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”)\n26 / 30" }, { "id": "lecture-01-p027", "sourceId": "lecture-01", "lectureId": "lecture-01", "page": 27, "title": "and class labels); sufficient in", "text": "About this Class\nAdministrative Stuff\nThe UE Project\nIntroduction to Classification\nPreview\nLiterature\nBuildin...
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lower-case", "bold", "functions", "over" ], "retrievalText": "Bayesian Decision Theory. Notation. Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nNotation\nSome Notational Conventions\nNOTATION\nT...
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1] ⊂R+ with P\ni P(xi) = 1.0\nThe Joint Distribution over a set of d discrete variables {X1, ..., Xd} is a\nfunction P : X1 × ... × Xd 7→[0, 1] that assigns a probability to each possible\ncombination of values of the variables\n\u0011 A distribution can be thought of as a list of n numbers.\nA joint distribution is a ...
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7a251922d8f3110981086ec7ffa784a62a1439c27bbd26e0dfcf08a96e5ad554
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function p : X 7→R+ is a Probability Density Function (PDF) for a\ncontinuous variable X if\n▶p(x) ≥0 for all x ∈Val(X)\n▶p(x) = 0 for all x /∈Val(X)\n▶R +∞\n−∞p(x)dx = 1.0\nImportant:\n▶A PDF itself is not a probability distribution!\n▶p(x) must not be interpreted as a probability (can even be > 1!)\n▶Probabilities ar...
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(a ≤X ≤b) = P(X ≤b) −P(X ≤a) =\nZ b\na\np(x)dx\n\u0011\nPDFs permit us to compute probabilities for (arbitrarily small) intervals of\nvalues.\n8 / 53" }, { "id": "lecture-02-p009", "sourceId": "lecture-02", "lectureId": "lecture-02", "page": 9, "title": "p(x) =", "text": "Mat...
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c4d9c07ff9eb40cde3328506f8f25ac9299cf02c00b5ac4d1b7d0a18ca69bffe
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(Gaussian) Distribution\nDefinition\nA variable X has a Normal Distribution with mean µ and variance σ2, denoted\nX ∼N(µ, σ2), if it has the Gaussian PDF\np(x) =\n1\n√\n2πσ\ne−(x−µ)2\n2σ2\n0.45\n0.4\n0.35\n0.3\n0.25\n0.2\n0.15\n0.1\n0.05\n0\n–10\n–5\n10\n5\n0\n(0,42)\n(0,1)\n(5,22)\n10 / 53" }, { "id": "le...
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are", "bayesian", "theory", "mathematical", "basics", "recap", "problem" ], "retrievalText": "Bayesian Decision Theory. P(x,y). Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiteratu...
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5919a9c6aedee9fca1a025b5a3753b2a28adbdde645fc414b423cba9249811a2
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▶“Sort incoming fish on a conveyor according to species, using optical\nsensing”\n⇒Species: Sea Bass or Salmon?\n1Due to [Duda, Hart & Stork, 2001].\n14 / 53" }, { "id": "lecture-02-p015", "sourceId": "lecture-02", "lectureId": "lecture-02", "page": 15, "title": "(e.g., length, ligh...
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96fff806b00d90201d6beb9e7c7da25f54671b5bba9c5ec25920ee7ef989ff3a
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Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nDescribing Objects via Features\nExample:\n▶Use two features lightness and width to describe a fish\n▶A fish is now represented as a list (vector) x of two features values:\nx =\n\u0012x1\nx2\n\u0013\nwhere\nx1 ∈R+ is ...
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7d6447ac3ba4bd6e91fb9ddfffc05ea25e994e14648523062f394dcb1f952369
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"text": "Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nClass-conditional Probabilities\nFigure : Class-conditional probability density functions p(x | ωi) show the probability\ndensity of measuring a particular value x given that ...
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-02", "lectureId": "lecture-02", "page": 20, "title": "first used probability inductively and established a mathematical basis for probability", "text": "Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nBayes’ ...
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c2634c96f6ade828553a0f3ca13ac7ead54c5367ff416490a2d2b35f00dd2548
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of London. Also buried in Bunhill Fields is Bayes’s friend Richard Price, a\npioneer of insurance, who presented Bayes’s famous paper on probability to the Royal\nSociety in 1763, two years after Bayes’s death.\n21 / 53", "keywords": [ "the", "bayes", "bayesian", "for", "de...
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1fcfb86f8399adf4382c2dda1eb326e6f30a111a5ed6d43eda486e33532639bd
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Rule\nUsing Bayes’ Rule for Classification\nView Classification as a Conditional Probability Question:\n▶Given that we have observed certain feature values x about an object (e.g.,\na fish), what is the probability that the object belongs to class ωi?\nP(ωi | x) = ?\n▶Assume we know the prior class probabilities P(ωi) and...
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652e72e1c6016506d03811d46beca54bdead394b829dd20fa021ae5daabcdb91
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i)P(ωi)\nNames of the components:\nP(ωi)\n=\na priori probability (“prior”) of class ωi\nP(ωi | x)\n=\na posteriori probability (“posterior”) of class ωi\np(x | ωi)\n=\nprobability of finding something like x in class ωi\n=\n“likelihood” of class ωi in the context of evidence x\np(x)\n=\nprobability of observing somethi...
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c3f8eb1a9e247db9e218d5e4021c8019e871b37cbeb7f14292d20dd9e1f40ea6
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) = .67\n▶consequently, medium and large fish are equally frequent:\nP(X) :\nP(med) = P(large) = 0.5\nWe catch a fish of medium size. What is it?\nP(b | med)\n=\n[P(med | b) · P(b)] /P(med) = .33 × .5/.5 = 0.33\nP(s | med)\n=\n[P(med | s) · P(s)] /P(med) = .67 × .5/.5 = 0.67\n25 / 53" }, { "id": "lecture-02...
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58f85f5e7bcd9b63c3cd70fd389413dcf224843f9c2b8ce6cbb816f4be2192bd
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(med) = .42; P(large) = 0.58\nWe catch a fish of medium size. What is it?\nP(b | med)\n=\n[P(med | b) · P(b)] /P(med) = .33 × .75/.42 = 0.60\nP(s | med)\n=\n[P(med | s) · P(s)] /P(med) = .67 × .25/.42 = 0.40\n26 / 53" }, { "id": "lecture-02-p027", "sourceId": "lecture-02", "lectureId": "lecture...
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", "cation", "class-conditional", "probabilities" ], "retrievalText": "Bayesian Decision Theory. ω2 is roughly 0.08, and that it is in ω1 is 0.92. At every x, the posteriors sum to 1.0.. Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classifica...
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b71db5060d83b0a2349e6d5d6b00a5fcb2b26b363a97ba2e80282fff5e048f41
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calculate posterior class probabilities and predict class as follows:\nif P(ω1 | x) > P(ω2 | x)\n⇒predict ω1\nif P(ω1 | x) ≤P(ω2 | x)\n⇒predict ω2\nResulting probability of error:\n▶Probability that our prediction is wrong =\nprobability that true class is the other one:\nIf we decided ω1 because P(ω1|x) > P(ω2|x), P(e...
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6f8ffb45307b6002f3bc4a08ed8658116d781a229918973abb112ddb2c429706
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don’t know”, e.g., in close or risky cases)\n▶The loss function permits us to distinguish between different kinds of errors.\n(Remember example from last slide set: cost of misclassifying sea bass as\nsalmon may be higher than misclassifying salmon as sea bass)\n32 / 53", "keywords": [ "the", "act...
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5856653fbf94399365f40d71ec8d52315ad11522b1b4cb97a9528e901211647c
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c categories we wish to distinguish\n▶A = {α1, α2, ..., αa} a set of possible actions / decisions\n▶λ(αi|ωj) the cost (‘loss’) of deciding on action αi when the true class is ωj\n\u0011 The a × c function (table) λ is called a Loss Function (or Cost Function).\nDefinition\nThe Conditional Risk or Expected Loss R(αi | x)...
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6563e1c1e913275fd867ee2a4f7a9620ef8f499d9c4a4234a8ac89f81687f9cf
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classifications have cost 0", "text": "Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nA Special Case: Minimum Error Rate Classification\nMAP Classification as a Special Case of Risk Minimisation\nSuppose:\n▶All types of errors a...
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4b5d576de4777c1bad4d73706b90c5596e99baf265c2853a989dec4fc0a2d2dd
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", "max", "bayesian" ], "retrievalText": "Bayesian Decision Theory. if. Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nA Special Case: Minimum Error Rate Classification\nMinimum Error Rate Classification\n...
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bdb5320d5cfda47a8dc51c684428cdc5870c4795471b567674dd03fbf05d1a4e
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11)p(x|ω1)P(ω1) > (λ12 −λ22)p(x|ω2)P(ω2)\n37 / 53" }, { "id": "lecture-02-p038", "sourceId": "lecture-02", "lectureId": "lecture-02", "page": 38, "title": "(λ21 −λ11)p(x|ω1)P(ω1) > (λ12 −λ22)p(x|ω2)P(ω2)", "text": "Mathematical Basics\nRecap: The Problem\nClassification as a P...
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1f9899decefaf96a9f8488dda399d0b288f4ee69cae42d9a05b67d8413d34e3b
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\n⇒\nθb = 3P(ω2)\nP(ω1)\nDecision Regions Ri = regions in feature space where the decision rule\nwould predict a particular class ωi\n39 / 53", "keywords": [ "decision", "classi", "cation", "boundaries", "the", "likelihood", "and", "rule", "r...
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20b64e304025851b1d39b626a6b2e30b75ef872b5b053b3db9bd12e70998cc0f
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nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nDecision Boundaries for Gaussian Distributions: Some Special Cases\nIn General, Decision Boundaries of Bayesian Classifiers\nare not Linear\nFigure : In this two-dimensional two-category classifier, the probability densities a...
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60dd0268f82210b01ae7208da379c6d5e3e5e733dec4fd72232900242a836add
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..\n0\n. . .\nσ2\n\n\n\n(spherical Gaussians of equal size)\nTHEN\n▶the decision boundary between two regions Ri and Rj is linear (a\nhyperplane) and orthogonal to the line linking the means µi and µj\n42 / 53" }, { "id": "lecture-02-p043", "sourceId": "lecture-02", "lectureId": "lecture-02...
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46d35178e53e662ebd60aa69d7cf7e77a833151b4745608ad9db25651a91d43c
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boundary shifts; for sufficiently disparate\npriors the boundary will not lie between the means of the Gaussian distributions any more.\n(From [Duda et al., 2001].)\n44 / 53" }, { "id": "lecture-02-p045", "sourceId": "lecture-02", "lectureId": "lecture-02", "page": 45, "title": "45 ...
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7ce687f51b614b09a2e439ee1f003c4b6050afe657e278a63d30fd1aa5ecb12d
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(From [Duda et al., 2001].)", "text": "Mathematical Basics\nRecap: The Problem\nClassification as a Probabilistic Task\nOptimal Classification\nDecision Boundaries\nLiterature\nDecision Boundaries for Gaussian Distributions: Some Special Cases\nDecision Boundaries: Special Case 2\nFigure : Probability densities (in...
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, "distributions", "contours", "constant", "density", "duda", "classi", "cation", "general" ], "retrievalText": "Bayesian Decision Theory. contours of constant probability density. (From [Duda et al., 2001].). Mathematical Basics\nRecap: The Pr...
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81aac20b48fb776c5a60405853163c45995f07c1c7c95d897ef08d6f42d351fa
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\n▶The posterior can never be exactly 1.0\n▶The posterior can never be exactly 0.0\n▶The posterior can never be larger than the prior\n▶The posterior can never be larger than the likelihood\n▶If we have a prior that puts a probability of 1.0 on one class and 0.0 on all others, the\nclassifier can never predict any of th...
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952aac77e7776b78bf94cb40741c2c6f1b5b2ee7a5e00c1b9ee67fb18144d243
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(i.e., discrete classes with a natural\nordering among them, such as Ω= {tiny, small, medium, large, huge}). Assume we\nencode these classes with consecutive integers (Ωnum = {1, 2, 3, 4, 5}). We want to learn\na classifier that, when it does make errors, tends to make small errors (e.g., misclassifies\nlarge as medium, ...
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5c5759bbcde56d829f20abfca52b4eb347e6a74d2774ce48c1e20954bc79578f
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: Wiley & Sons.\n53 / 53", "keywords": [ "classi", "cation", "decision", "literature", "bayesian", "theory", "mathematical", "basics", "recap", "the", "problem", "probabilistic" ], "retrievalText": "Bayesia...
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219aaee67c83984c80eca752d3d805000542792dadfe3d07a45c810171288578
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: "Motivation\nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nOutline\n1 Motivation\n2 Parametric Density Estimation\nThe Likelihood Function\nMaximum Likelihood (ML) Parameter Estimation\nML Estimates for Discrete Distributions\nML Estimates for ...
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we generally do not have this information ...\n4 / 72" }, { "id": "lecture-03-p005", "sourceId": "lecture-03", "lectureId": "lecture-03", "page": 5, "title": "▶Prior probability distribution P(Ω) over classes: what is the probability", "text": "Motivation\nParametric Density ...
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63270c2e4945ed7b552975e656c72801ef5ba7f1be37f62848fc646dc7477033
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ncombination x locally, on demand (when needed for classification)\n6 / 72", "keywords": [ "density", "estimation", "the", "non-parametric", "parametric", "classi", "set", "parameters", "try", "estimate", "shape", "dist...
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db8f9f8a067ae765c541f91826ad05747d8fd7d83e087de4772183c552214df4
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Density Estimation to NN Classifiers\nLiterature\nParameter Estimation: Two Fundamental Approaches\n1. Maximum Likelihood Estimation:\n▶View parameters θ as quantities with fixed but unknown values\n▶Best estimate ˆθ is the one that gives a distribution that best ‘agrees with’ the\nobserved data D\n▶‘Agreement’ with the ...
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2d68315c8f26576b7d5aa2ff9125cf3641eb6ab7fa159be0d2bbf969614217ab
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nLiterature\nMaximum Likelihood Parameter Estimation\nTask:\n▶Given: a set of training observations D = {x1, ..., xn}\n▶Assumption: a certain family of distributions underlying the data,\nparametrisable via set of parameters θ\n▶Task: Find values ˆθ that have the highest likelihood in view of the given\ndata D\nDefiniti...
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68874c8cafb7b243f2ad47a4486ecae66e09558f95458f5ce43c7d5aea2d99bf
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\nFigure : The top graph shows several training points in one dimension, known or assumed\nto be drawn from a Gaussian of a particular variance, but unknown mean. Four of the\ninfinite number of candidate source distributions are shown in dashed lines. The middle\nfigure shows the likelihood p(D|θ) as a function of the m...
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, "retrievalText": "Density Estimation. ▶Parameter values ˆθ that maximise the likelihood function relative to D:. Motivation\nParametric Density Estimation\nNon-parametric Density Estimation\nFrom Density Estimation to NN Classifiers\nLiterature\nMaximum Likelihood (ML) Parameter Estimation\nCalculating the Maxim...
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3f59430421748ed67d5d091d730b8b7b39bc5f0071b5241d0f910e09276c20c2
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= 1 −θ\nQuestion:\n▶After having observed set D consisting of b bass and s salmon, what is our\nbest estimate of θ = P(B)?\n13 / 72" }, { "id": "lecture-03-p014", "sourceId": "lecture-03", "lectureId": "lecture-03", "page": 14, "title": "Y", "text": "Motivation\nParametric De...
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94f737c479de2aef3b0acc00047db0fb21f7e3a7f4bc4e5f60f8488d801f562c
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\nˆθi = ˆP(xi) of the probability distribution P(X) = {P(x1), ..., P(xk)} is\nˆθi = ˆP(xi) =\nN[xi]\nP\nj N[xj] = N[xi]\n|D|\nIn Words:\n▶The ML estimate for a discrete probability is the proportion of occurrences of\nthe respective event in the data set\n▶Not surprising? Have proven in what sense relative frequencies ...
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78172cd30840a43e2d991c4aff14f83665c86169437fbfd25c9eb595f8e9d3ff
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\nFrom Density Estimation to NN Classifiers\nLiterature\nML Estimates for Gaussian Density Functions\nExample 2: ML Estimation for Univariate Normal Densities\nTask: Estimate Class-conditional Distributions p(x | ωi)\n▶Assume our fish are described by one (numeric) feature only: Length\n▶Assume we believe that fish length...
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d7e14a25b5da51f16dcdc3ec62b358d89aae105b8fef51daf95703bd04c56f0b
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\ne−(xi−µ)2\n2σ2\n=\nM(−log\n√\n2π −log σ) −\nM\nX\ni=1\n(xi −µ)2\n2σ2\n2 Compute the derivative w.r.t. the parameters and set to 0:\n∂ℓ(θ)\n∂µ\n= −1\nσ2\nM\nX\ni=1\n(xi −µ) = 0\n∂ℓ(θ)\n∂σ\n= −M\nσ + 1\nσ3\nM\nX\ni=1\n(xi−µ)2 = 0\n3 Search maximum: Solve the system of 2 equations to obtain:\nˆµ =\nP\ni xi\nM\nˆσ =\nsP\...
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0bf80138ba6efefffdf86fcb69c27ac67f667ae7cf2a0002f098702c27599fa5
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i (the mean vector)\n▶Σi = {σijk} is the d × d matrix of pairwise correlations between features j\nand k in class i (the covariance matrix)\nQuestion:\n▶If our training set D contains M salmon x1, ..., xM, what is our estimate of\nthe parameters θ = {µs, Σs}?\n19 / 72", "keywords": [ "density", "t...
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56e1121c3272b03ede16e60cc65765014bc0dafac4908253fcb204dd8ea42648
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1/2 e[−1\n2 (xi−µ)T Σ−1(xi−µ)]\nℓ(θ : D) = log p(D | θ)\n=\n. . .\n2 Compute the derivative w.r.t. the parameters and set to 0:\n. . . abracadabra . . .\n3 Resulting ML Estimates:\nˆµ = 1\nM\nX\ni\nxi\nˆΣ = 1\nM\nX\ni\n(xi −ˆµ)(xi −ˆµ)T\n\u0011 So now you know how to estimate multivariate Gaussians from data.\n20 / 72"...
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0ff6d6eb8834a8db2ea7b1a60272303b57d539b64b9e7dcc071ed32d6d4d301d
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