2d:null 2e:["$","title",null,{"children":"The Art of Doing Science | Debanjum | Observable"}] 33:T488,belief_updates = { // Setup Iteration Parameters const learning_rate = 0.01 const steps = 10; const theoretical_scientist = make_theoretical_scientist(sampled_positions); const actual = reality(sampled_positions) // Our first (random) theory to explain reality var theory_parameters = [2, 2]; var belief_updates = []; var latest_predictions; // The Scientific Method for (let step = 0; step < steps; step++) { // Predictions by the latest theory var latest_predictions = theoretical_scientist(...theory_parameters); // How close are the predictions to reality? var latest_loss = get_loss(actual, latest_predictions).arraySync() // Collate results sampled_positions.map((p, id) => { belief_updates.push({ "predictions": latest_predictions[id], "loss": `i=${step}, error=${latest_loss.toFixed(2)}`, "sampled_positions": p, "iteration": step }) }); // Update theory to make predictions that better match our experimental data of reality theory_parameters = update(theory_parameters, sampled_positions, learning_rate); } return belief_updates }34:T59b,## 🤔 Next Steps Play with this posts [interactive notebook](https://observablehq.com/@debanjum/the-art-of-doing-science).${htl.html`
`} Extend it to test your own ideas of doing science or implement the ideas below: - The scientific method requires theory and experiment improve in lock step - The theory doesn't just iterate on a static dataset collected by experimentalists like we modeled above - The dataset is made richer and less noisy by experimentalists by creating better techniques and equipment - *Could we train on a slowly changing, improving dataset? Does that increase or decrease the speed of learning?* - New theories should make testable hypothesis - These are predictions for which we have no experimental data yet. But which can be obtained with practical amounts of effort soon. - Experimental validation of theories provides evidence that the theories predictions generalize - In machine learning, the validation dataset is hidden from the model during training for similar reasons - *Can we change the code to have theories make predictions from outside the data it trained on? How well do theories generalize now?* - Machine learning follows a very similar process as the scientific method. Neural networks are [universal approximators](https://en.wikipedia.org/wiki/Universal_approximation_theorem). *Can we use them to automate the process of doing actual science?!*35:[] 31:{"id":"6a34553928939266","slug":"art-of-doing-science","title":"The Art of Doing Science","thumbnail":"804867c277a8734f623ff0c306ebdd441b9557bc52d9f296d51e82cb204383d2","access_level":"public","create_time":"$D2022-12-28T03:56:52.566Z","update_time":"$D2022-12-29T20:42:49.484Z","view_time":null,"first_public_time":"$D2022-12-28T03:56:52.566Z","has_unpublished_edits":null,"fork":false,"stars":2,"starred":false,"owner":{"type":"team","id":"73b8e7a6b35f81ef","login":"debanjum","name":"Debanjum","avatar_url":"https://avatars-next.observableusercontent.com/71f857e5cd72989932412a06de3d43225f0fef328029a743310bec2ea8b7998b","tier":"starter_2024"},"version":832,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"068085ccdab1075c","login":"debanjum","name":"Debanjum","avatar_url":"https://avatars-next.observableusercontent.com/71f857e5cd72989932412a06de3d43225f0fef328029a743310bec2ea8b7998b","tier":null}],"members":[{"type":"individual","id":"068085ccdab1075c","login":"debanjum","name":"Debanjum","avatar_url":"https://avatars-next.observableusercontent.com/71f857e5cd72989932412a06de3d43225f0fef328029a743310bec2ea8b7998b","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":3,"collection":{"id":"67850cef8fae34ff","slug":"taksati","title":"Taksati","owner":{"id":"73b8e7a6b35f81ef","login":"debanjum","name":"Debanjum"}},"type":"notebook","body":{"title":"The Art of Doing Science","cells":[{"id":798,"value":"# The Art of Doing Science","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":795,"value":"This post is a short, [explorable explanation](https://explorabl.es/) of the art of doing science. We'll build a simple universe, perform experiments and create theories that better model reality. In the process we hope to get a better understanding of the scientific method. See the posts [interactive notebook](https://observablehq.com/@debanjum/art-of-doing-science) for a deeper dive.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":797,"value":"\"Note: This notebook was created for a blog post on taksati: https://taksati.xyz/the-art-of-doing-science.html\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"toc({skip: [\"Appendix\", \"The Art of Doing Science\", \"Testing\"], headers: \"h2, h3, h4\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"## ♾️ Generate Reality\n\nOur universe will be **1-dimensional** and **defined by a single scalar property** for the sake of illustration.\nThis universe can be modeled by ${tex`y = reality(x)`}, where\n- ${tex`x`} is a point in the universe\n- ${tex`y`} is the value of the property that defines the universe at that point\n\nFor simplicity, we assume ${tex`reality(x)`} to be linear. That is, ${tex`reality(x) = ax + b`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":39,"value":"// Generate a Linear Reality\nreality = make_linear(a, b)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":41,"value":"// Initialize Generator of Linear Realities","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":47,"value":"function make_linear(slope, intercept) {\n return function(position) {\n return linear(slope, intercept, position);\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"function linear(...params) {\n // normalize parameters to tensors\n let anyTensors = params.filter(isTensor).length > 0\n params = params.map(toTensor)\n let [slope, intercept, position] = params\n\n // Calculate y = f(position) = slope*position + intercept\n let y = position.mul(slope).add(intercept)\n\n // return y with same type as original params\n return anyTensors ? y : y.arraySync()\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":51,"value":"### 🌈 A Rainbow, String Universe\nA rainbow colored, string universe is an example of a 1d, scalar universe. ${htl.html`
`}\nThe colors at each point in this string universe are determined by:\n\n${tex`color = rainbox\\_universe(point\\_on\\_string)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":53,"value":"viewof a = slider(-5, 5, 0.1, 0.5, \"slope\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":55,"value":"viewof b = slider(-5, 5, 0.1, 0.5, \"intercept\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":57,"value":"// A rainbow colored, string universe\n{\n let rainbow_universe = make_linear(a, b)\n return Plot.plot({\n grid: true,\n x: {label: \"position in string universe →\", domain: [_.min(sampled_positions), _.max(sampled_positions)]},\n y: {label: \"↑ color in rainbow universe\", domain: [-2, 2]},\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.dot(\n _.range(_.min(sampled_positions), _.max(sampled_positions), 0.01),\n {\n x: point_on_string => point_on_string,\n y: point_on_string => rainbow_universe(point_on_string),\n fill: point_on_string => rainbow_universe(point_on_string)\n })\n ]\n })\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"## 👩🏻‍🔬 Perform Experiments\n\nExperimentalists capture data points about our universe by performing experiments.\n- Identify locations in the universe from which to capture experimental data: ${tex`_.range(start: -2, end: 2, step: 0.1)`}\n- Model the noise introduced in capturing the experimental data: ${tex`experimental\\_data = reality(position) + \\mathcal{N}(0, \\sigma^2)`}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":64,"value":"// Perform Scientific Experiments to Sample Data from Reality","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"sampled_positions = _.range(-2, 2, 0.1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":72,"value":"sampled_reality = sampled_positions.map(position => reality(position))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":149,"value":"viewof noise = slider(0, 2, 0.1, 0.2, \"add noise\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":74,"value":"noisy_sampled_reality = add_noise(sampled_reality, noise);","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":76,"value":"// Visualize data samples of reality collected from our experiment\n// It gives us a glimpse into the true nature of reality\nPlot.plot({\n grid: true,\n x: {label: \"position in string universe →\", domain: [_.min(sampled_positions), _.max(sampled_positions)]},\n y: {label: \"↑ color in rainbow universe\", domain: [-2, 2]},\n marks: [\n Plot.ruleY([0]),\n Plot.ruleX([0]),\n Plot.dot(\n sampled_positions,\n {\n x: sampled_positions,\n y: noisy_sampled_reality,\n fill: sampled_positions,\n })\n ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":88,"value":"## 👨🏻‍⚕️ Do Theoretical Research\n\nNow let us do some theoretical research. ${htl.html`
`}\nThe true nature of the universe is visible to you. ${htl.html`
`}\nCreate a theory that best fits reality. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":186,"value":"viewof slope = slider(-5, 5, 0.1, 0, \"slope\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":193,"value":"viewof intercept = slider(-5, 5, 0.1, 0, \"intercept\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":460,"value":"error = {\n const latest_theory = make_linear(slope, intercept);\n const predictions = sampled_positions.map(latest_theory)\n const loss = get_loss(predictions, reality(sampled_positions));\n return htl.html`${tex`error: ${loss.arraySync().toFixed(2)}`}`\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":202,"value":"{\n const latest_theory = make_linear(slope, intercept);\n const theory_plot = sampled_positions.map(position => { return {\"x\": position, \"y\": latest_theory(position), \"stroke\": \"#e15759\"}})\n const reality_plot = sampled_positions.map(position => { return {\"x\": position, \"y\": reality(position), \"stroke\": \"#4e79a7\"}})\n\n return Plot.plot({\n grid: true,\n x: {label: \"position →\", domain: [_.min(sampled_positions), _.max(sampled_positions)]},\n y: {label: \"↑ value\", domain: [-2, 2]},\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n Plot.line(reality_plot, Plot.windowY({k: 1, x: \"x\", y: \"y\", stroke: \"stroke\"})),\n Plot.line(theory_plot, Plot.windowY({k: 1, x: \"x\", y: \"y\", stroke: \"stroke\"}))\n ]\n })\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":90,"value":"Let us try theoretical research again. ${htl.html`
`}\nBut now the true nature of the universe is hidden from you. ${htl.html`
`}\nYou only have the noisy, sampled experimental data to guide you. ${htl.html`
`}\nCreate a theory that best fits reality.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":369,"value":"viewof slope_n = slider(-5, 5, 0.1, 0.5, \"slope\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":370,"value":"viewof intercept_n = slider(-5, 5, 0.1, 0.5, \"intercept\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":401,"value":"{\n const latest_theory = make_linear(slope_n, intercept_n);\n const predictions = sampled_positions.map(latest_theory)\n const theory_plot = sampled_positions.map((position, id) => { return {\"x\": position, \"y\": predictions[id], \"stroke\": \"#e15759\"}})\n const loss = get_loss(predictions, noisy_sampled_reality);\n return htl.html`${tex`error: ${loss.arraySync().toFixed(2)}`}`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":359,"value":"{\n const latest_theory = make_linear(slope_n, intercept_n);\n const predictions = sampled_positions.map(latest_theory)\n const theory_plot = sampled_positions.map((position, id) => { return {\"x\": position, \"y\": predictions[id], \"stroke\": \"#e15759\"}})\n const loss = get_loss(predictions, noisy_sampled_reality)\n return Plot.plot({\n grid: true,\n x: {label: \"position →\", domain: [_.min(sampled_positions), _.max(sampled_positions)]},\n y: {label: \"↑ value\", domain: [-2, 2]},\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n Plot.dot(sampled_positions, {x: sampled_positions, y: noisy_sampled_reality, fill: \"#4e79a7\"}),\n Plot.line(theory_plot, Plot.windowY({k: 1, x: \"x\", y: \"y\", stroke: \"stroke\"})),\n ]\n })\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":92,"value":"Observe:\n1. The error never drops to 0 now\n2. The theory that best fits the experimental data may not match the true nature of reality exactly","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":79,"value":"## 🥼 Understand the Scientific Method\nThe [scientific method](https://en.wikipedia.org/wiki/Scientific_method) is an iterative feedback loop of performing experiments and creating theories to better fit (and predict) the nature of reality.\n\nAt our current scale, this iterative feedback loop requires participation by folks from all over the world. From individuals and civil society, to governments and militaries to researchers and technologists. It is a societal endeavor worked on, funded and benefited by society (in unequal proportions).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":81,"value":"$33","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":83,"value":"Plot.plot({\n grid: true,\n x: {label: \"position →\", domain: [_.min(sampled_positions), _.max(sampled_positions)]},\n y: {label: \"↑ value\", domain: [-10, 10]},\n marks: [\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Visualize the true nature of our reality\n Plot.line(sampled_positions.map(position => [position, reality(position)])),\n // Visualize data samples of reality collected from our experiment\n Plot.dot(sampled_positions, \n {x: sampled_positions, y: noisy_sampled_reality, fill: \"red\"}),\n // Visualize our (hopefully) improving theories with each iteration of the scientific process \n // Color of each newer theory is darker \n Plot.line(belief_updates, \n {x: \"sampled_positions\", y: \"predictions\", z: \"iteration\", stroke: \"iteration\"}),\n // Label loss/error, step of each belief update\n Plot.text(belief_updates, \n Plot.selectLast({x: \"sampled_positions\", y: \"predictions\", z: \"iteration\", text: \"loss\", textAnchor: \"start\", dx: -80})),\n ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":94,"value":"$34","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":43,"value":"## Appendix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":10,"value":"tf = require('@tensorflow/tfjs@3.0')","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"function make_noise(x, scale) {\n return tf.randomNormal([x.length], 0, scale).arraySync();\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":14,"value":"function add_noise(points, scale) {\n const noise_in_sample = make_noise(points, scale)\n return points.map((point, id) => point + noise_in_sample[id]) \n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"function mean_absolute_error(prediction, actual) {\n return tf.sub(prediction, actual).abs().mean()\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":18,"value":"function get_loss(predictions, actual) {\n return mean_absolute_error(predictions, actual);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":20,"value":"function make_loss(theory, experiment) {\n return (sample) => get_loss(theory(sample), experiment);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":22,"value":"function make_theoretical_scientist(experimental_data) {\n return (a, b) => linear(a, b, experimental_data);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"function isTensor(item) {\n return tf.tensor_util.getTensorsInContainer(item).length > 0\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"function toTensor(item) {\n return isTensor(item) ? item : tf.tensor(item)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"function update(theory_params, sample_points, learning_rate) {\n // Setup\n // Initialize, normalize variables\n sample_points = toTensor(sample_points)\n learning_rate = tf.scalar(learning_rate)\n const theoretical_scientist = make_theoretical_scientist(sample_points)\n const gradient_generators = tf.grads(theoretical_scientist);\n\n // Calculate loss\n // How off were the theoretical scientists beliefs from reality?\n let predictions = theoretical_scientist(...theory_params)\n let actual = reality(sample_points)\n let loss = get_loss(predictions, actual)\n\n // Calculate gradients\n // Find the direction, magnitude of updates to the theories required\n let gradients = gradient_generators(theory_params)\n gradients = tf.tensor(gradients.map(g => g.arraySync()))\n\n // Update theory paramaters\n // Move the theoretical scientists beliefs closer to reality\n theory_params = tf.tensor(theory_params)\n .sub(gradients\n .mul(learning_rate)\n .mul(loss))\n .arraySync()\n return theory_params\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":168,"value":"slider = (min, max, step, value, labelText=\"\") => {\n const form = html`\n `;\n \n const label = form.querySelector('span.value');\n form.value = form.iters.valueAsNumber;\n form.iters.oninput = () => {\n form.value = form.iters.valueAsNumber;\n label.innerHTML = form.iters.valueAsNumber;\n };\n \n return form; \n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"import {toc} from \"@nebrius/indented-toc\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"import {show} from \"@zaidalyafeai/show-tensor\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 32:"$31:body"