14_ベイズ推論_ベイズ線形回帰.nextjournal.ipynb

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\u7b2c14\u56de \u30d9\u30a4\u30ba\u63a8\u8ad6\u305d\u306e2: \u30d9\u30a4\u30ba\u63a8\u5b9a\u306e\u5fdc\u7528\n\n\u3044\u3064\u3082\u306e\u3088\u3046\u306b\u53f3\u4e0a\u306e\u300cRemix\u300d\u30dc\u30bf\u30f3\u3092\u62bc\u3057\u3066\u81ea\u5206\u7528\u306e\u8907\u88fd\u3092\u4f5c\u6210\u3059\u308c\u3070\u3001\u5404\u30bb\u30eb\u3092\u300cShift+Enter\u300d\u3067\u5b9f\u884c\u3057\u306a\u304c\u3089\u8aad\u3080\u3053\u3068\u304c\u3067\u304d\u307e\u3059\n\n\u7de8\u96c6\u3092\u884c\u306a\u3063\u305f\u5f8c\u306f\u3001Windows\u3067\u306f\u300cCtrl+s\u300d\u3001Mac\u3067\u306f\u300cCmd+s\u300d\u3067\u4fdd\u5b58\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\n\n\u6700\u521d\u306b\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092\u8aad\u307f\u8fbc\u307f\u307e\u3059"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"298eccd5-2d6d-42ab-983d-156d08014294","kind":"code","language":"julia"}},"source":["using LinearAlgebra: inv, logdet, I, diag\nusing Plots"],"outputs":[],"execution_count":1},{"cell_type":"markdown","metadata":{},"source":["# \u30d9\u30a4\u30ba\u7dda\u5f62\u56de\u5e30"]},{"cell_type":"code","metadata":{"nextjournal":{"id":"3d9ead12-a353-48c8-8ad1-f055341c225a","kind":"code","language":"julia"}},"source":["# X\u306e\u591a\u9805\u5f0f\u306e\u57fa\u5e95\u95a2\u6570\nfunction polynomial_basis(x, M) # M\u306f\u6b21\u6570\n    N = length(x) # N\u306f\u30c7\u30fc\u30bf\u6570\n    X = zeros(Float64, N, M)\n    for m = 1:M\n        X[:, m] = x.^m\n    end\n    return X\nend"],"outputs":[{"output_type":"execute_result","data":{"text\/plain":"polynomial_basis"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"code","metadata":{"nextjournal":{"id":"2d98e6a3-a518-4e79-9cc3-ae954f16037c","kind":"code","language":"julia"}},"source":["# \u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306e\u4f5c\u6210\nN = 30\n#x = range(0.0, stop=1.0, length=N)\nx = 2.0.*(rand(N).-0.5)\nx = sort(x)\nsigma2_y = 0.01\nX = polynomial_basis(x, 3)\nw_truth = [0.0, 0.0, 1.0]\nY = X * w_truth .+ sqrt(sigma2_y) * randn(N);"],"outputs":[],"execution_count":1},{"cell_type":"code","metadata":{"nextjournal":{"id":"fc26d954-9cf2-437a-812a-329e5e6f9548","kind":"code","language":"julia"}},"source":["# \u30c6\u30b9\u30c8\u30c7\u30fc\u30bf\u306e\u30d7\u30ed\u30c3\u30c8\nx_range = range(-1, stop=1, length=100)\nscatter(x, Y, xlabel=\"x_1\", ylabel=\"y\", ms=8, label=\"y=f(x)+e\")\nplot!(x_range, x_range.^3, label=\"f(x)\", legend=:topleft)"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n<defs>\n  <clipPath id=\"clip660\">\n    <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip660)\" d=\"\nM0 1600 L2400 1600 L2400 0 L0 0  Z\n  \" fill=\"#ffffff\" 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\/><\/svg>\n"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"code","metadata":{"nextjournal":{"id":"3245c6f2-b0d4-42a7-aee4-50b442f26372","kind":"code","language":"julia"}},"source":["# \u30d9\u30a4\u30ba\u63a8\u5b9a\u306b\u3088\u308a\u3001\u4e88\u6e2c\u5206\u5e03\u3068\u30a8\u30d3\u30c7\u30f3\u30b9\u3092\u8a08\u7b97\u3059\u308b\u95a2\u6570\u3092\u5b9a\u7fa9\u3059\u308b\nfunction bayes_inference(x, Y, M, sigma2_y, sigma2_w, x_range)\n    X = polynomial_basis(x, M)\n    N = length(x)\n    X_t = X'\n    \n    # posterior\n    sigma2_w_new = inv(X_t*inv(sigma2_y*I(N))*X_t' + inv(sigma2_w * I(M)))\n    mu_w_new = sigma2_w_new * (X_t * inv(sigma2_y * I(N)) * Y)\n\n    # predictive\n    X_range = polynomial_basis(x_range, M)\n    X_range_t = X_range'\n    Y_prediction = (mu_w_new' * X_range_t)\n    sigma2_y_prediction = sigma2_y .+ diag(X_range_t' * sigma2_w_new * X_range_t)\n    \n    # evidence\n    evidence = -0.5* (sum(Y) * inv(sigma2_y) + N*log.(sigma2_y) + N*log.(2*pi)\n         + logdet(sigma2_w) - (mu_w_new'*inv(sigma2_w_new)*mu_w_new)[1] - logdet(sigma2_w_new))\n\n    return Y_prediction[:], sqrt.(sigma2_y_prediction[:]), evidence\n  end"],"outputs":[{"output_type":"execute_result","data":{"text\/plain":"bayes_inference"},"execution_count":1,"metadata":{}}],"execution_count":1},{"cell_type":"code","metadata":{"nextjournal":{"id":"d2fdb103-5472-481e-a9f2-e764d21442c9","kind":"code","language":"julia"}},"source":["# \u4e88\u6e2c\u5206\u5e03\u306e\u30d7\u30ed\u30c3\u30c8\nM=3 #\u591a\u9805\u5f0f\u306e\u6b21\u6570\nx_range = range(-1.0, stop=1.0, length=200)\nsigma2_w = 1.0\nbayes = bayes_inference(x, Y, M, sigma2_y, sigma2_w, x_range)\nscatter(x, Y, xlabel=\"x_1\", ylabel=\"y\", ms=8, label=\"y=f(x)+e\")\nplot!(x_range, x_range.^3, label=\"f(x)\", locatoin=4)\nplot!(x_range, bayes[1], ylims=(-1.5,1.5), ribbon=(2*bayes[2], 2*bayes[2]), label=\"bayesian prediction\", legend=:topleft, title=\"M=$(M)\")"],"outputs":[{"output_type":"execute_result","data":{"image\/svg+xml":"<?xml version=\"1.0\" encoding=\"utf-8\"?>\n<svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n<defs>\n  <clipPath id=\"clip700\">\n    <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip700)\" d=\"\nM0 1600 L2400 1600 L2400 0 L0 0  Z\n  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"\/>\n<defs>\n  <clipPath id=\"clip701\">\n    <rect x=\"480\" y=\"0\" width=\"1681\" height=\"1600\"\/>\n  <\/clipPath>\n<\/defs>\n<path clip-path=\"url(#clip700)\" d=\"\nM242.922 1423.18 L2352.76 1423.18 L2352.76 123.472 L242.922 123.472  Z\n  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"\/>\n<defs>\n  <clipPath id=\"clip702\">\n    <rect x=\"242\" y=\"123\" width=\"2111\" height=\"1301\"\/>\n  <\/clipPath>\n<\/defs>\n<polyline 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