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+"h i l=−(1−y)log 1−sig[f[x,ϕ]] −ylog sig[f[x,ϕ]] , (5.33) wheresig[•]isdefinedinequation5.32. plotthislossasafunctionofthetransformednetwork output sig[f[x,ϕ]]∈[0,1] (i) when the training label y=0 and (ii) when y=1. problem5.3∗ supposewewanttobuildamodelthatpredictsthedirectiony inradiansofthe prevailing wind based on local measurements of barometric pressure x. a suitable distribution over circular domains is the von mises distribution (figure 5.13): (cid:2) (cid:3) exp κcos[y−µ] pr(y|µ,κ)= , (5.34) 2π·bessel [κ] 0 this work is subject to a creative commons cc-by-nc-nd license. (c) mit press.notes 75 figure 5.14 multimodal data and mixture of gaussians density. a) example training data where, for intermediate values of the input x, the corresponding output y follows one of two paths. for example, at x = 0, the output y might be roughly −2 or +3 but is unlikely to be between these values. b) the mixture of gaussians is a probability model suited to this kind of data. as the name suggests, the model is a weighted sum (solid cyan curve) of two or more normal distributionswithdifferentmeansandvariances(here,twoweighteddistributions, dashed blue and orange curves). when the means are far apart, this forms a multimodal distribution. c) when the means are close, the mixture can model unimodal but non-normal densities. where µ is a measure of the mean direction and κ is a measure of the concentration (i.e., the inverse of the variance). the term bessel [κ] is a modified bessel function of order 0. 0 use the recipe from section 5.2 to develop a loss function for learning the parameter µ of a model f[x,ϕ] to predict the most likely wind direction. your solution should treat the concen- tration κ as constant. how would you perform inference? problem 5.4∗ sometimes, the outputs y for input x are multimodal (figure 5.14a); there is morethanonevalidpredictionforagiveninput. here,wemightuseaweightedsumofnormal componentsasthedistributionovertheoutput. thisisknownasamixtureofgaussiansmodel. for example, a mixture of two gaussians has parameters θ={λ,µ ,σ2,µ ,σ2}: 1 1 2 2 (cid:20) (cid:21) (cid:20) (cid:21) λ −(y−µ )2 1−λ −(y−µ )2 pr(y|λ,µ ,µ ,σ2,σ2)= p exp 1 + p exp 2 , (5.35) 1 2 1 2 2πσ2 2σ2 2πσ2 2σ2 1 1 2 2 where λ ∈ [0,1] controls the relative weight of the two components, which have means µ ,µ 1 2 and variances σ2, σ2, respectively. this model can represent a distribution with two peaks 1 2 (figure 5.14b) or a distribution with one peak but a more complex shape (figure 5.14c). usetherecipefromsection5.2toconstructalossfunctionfortrainingamodelf[x,ϕ]thattakes input x, has parameters ϕ, and predicts a mixture of two gaussians. the loss should be based on i training data pairs {x ,y }. what problems do you foresee when performing inference? i i problem5.5considerextendingthemodelfromproblem5.3topredictthewinddirectionusing a mixture of two von mises distributions. write an expression for the likelihood pr(y|θ) for this model. how many outputs will the network need to produce? draft: please send errata to [email protected] 5 loss functions figure 5.15 poisson distribution. this discrete distribution is defined over non- negative integers z �"