of books linked to both red and blue among books linked to either is calculated in the randomized network. We carry out 100 such random simulations to obtain a distribution of the overlap between blue and red (the fraction of books linked to both blue and red) in the random model. Finally, polarization is measured as the difference between the expected overlap in the random null model, E[O], and the observed overlap: $E[O]−ovar[O]$

If the observed overlap is smaller than what would be expected at random, polarization is positive and increases with the difference between expected and observed overlaps. Polarization of a topic equals zero when red and blue books are co-purchased with books within the topic uniformly at random, but increases as the sets of red- and blue-linked books diverge, indicating red and blue preferences for distinct books.

## Applied index

To quantify the extent to which the political alignment of scientific fields is correlated with their application, we developed an index measuring the extent to which a field is commercially applied.

Among many possible measures of commercial application, a reasonable and tractable one is the degree to which patents build upon knowledge produced by the field. To that end, we use the US patent database from Google, 1976–2014, which is the most digitally complete. For each journal, we tabulate the number of times that journal was cited by all patents in the patent database. We aggregated these within the Dewey Decimal and Library of Congress categories in which Amazon books are categorized, such that the number of citations received by a field is computed as the sum of citations received by all the journals in that field.

Citations from patents meaningfully reflect gross contributions of commercial relevance to the real world, but they do not fully capture the degree to which a field is applied, because the number of citations is strongly influenced by field size and activity. Imagine a small, focused field that is commercially applied, compared with a large, broad field. Perhaps the smaller, more focused field receives fewer citations from patents only because it produces fewer total patents, but it is actually more application-oriented because all knowledge in the field is transferred to technology. Therefore, we build a field-level citation network to capture how active or impactful a field is in the scientific space.

We find that the number of citations that a sub-discipline (as defined by our classification of science categories) receives from patents is strongly correlated (r = 0.8, p < 0.001) with the number of citations that it receives from other sub-disciplines (Supplementary Fig. 3). This correlation reveals that most sub-disciplines are proportionally active in both patent and academic domains, and either of the two types of citations alone is not sufficient to estimate how applied the field might be. Accordingly, we constructed a commercial applied index that combines both kinds of citations and measures how much each sub-discipline is cited by patents relative to articles. Specifically, for each sub-discipline i, we denote the number of citations from patents by yi and the number of citations from articles in other sub-disciplines by xi. The expected number E[Y] of citations from patents given the number X of citations from other sub-disciplines is modelled E[Y] = exp(aX + b), which explains the correlation illustrated in Supplementary Fig. 3. Finally, the commercial application index A of a sub-discipline is calculated as the number of citations from patents normalized by its expected number of patent citations given the number of citations from other sub-disciplines: Ai = yi/exp(axi + b).

## Scientific breadth

Scientific breadth measures the breadth of interests in science from conservatives and liberals. For example, if red books link to a narrow subset of books within a discipline, while an equal number of blue books connect to a large and diverse subset of disciplinary books, those purchasing blue books have exposure to a wider range of science books — and probably a wider range of scientific perspectives — than those purchasing red books.

With respect to each discipline, the scientific breadth of conservatives (or liberals) is defined by the number of distinct titles within the discipline linked to red (or blue) books divided by the number of red (or blue) books linked to the discipline.

## Centrality

The network location of each book with links to red and blue relative to the core or periphery of a disciplinary book network was quantified by assessing the closeness centrality41 of each book with respect to the given disciplinary network. Centralities of blue-linked books were then compared with those of red-linked books within each discipline to assess core location of blue and red with respect to the discipline. Books at the centre of a disciplinary book network can be interpreted as having greater disciplinary relevance, given their co-presence in personal libraries with many other books in the discipline. In contrast, books at the periphery are rarely purchased with other books in the discipline.

To compare the difference between centralities of blue- and red-linked books across disciplines, we calculated the widely used t-score $E[X]−E[Y]S2(1/nx+1/ny)$

for each discipline, where X corresponds to centralities of blue-linked books, Y to centralities of red-linked books, nx to the number of blue-linked books and ny to the number of red-linked books; S2 is the pooled sample variance of blue- and red-linked books.

The standardized difference (t-score) between centralities of blue- and red-linked books is shown in Supplementary Fig. 4. The mean centrality of blue-linked books is larger than that of red-linked books for most disciplines, and as polarization increases, blue-linked books are more central than red-linked books. Exceptions to this pattern include history and economics.

## Data availability

All data that support the findings of this study are publicly available at http://www.knowledgelab.org/research_grants/research/amazon/

## Code availability

Python code for data processing, analysis and visualization is publicly available at http://www.knowledgelab.org/research_grants/research/amazon/

## Additional information

How to cite this article: Shi, F., Shi, Y., Dokshin, F. A., Evans, J. A. & Macy, M. W. Millions of online book co-purchases reveal partisan differences in the consumption of science. Nat. Hum. Behav. 1, 0079 (2017).

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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## Acknowledgements

We are grateful for comments from participants in seminars at Microsoft Research–New England, MSR-NY, GESIS-Koln, University of Michigan School of Information, Duke University DNAC, the first International Conference on Computational Social Science, and the Computational Social Science Summit at Northwestern. We acknowledge funding from the John Templeton Foundation to the Metaknowledge Network, NSF SES 1303533, SES 1226483, SES 1158803, National Research Foundation of Korea NRF-2013S1A3A2055285 and Air Force Office of Scientific Research FA9550-15-1-0162, and computation support from the Open Science Data Cloud. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

## Author notes

• Feng Shi
•  & Yongren Shi

These authors contributed equally to this work.

## Affiliations

1. ### Computation Institute, University of Chicago, 5735 South Ellis Avenue, Chicago, Illinois 60637, USA

• Feng Shi
•  & James A. Evans
2. ### Yale Institute for Network Science, Yale University, 17 Hillhouse Avenue, New Haven, Connecticut 06511, USA

• Yongren Shi
3. ### Department of Sociology, Cornell University, 323 Uris Hall, Ithaca, New York 14853, USA

• Fedor A. Dokshin
•  & Michael W. Macy
4. ### Department of Sociology, University of Chicago, 1126 East 59th Street, Chicago, Illinois 60637, USA

• James A. Evans

## Contributions

All authors designed the research and drafted the paper. F.S., Y.S. and F.A.D. gathered and analysed the data.

## Competing interests

The authors declare no competing interests.

## Corresponding authors

Correspondence to James A. Evans or Michael W. Macy.

## PDF files

1. 1.

### Supplementary Information

Supplementary Figures, Supplementary Notes, Supplementary Tables 1 and 2.

1. 1.

2. 2.

1. 1.

2. 2.

format: plain


text:
A note on the number of squares in a partial word with one hole
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 43 (2009) no. 4, pp. 767-774.

A well known result of Fraenkel and Simpson states that the number of distinct squares in a word of length $n$ is bounded by $2n$ since at each position there are at most two distinct squares whose last occurrence starts. In this paper, we investigate squares in partial words with one hole, or sequences over a finite alphabet that have a “do not know” symbol or “hole”. A square in a partial word over a given alphabet has the form $uv$ where $u$ is compatible with $v$, and consequently, such square is compatible with a number of words over the alphabet that are squares. Recently, it was shown that for partial words with one hole, there may be more than two squares that have their last occurrence starting at the same position. Here, we prove that if such is the case, then the length of the shortest square is at most half the length of the third shortest square. As a result, we show that the number of distinct squares compatible with factors of a partial word with one hole of length $n$ is bounded by $\frac{7n}{2}$.

DOI: 10.1051/ita/2009019
Classification: 68R15,  05A05
Keywords: combinatorics on words, partial words, squares
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Blanchet-Sadri, Francine; Mercaş, Robert. A note on the number of squares in a partial word with one hole. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 43 (2009) no. 4, pp. 767-774. doi : 10.1051/ita/2009019. http://www.numdam.org/articles/10.1051/ita/2009019/

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[2] F. Blanchet-Sadri, Algorithmic Combinatorics on Partial Words. Chapman & Hall/CRC Press, Boca Raton, FL (2008). | MR

[3] F. Blanchet-Sadri and D.K. Luhmann, Conjugacy on partial words. Theoret. Comput. Sci. 289 (2002) 297-312. | MR | Zbl

[4] F. Blanchet-Sadri, R. Mercaş and G. Scott, Counting distinct squares in partial words, edited by E. Csuhaj-Varju, Z. Esik, AFL 2008, 12th International Conference on Automata and Formal Languages, Balatonfüred, Hungary (2008) 122-133, www.uncg.edu/cmp/research/freeness | MR

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Cited by Sources:


text:
Batch Selection for Parallelisation of Bayesian Quadrature

Batch Selection for Parallelisation of Bayesian Quadrature

Ed Wagstaff
University of Oxford
University of Oxford
Michael Osborne
University of Oxford
Abstract

Integration over non-negative integrands is a central problem in machine learning (e.g. for model averaging, (hyper-)parameter marginalisation, and computing posterior predictive distributions). Bayesian Quadrature is a probabilistic numerical integration technique that performs promisingly when compared to traditional Markov Chain Monte Carlo methods. However, in contrast to easily-parallelised MCMC methods, Bayesian Quadrature methods have, thus far, been essentially serial in nature, selecting a single point to sample at each step of the algorithm. We deliver methods to select batches of points at each step, based upon those recently presented in the Batch Bayesian Optimisation literature. Such parallelisation significantly reduces computation time, especially when the integrand is expensive to sample.

\DeclareLanguageMapping

1 Introduction

Both within and without machine learning, we encounter integrals of the form

 Z=∫ℓ(θ)π(θ)dθ, (1)

where both (e.g. a likelihood), and (e.g a prior) are non-negative. These integrals are usually intractable, and we must therefore turn to approximation to solve them.

One possibility is the Laplace approximation. For multimodal distributions it can be repeated on the residuals – the difference between the approximation and the true distribution – to obtain a better approximation \parenciteBornkamp2011. However, this requires optimisation and is not invariant to changes of representation. Another possibility is variational inference which provides a lower bound for the value of the integral, but there is limited underpinning theory and optimisation of the lower bound can be difficult \parenciteBlei2018. Markov Chain Monte Carlo (MCMC) methods have also seen broad use and success, but they are subject to a number of objections \parenciteOHagan1987, the most pertinent of which is that they ignore the locations of samples from the likelihood when computing the integral. They therefore suffer from low sample efficiency and a poor convergence rate. All of these approximation methods are unsuitable for applications where evaluating the desired likelihood is expensive (e.g. if we must make an estimate based on only a few dozen evaluations).

Bayesian Quadrature (BQ) offers an alternative framework for approaching the same integral problem. It belongs to a family of methods termed probabilistic numerics \parenciteHennig2015, which view the approximation of intractable quantities as a problem of inference from limited data, to which the tools of probability theory can be applied. In BQ, a probabilistic model of the integrand (typically a Gaussian Process) is maintained, from which the integral is inferred. Gunter et al. \parencite*Gunter2014 showed that BQ can achieve faster convergence – both in wall-clock time and sample complexity – than state of the art MCMC methods. Some deterministic quadrature techniques can be reinterpreted in terms of Bayesian Quadrature – for instance, the trapezoid rule (with convergence) arises from BQ with a linear spline kernel \parenciteKarvonen2017.

One drawback of BQ as developed thus far is that is not easily parallelised, in contrast to the comparatively straightforward situation with MCMC methods where, for instance, several independent Markov chains may be run in parallel. While the probabilistic model allows us to explore the space more efficiently, determining where next to evaluate the integrand depends on the current state of the model, i.e. on all evaluations made so far. Nonetheless, parallelisation is essential to realising competitive performance using modern computational hardware.

Using ideas applied in Batch Bayesian Optimisation, particularly those by Gonzalez et al. \parencite*Gonzalez2015, we propose a novel technique for parallelising BQ by selecting new points in batches rather than one-by-one.

In Bayesian Quadrature, we work with a probabilistic model of an integrand which induces a probability distribution over the value of the integral. The early work on BQ \parenciteDiaconis1988, OHagan1991, Minka2000, Rasmussen2003 identified that Gaussian Processes offer a well-studied method for placing a probability distribution over functions, so we work with these to provide our probabilistic model.

Recall from Equation (1) that we are interested in the quantity . We typically choose to place our Gaussian process prior on the likelihood , rather than on the product (though we could choose to do either).

Given a Gaussian prior and a GP model for our likelihood,

 ℓ∣y ∼GP(my,Cy) (2a) my(x) =k(x,X)k(X,X)−1y (2b) Cy(x,x′) =k(x,x′)−k(x,X)k(X,X)−1k(X,x′) (2c)

the value of the integral is Gaussian distributed – since Gaussian Processes are closed under linear operators (see e.g. [Rasmussen2006]), the GP on the function induces a normal distribution on the value of the integral . By linearity of expectation and bilinearity of covariance, the mean and variance of are simply \parenciteBriol2015:

 E(Z∣y) =∫my(x)π(x)dx (3a) var(Z∣y) =∬Cy(x,x′)π(x)π(x′)dxdx′ (3b)

These integrals are, for certain choices of prior, , and kernel, , analytic. In particular, this is true when is a Gaussian and is the squared exponential covariance, and we use this combination throughout this paper.

At a high level, the advantages of Bayesian Quadrature are two-fold. First, with the stronger prior of BQ, evaluations provide stronger constraints on the integrand, and therefore more information about the integral. Secondly, we can use the uncertainty of the GP model to inform our choice of where next to evaluate the integrand, thus exploring the space more efficiently. We do this by defining and maximising an acquisition function which is high-valued at points which, according to our model, will reduce the uncertainty in the integral. In the standard BQ model, this second advantage does not apply – note that the posterior covariance only depends on the locations sampled, not on the function value at those points \parenciteOsborne2012. However, as we shall see below, it is possible to develop more complex models where, among other advantages, the covariance of better-informed posteriors does depend on the sampled function values.

The Gaussian process framework has some restrictions which interfere with our ability to model our true beliefs about the functions we wish to work with. Standard Bayesian Quadrature does not allow us to express our belief that the function to be modelled (in our case a likelihood) is non-negative, since the distribution at any point is a Gaussian. Recent work has investigated applying a warping to the output space of the GP to address this \parenciteOsborne2012, Gunter2014, Chai2018. Rather than modelling the likelihood itself as a GP, we model a transformed likelihood such that the true likelihood is non-negative (i.e. such that the range of is the positive reals). For example, and have been investigated. Though the final posterior on is not a GP, it is possible to derive a GP which closely approximates it.

For the purposes of this investigation we follow the method of Gunter et al. \parencite*Gunter2014, dubbed WSABI, which makes use of the transformation . In particular we use the approximation dubbed WSABI-L, which has the virtue of providing a closed-form estimate for the final integral rather than requiring Monte Carlo estimation. In WSABI, the acquisition function is simply the variance of the posterior model – due to the form of the model, this automatically balances exploring areas of high uncertainty and exploiting in areas where the objective function is believed to be high-valued. WSABI has so far been considered only as a sequential process, selecting and evaluating one point at a time. We propose its parallelisation by turning to batch methods.

3 Batch methods

Bayesian Quadrature as described above is an essentially serial process. That is, we must sample a single point at each step, before updating our model (and therefore our acquisition function) accordingly in order to recommend the next point. Since we expect each sample to be expensive, this is a serious bottleneck – we would rather be able to work in parallel, taking several samples at once in order to gain the maximum amount of information about our integrand. In order to enable this, we turn to ideas which have been successfully used in Bayesian Optimisation.

The field of Bayesian Optimisation has a clear relationship to Bayesian Quadrature. In particular, both fields use a probabilistic model to guide exploration of a search space by defining an acquisition function to be maximised. In Bayesian Optimisation, so-called batch methods have been proposed as an effective way of allowing parallelisation of this exploration (see e.g. Ginsbourger et al. \parencite*Ginsbourger2010, Gonzalez et al. \parencite*Gonzalez2015, Nguyen et al. \parencite*Nguyen2017).

In batch methods, rather than selecting one point to sample at each step, we select a batch of points, usually of a fixed size . Naively one could try to select the highest points of the acquisition function, but this has the obvious issue that these will all be essentially equal to the maximum of the acquisition function, leading to an uninformative batch – the quality of the batch as a whole is more complex than simply the sum of the quality of each point individually. One way of viewing this issue is that we require an informative batch, and therefore we require a diverse batch.

To find the optimal batch we could, in principle, define an acquisition function over all sets of points (e.g. by considering the expected reduction in variance of our integral estimate) and maximise it, but in practice this is too expensive – it increases the dimensionality of the problem by a factor of and the acquisition function is also likely to be more costly to compute. It is therefore common to turn to heuristics where each point is selected sequentially, with the acquisition function modified after each point is selected.

A simple method for performing this sequential selection is the so-called Kriging Believer strategy \parenciteGinsbourger2010. We “sample” our chosen point by setting its value to the posterior mean, update our Gaussian process model to take the new evidence into account, and select the most informative point from the posterior. We repeat this process until we have a sufficiently large batch, and then make the (real) evaluations at selected points and input them as observations into our model. This naturally forces further exploration of the space, as the region immediately around each “sampled” point will no longer be considered informative.

We may also consider a local penalisation strategy which directly penalises the acquisition function around the chosen point (in Bayesian Optimisation, this strategy was introduced by Gonzalez et al. \parencite*Gonzalez2015). With this strategy, we don’t use our full model, as in Kriging Believer, but instead directly modify the acquisition function to lower its value around each selected point, encouraging diversity in the final batch.

In the next section, we will consider the application of batch methods to Bayesian Quadrature.

4 Method

In this section we introduce, to our knowledge, the first Batch Bayesian Quadrature routine. We also introduce a novel version of local penalisation.

The core algorithm is simple – we repeatedly select a batch of points, evaluate the integrand at these points, and input the returned data into our model. At each iteration, we can calculate an estimate of the integral using BQ. There is a question here of the stopping criterion. Theoretically, Bayesian Quadrature offers a principled stopping criterion since it returns a variance for its estimate of the integral, and we can simply evaluate the variance at each iteration and stop once it has reached a certain threshold. For the specific case of WSABI-L, the model is overconfident away from the data, and we expect that the BQ variance will therefore be erroneously low and unsuitable as a stopping criterion. This is a question that clearly merits further investigation, but for the purposes of this paper we simply set a fixed computational budget.

The most important step of the algorithm is the batch selection process. Above we described the Kriging Believer approach, and in general terms described local penalisation. We now outline our approach to local penalisation.

4.1 Local Penalisation

With local penalisation methods, there is a balance to be struck in how wide the penalisers are – if our penalisation affects a broad region, we may erroneously exclude genuinely informative points. If, however, our penalisation affects a narrow region, we may not sufficiently encourage diversity, and produce a batch of points which are clustered too closely together. Gonzalez et al. \parencite*Gonzalez2015 propose a multiplicative penaliser with a minimum at the sampled point. To determine how wide the penaliser should be, they make use of a Lipschitz constant, a measure of the maximum absolute slope of the acquisition function. In this method the Lipschitz constant is incorporated into a probablility density function which is used as part of the penaliser. The authors’ proposition is less suited to quadrature than to optimisation, as it is not sufficiently exploratory – it heavily penalises regions surrounding a selection which has a high variance. Thorough exploration is more important in quadrature, since the result depends on the global behaviour of the function rather than a single optimum. We therefore consider an alternative approach, attempting to place an upper bound on the true acquisition function (i.e. the acquisition function had the integrand been evaluated at the previous selections).

Recall that a function is said to be Lipschitz continuous if there is a constant such that, for all , ,

 ||f(x)−f(y)||≤L||x−y||. (4)

In our method, we apply the Lipschitz constant more directly. We expect our true acquisition function to be zero at the sampled point and increase away from it, and wish for our penalised function to reflect this. We find a local Lipschitz constant for a given point by locally optimising the norm of the gradient of the acquisition function. Viewing this as a bound on the slope of the acquisition function around that point, we then define a Lipschitz Cone which is zero at the point and increases linearly away from the point. i.e. for a Lipschitz constant and point , we define the Lipschitz Cone,

 C(x)=L||x−x0||. (5)

Our acquisition function is then taken to be the minimum of the original acquisition function and the Lipschitz Cones of all points so far added to the batch.

This minimum, however, is not differentiable everywhere. In particular, it is not differentiable where the cone intersects the original acquisition function (or where two cones intersect). These intersections may in fact be maxima of the penalised acquisition function, so ideally the penalised function would have stationary points on these intersection surfaces. We address this by applying a soft minimisation, rather than the true minimum, which leaves us with an acquisition function which is differentiable everywhere except at the centres of the Lipschitz cones (i.e. the already selected batch points).

Recall that the norm is defined by

 ||x||p=(∑ixpi)1p. (6)

It is well known that the -norm for large approaches the norm, . Large values of can therefore be viewed as applying a kind of soft maximisation. Large negative values of , on the other hand, apply a soft minimisation to the input vector.111For negative this is no longer a norm, but we refer to it as an -norm for brevity. We therefore take our final acquisition function to be an -norm of the vector whose components are the function evaluation for each of the penalisers and the original acquisition function. There is a balance to be struck in the choice of here – large will more closely resemble the true minimum of all the functions, but will also be more prone to numerical errors (these errors will appear particularly around the important regions where the true minimum is not differentiable). We found that small values of already came reasonably close to the true minimum function, and selected for our experiments.

Since we expect the true acquisition function to be smooth, rather than resembling a sharp cone near evaluated points, we set the gradient of our Lipschitz cone to be a fraction of the Lipschitz constant – this has the effect of encouraging (or rather not discouraging) exploration. We found to work well in practice. In principle, this parameter could be tuned to help determine the degree to which the algorithm explores diverse regions of the space, with smaller values encouraging exploration more strongly, but we did not investigate this possibility as part of this work.

4.2 Optimising the acquisition function

To optimise the acquisition function, we run a local optimiser multiple times from randomly selected starting locations. We sampled these locations according to the prior against which we are integrating. We chose the number of starting locations to be 10 times the dimensionality of the integrand.

4.2.1 Implementing Multi-start Optimisation

Our code for this method was written in python, using numpy and scipy. The most straightforward way to implement multi-start optimisation is to simply loop over the starting points and perform the optimisation many times. It is however possible to speed up this process when using numpy by taking advantage of fast vectorised computation in numpy rather than using slow python loops. The acquisition function can be evaluated at many points by passing these points as a numpy array, and this is many times faster than passing the points one at a time in a python loop. To take advantage of this in the optimisation process, we posed our multi-start optimisation problem as a single-start optimisation problem in higher dimensions. Each point in our high-dimensional space is the concatenation of multiple points in the original low-dimensional space. Denoting concatenation by , we define a new function . The Jacobian of this function is then obtained by concatenating the Jacobian of at each , and we eliminate the python loop over starting points by simply starting this high-dimensional optimisation at .

5 Experiments

Prior work \parenciteOsborne2012, Gunter2014, Chai2018 has demonstrated that in terms of wall-clock time, BQ methods are competitive with MCMC. Here we investigate the degree to which adding batch methods can improve the performance of BQ in terms of the number of batches required (which, with an objective function which can be effectively parallelised, is a better indicator of actual time taken than the number of samples taken). For a batch of size , the best we can reasonably hope for is a factor- decrease in batch complexity against the sequential (i.e. batch size 1) method. We test both of our proposed batch selection methods on synthetic problems to assess the improvement achieved by batching, and compare both batch selection methods against MCMC methods in a test computing the model evidence of a Gaussian Process.

Our first experiment is on an in-model integral, i.e. a sample from a Gaussian Process prior with the WSABI warping applied to it. We integrated this function with WSABI for a number of different batch sizes. Each sample was taken in 2 dimensions, and integrated against an isotropic Gaussian prior. Because we are working in 2 dimensions, the ground truth was obtained by simply evaluating the function on a fine grid and evaluating the integral naively. We plot the average absolute error over 10 runs of this experiment in Figure 2.

The in-model test demonstrates effective parallelisation of the process – we can clearly see that larger batch sizes converge much more quickly to the true integral. We also see that both methods are similarly effective – Kriging Believer and Local Penalisation show comparable performance.

Our second experiment is on a synthetic integral. Here we construct our integrand as a mixture of isotropic Gaussians.

We do this in 4 dimensions, varying the number of components between 10 and 15, setting their variance uniformly at random between 1 and 4, and setting their means uniformly at random within the box bounded by in all dimensions. We set the hyperparameters of the underlying GP in our model (which uses an isotropic squared exponential kernel) by optimising the marginal likelihood.

Again we test both of our batch selection methods, with a varying number of batch sizes. We plot the average absolute error over 10 runs of this experiment in Figure 3. The results are similar to the in-model test, demonstrating an effective speedup with increasing batch size, and again showing comparable performance between the two batch selection methods.

Our final experiment is computing the model evidence for a Gaussian Process. The data for the GP was collected by running 20 iterations of Bayesian Optimisation on the Branin-Hoo function, a common benchmark function for Bayesian Optimisation \parenciteJones2001. For the optimisation we used a GP with an isotropic squared exponential covariance, giving us two hyperparameters to marginalise over – we marginalise against a Gaussian prior on the hyperparameters. Since this integral is 2D, we once again obtained ground truth by evaluating on a fine grid.

In this experiment we also compare against two MCMC methods – Metropolis-Hastings (Metropolis et al., [*]Metropolis1953; [Hastings1970]), a popular standard method, and NUTS \parenciteHoffman2014, a newer state-of-the-art method. Both of these methods were run with 500 samples of burn-in, and when comparing against our batched method with a batch size of , we take the result obtained from running parallel chains. Figure 4 shows the average absolute error in the log of the model evidence, over 50 runs of this experiment. Once again it is clear that larger batch sizes converge more quickly for our method, and our method clearly outperforms the MCMC benchmarks.

All three experiments show improved performance as batch size increases, demonstrating that the selected batches are genuinely informative sets of points.

6 Conclusion

In this paper we have presented a routine for performing Bayesian Quadrature as a batched method, and introduced a novel form of local penalisation as part of the batch selection process. Our method allows parallelisation of methods which have so far only been available as serial processes. We have demonstrated that the performance of our method improves as the degree of parallelism increases, and shown that the method is competitive with state-of-the-art MCMC when the sampling budget is small.

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# Trace Resistance Calculator

This calculator calculate the resistance of a copper trace based on width, length, thickness and temperature. The resistance of a trace is important to estimate the power dissipate in a trace or to estimate roughly the voltage drop from a voltage supply based on the current flowing through the trace. The resistance in real life will vary a bit since conductor are not exactly rectangular. The trace on a printed circuit boards is more of a trapezoidal shape.

Width :

Length :

Thickness :

Temperature :

Resistance: Ohms

The equation for resistance of a rectangular copper trace is :

$Resistance (\Omega) = \rho * \cfrac{L}{T*W}*(1+Tc*(Temp-25))$

where :
$\rho$ = Resistivity
L = Length
W = Trace width
T = Trace thickness (or height)
Tc = Temperature Coefficient
Temp = Temperature

This calculator used the following resistivity and temperature coefficient for copper :

Resistivity of copper : $1.7*10^{-6}\ \Omega \ cm$

Temperature Coefficient of copper : $3.9*10^{-3}\ \Omega / \Omega / \textdegree C$


text:
## Degeneracy

Boltzmann Equation for Entropy: $S = k_{B} \ln W$

905085650
Posts: 62
Joined: Fri Sep 28, 2018 12:19 am

### Degeneracy

Can someone explain to me what degeneracy is and how it relates to entropy? Thank you!!

Nahelly Alfaro-2C
Posts: 59
Joined: Wed Nov 15, 2017 3:04 am

### Re: Degeneracy

Degeneracy (w) is the number of ways of achieving a given energy state. The relationship shared between entropy and degeneracy is the Boltzmann's equation- S= kB ln W. The way degeneracy would be calculated is W=2^N in which N represents the particles that can be found in one of two states. When connecting back to Boltzmann's equation it would be:
S=kB lnW
= kB ln(2^N)
=N kB ln2
S= Rln2

Michelle Nwufo 2G
Posts: 61
Joined: Fri May 18, 2018 3:00 am

### Re: Degeneracy

Also, it’s kind of confusing why work and degeneracy have the same constant. Are they somehow related?

Cameron_Greenberg_3C
Posts: 60
Joined: Fri Sep 28, 2018 12:26 am

### Re: Degeneracy

Michelle Nwufo 2G wrote:Also, it’s kind of confusing why work and degeneracy have the same constant. Are they somehow related?

Work has a lowercase w, while degeneracy is an uppercase W. They are not related.

Return to “Third Law of Thermodynamics (For a Unique Ground State (W=1): S -> 0 as T -> 0) and Calculations Using Boltzmann Equation for Entropy”

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text:
# Insert one rational numbers between $$\dfrac{1}{4}\:and\:\dfrac{3}{5}.$$

This question was previously asked in
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1. $$\dfrac{1}{2}$$
2. $$\dfrac{11}{5}$$
3. $$\dfrac{3}{2}$$
4. $$\dfrac{4}{5}$$

Option 1 : $$\dfrac{1}{2}$$

## Detailed Solution

Given

Two rational numbers 1/4 and 3/5

Concept used

Rational numbers :  A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.

Calculation

1/4 can be simplified as

⇒ 1/4 × 100 = 25

3/5 can be written as

⇒ 3/5 × 100 = 60

Go through the options

Option A = 1/2

⇒ 1/2 × 100 = 50

Option B = 11/5

⇒ 11/5 × 100 = 220

Option C = 3/2

⇒ 3/2 × 100 = 150

Option D = 4/5

⇒ 4/5 × 100 = 80

Between 25 and 60 only 50 comes

Ans : 1/2


text:
## Are there many dex based twf'r out there? How are the enhancement working for you?

Specifically my main is a drow 18 tempest ranger/1 rogue /1 fighter.

I like the tempest tree, mostly. I think the dex for damage should include rapiers. The requirement to go into deepwood sniper to get dex to damage on rapiers is, in my opinion, a flaw.

I've delved into the AA tree which is something I've never done before. I got a nice boost to my ranged effectiveness.

The drow racial tree is horribly broken though it looks like it should have great synergy with tempest if it ever gets fixed.

I had high hopes for this character, but overall I'm underwhelmed by the changes to it. I think I came out about even, when on paper it should of been a nice boost to dps. (i.e. I added dex to damage and all the xendrik and tempest damage enhancements but came out with the same dps as before.)

How is it going for the rest of you dex fighters?