31:T41d,md`# Laplace Approximation The [Laplace Approximation](https://en.wikipedia.org/wiki/Laplace%27s_method) concerns integrals of the form ${tex`\int e^{Mf(x)}`}, where ${tex`f(x)`} is a twice differentiable function. As ${tex`M`} increases, the integral of ${tex`e^{Mf(x)}`} is well approximated by the integral of a Gaussian function centered at the function's mode. The plot below shows ${tex`e^{Mf(x)}`} with ${tex.block`f(x) = \frac{\sin(x)}{x}`} compared to the approximation ${tex.block` e^{-\frac{1}{2} M |f^{\prime\prime}(x_0)|(x-x_0)^2} `} where ${tex`x_0`} is the global maximum of ${tex`f(x)`}. `32:T50d,{ const height = 450 const svg = d3.select(DOM.svg(width, height)); const margin = { top: 20, right: 20, bottom: 20, left: 40 }; const xlim = [-15, 15]; const data = _.range(xlim[0], xlim[1], 0.01).map((x) => ({ x: x, y: e_Mf(x, M) })); const data_approx = _.range(xlim[0], xlim[1], 0.01).map((x) => ({ x: x, y: approx_f(x, 0, M, f, f_primeprime) })); const x = d3.scaleLinear() .domain(xlim) .range([margin.left, width - margin.right]) const y = d3.scaleLinear() .domain([Math.min(0, d3.min(data.map((d) => d.y))), d3.max(data.map((d) => d.y))]) .range([height - margin.top, margin.bottom]); svg.append("g") .attr("class", "axis axis--x") .attr("transform", `translate(${-0.5},${y(0)})`) .call(d3.axisBottom(x)); svg.append("g") .attr("class", "axis axis--y") .attr("transform", `translate(${x(0)},0)`) .call(d3.axisLeft(y)); const line = d3.line() .x((d) => x(d.x)) .y((d) => y(d.y)); svg.append('path') .datum(data) .attr('d', line) .attr('stroke', palette.green_sea) .attr('fill', 'transparent'); svg.append('path') .datum(data_approx) .attr('d', line) .attr('stroke', palette.wisteria) .attr('fill', 'transparent'); return svg.node(); }33:[] 2f:{"title":"Laplace Approximation","cells":[{"id":0,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"viewof M = slider({ min: 0.2, max: 10, value: 1.5, title: \"M\" })","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"f = (x) => x == 0 ? 1 : Math.sin(x) / x","pinned":false,"hidden":false,"mode":"ojs"},{"id":25,"value":"f_primeprime = (x) => x == 0 ? -1/3 : -(2 * x * Math.cos(x) + (-2 + Math.pow(x, 2)) * Math.sin(x))/Math.pow(x, 3)","pinned":false,"hidden":false,"mode":"ojs"},{"id":20,"value":"e_Mf = (x, M) => Math.exp(M * f(x))","pinned":false,"hidden":false,"mode":"ojs"},{"id":34,"value":"","pinned":true,"hidden":false,"mode":"ojs"},{"id":49,"value":"approx_integral = (x0, M, f, f_primeprime) => Math.sqrt(2 * Math.PI / (M * Math.abs(f_primeprime(x0)))) * Math.exp(M * f(x0))","pinned":false,"hidden":false,"mode":"ojs"},{"id":28,"value":"approx_f = (x, x0, M, f, f_primeprime) => Math.exp(M * f(x0)) * Math.exp(-1/2 * M * Math.abs(f_primeprime(x0)) * Math.pow(x - x0, 2))","pinned":false,"hidden":false,"mode":"ojs"},{"id":4,"value":"d3 = require('d3')","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"_ = require('lodash')","pinned":false,"hidden":false,"mode":"ojs"},{"id":15,"value":"import {select, button, slider, input} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":68,"value":"import { palette } from \"@herbps10/flat-ui-colors\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}