Maximum A Posteriori Chris Piech CS109, Stanford University

Maximum A Posteriori Chris Piech CS109, Stanford University

Previously in CS109

Game of Estimators Estimators Maximum Likelihood Non spoiler alert: this didn t happen in game of thrones

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sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> Think: observations of n IID random variables: Where: has likelihood (PDF) function: Likelihood of data L( ) = Y i X (1),...,X (n) <latexit sha1_base64="8ecrizo7nk5izr4/gabdstbkcjw=">aaacahicbvdlsgmxfl3js9bxqasxbojfakgugrf04algxmuf+4b2ljk004zmmkosecowg3/fjqtf3poz7vwb03yw2nogchlovte5x485u9pxvq2v1bx1jc3cvnf7z3dv3z44bkkokyq2scqj2fgxopwj2trmc9qjjcwhz2nbh99m/fyjlypf4l5pyuqfechywajwrurbx52htoxwsirqdsktqmh6f5wsb5ecmjmdwizutkqqo9g3v8wakoruamkxul3xibwxyqkz4tqr9hjfy0zgeei7hgocuuwlswuydgauaqoiay7qakb+7khxqnqk9e1lipvilxpt8t+vm+jgykuzibnnbzk/fcqc6qhn00adjinrfgiijpkzvyiywhitbtirmhdcxzwxseu85jo19+6ivl/o4yjaczxcgvy4hdrcqgoaqccdz3ifn+vjerhery956yqv9xzbh1ifpzgxlny=</latexit> <latexit sha1_base64="8ecrizo7nk5izr4/gabdstbkcjw=">aaacahicbvdlsgmxfl3js9bxqasxbojfakgugrf04algxmuf+4b2ljk004zmmkosecowg3/fjqtf3poz7vwb03yw2nogchlovte5x485u9pxvq2v1bx1jc3cvnf7z3dv3z44bkkokyq2scqj2fgxopwj2trmc9qjjcwhz2nbh99m/fyjlypf4l5pyuqfechywajwrurbx52htoxwsirqdsktqmh6f5wsb5ecmjmdwizutkqqo9g3v8wakoruamkxul3xibwxyqkz4tqr9hjfy0zgeei7hgocuuwlswuydgauaqoiay7qakb+7khxqnqk9e1lipvilxpt8t+vm+jgykuzibnnbzk/fcqc6qhn00adjinrfgiijpkzvyiywhitbtirmhdcxzwxseu85jo19+6ivl/o4yjaczxcgvy4hdrcqgoaqccdz3ifn+vjerhery956yqv9xzbh1ifpzgxlny=</latexit> <latexit sha1_base64="8ecrizo7nk5izr4/gabdstbkcjw=">aaacahicbvdlsgmxfl3js9bxqasxbojfakgugrf04algxmuf+4b2ljk004zmmkosecowg3/fjqtf3poz7vwb03yw2nogchlovte5x485u9pxvq2v1bx1jc3cvnf7z3dv3z44bkkokyq2scqj2fgxopwj2trmc9qjjcwhz2nbh99m/fyjlypf4l5pyuqfechywajwrurbx52htoxwsirqdsktqmh6f5wsb5ecmjmdwizutkqqo9g3v8wakoruamkxul3xibwxyqkz4tqr9hjfy0zgeei7hgocuuwlswuydgauaqoiay7qakb+7khxqnqk9e1lipvilxpt8t+vm+jgykuzibnnbzk/fcqc6qhn00adjinrfgiijpkzvyiywhitbtirmhdcxzwxseu85jo19+6ivl/o4yjaczxcgvy4hdrcqgoaqccdz3ifn+vjerhery956yqv9xzbh1ifpzgxlny=</latexit> <latexit sha1_base64="8ecrizo7nk5izr4/gabdstbkcjw=">aaacahicbvdlsgmxfl3js9bxqasxbojfakgugrf04algxmuf+4b2ljk004zmmkosecowg3/fjqtf3poz7vwb03yw2nogchlovte5x485u9pxvq2v1bx1jc3cvnf7z3dv3z44bkkokyq2scqj2fgxopwj2trmc9qjjcwhz2nbh99m/fyjlypf4l5pyuqfechywajwrurbx52htoxwsirqdsktqmh6f5wsb5ecmjmdwizutkqqo9g3v8wakoruamkxul3xibwxyqkz4tqr9hjfy0zgeei7hgocuuwlswuydgauaqoiay7qakb+7khxqnqk9e1lipvilxpt8t+vm+jgykuzibnnbzk/fcqc6qhn00adjinrfgiijpkzvyiywhitbtirmhdcxzwxseu85jo19+6ivl/o4yjaczxcgvy4hdrcqgoaqccdz3ifn+vjerhery956yqv9xzbh1ifpzgxlny=</latexit> f(x (i) = x (i) ) f(x (i) = x (i) ) Log Likelihood LL( ) = X i log f(x (i) = x (i) ) ˆ MLE = argmax LL( ) You have now estimated parameters Max Likelihood

Side Plot argmax argmax of log Gradient Ascent Mother of optimizations?

Linear Regression (simple) X = CO 2 level Y = Average Global Temperature N training datapoints (x (1),y (1) ), (x (2),y (2) ),...(x (n),y (n) ) Linear Regression Lite Model Y = X + Z Z N(0, 2 ) Y X N( X, 2 )

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Linear Regression (simple) Initialize: θ = 0 Repeat many times: gradient = 0 For each training example (x, y): gradient += 2(y - θ x)(x)! += h * gradient

Linear Regression (regular) X 1 = Temperature X 2 = Elevation X 3 = CO 2 level yesterday X 4 = GDP of region X 5 = Acres of forest growth Y = CO 2 levels

<latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">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</latexit> <latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">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</latexit> <latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">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</latexit> <latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">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</latexit> Linear Regression (regular) Problem: Predict real value Y based on observing variable X Model: Linear weight every feature Y = 1 X 1 + + m X m + Z = T X + Z Training: Gradient ascent to chose the best thetas to describe your data ˆ MLE = argmax n X i=1 (y (i) x (i) ) 2

Linear Regression (regular) Initialize: θ j = 0 for all 0 j m Repeat many times: gradient[j] = 0 for all 0 j m For each training example (x, y): For each parameter j: gradient[j] += (y θ T x)(-x[j])! j += h * gradient[j] for all 0 j m

<latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">aaacqnicbvbns8naen34wetx1aoxxaiiqkle0ioc4mvjbavvppbndmixd5owoxfl6g/z4i/w5g/w4kerrx7cpd34ntdm2zdvmnkxjfiydn0nz2x8ynjqujrtnp2bx1islc2fmtjvhbo8lrfubsyafbe0ukcezqkbqudcexbzlpfpb0ebeuen2e+grdh1jelbgvqqu7m4oowna+pjd5b1pnq0uux9bowmr0nawvrz5yx1/uidxljr+sqwoeiqox1ymdxhegbnqshtvkpuzs2c/gxecftjkoqdyqndy1mfexljjgl5boltjgkuxmkg7kcgeszv2dw0liyyatpocgsgdn0yxrrg2maetgc/t2rmmfxiq1qme+z3lyf/67vsdpfamyisfchiw0vhkingnpetdougjrjvaena2fsp7zhnofrxy9ye7/ex/4kz7zrn1rytnerh/sioelkla2stegsxhjjjuicnwsk9esav5m15cf6cd+djkb1zrjmr5ec4n193r6y0</latexit> <latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">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</latexit> <latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">aaacqnicbvbns8naen34wetx1aoxxaiiqkle0ioc4mvjbavvppbndmixd5owoxfl6g/z4i/w5g/w4kerrx7cpd34ntdm2zdvmnkxjfiydn0nz2x8ynjqujrtnp2bx1islc2fmtjvhbo8lrfubsyafbe0ukcezqkbqudcexbzlpfpb0ebeuen2e+grdh1jelbgvqqu7m4oowna+pjd5b1pnq0uux9bowmr0nawvrz5yx1/uidxljr+sqwoeiqox1ymdxhegbnqshtvkpuzs2c/gxecftjkoqdyqndy1mfexljjgl5boltjgkuxmkg7kcgeszv2dw0liyyatpocgsgdn0yxrrg2maetgc/t2rmmfxiq1qme+z3lyf/67vsdpfamyisfchiw0vhkingnpetdougjrjvaena2fsp7zhnofrxy9ye7/ex/4kz7zrn1rytnerh/sioelkla2stegsxhjjjuicnwsk9esav5m15cf6cd+djkb1zrjmr5ec4n193r6y0</latexit> <latexit sha1_base64="op4au8hhvbhdfsncb+88r15unku=">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</latexit> Predicting Warriors Y = Warriors points Y = 1 X 1 + + m X m + Z = T X + Z X 1 = Opposing team ELO X 2 = Points in last game X 3 = Curry playing? X 4 = Playing at home? X 5 = 1! 1 = -2.3! 2 = +1.2! 3 = +10.2! 4 = +3.3! 5 = +95.4

Supervised Learning Real World Problem Model the problem Formal Model! Training Data Learning Algorithm New Data Prediction Function! * Predictions

Modelling Real World Problem Model the problem Formal Model! Training Data Learning Algorithm New Data Prediction Function! * Predictions

Training Real World Problem Model the problem Formal Model! Training Data Learning Algorithm New Data Prediction Function! * Predictions

Make Predictions Real World Problem Model the problem Formal Model! Training Data Learning Algorithm New Data Prediction Function! * Predictions

Our Path Neural Networks Linear Regression Naïve Bayes Logistic Regression Unbiased estimators Maximizing likelihood Bayesian estimation

Episode 2 The Song of The Last Estimator

Something rotten in the world of MLE

Foreshadowing..

Need a Volunteer So good to see you again!

I have two envelopes, will allow you to have one One contains $X, the other contains $2X Select an envelope o Open it! Now, would you like to switch for other envelope? To help you decide, compute E[$ in other envelope] o Let Y = $ in envelope you selected 1 Y 1 E $in other envelope] = 2 2 + 2 2 Before opening envelope, think either equally good So, what happened by opening envelope? o Two Envelopes [ Y = And does it really make sense to switch? 5 4 Y

Thinking Deeper About Two Envelopes The two envelopes problem set-up Two envelopes: one contains $X, other contains $2X You select an envelope and open it o o Let Y = $ in envelope you selected Let Z = $ in other envelope 1 Y 1 5 E[ Z Y] = 2 2 + 2 2Y = 4 Y E[Z Y] above assumes all values X (where 0 < X < ) are equally likely o Note: there are infinitely many values of X o So, not true probability distribution over X (doesn t integrate to 1)

All Values are Equally Likely? p(x) Infinite powers of two 0 10 20 40 60 80 100 X

Subjectivity of Probability Belief about contents of envelopes Since implied distribution over X is not a true probability distribution, what is our distribution over X? o o Frequentist: play game infinitely many times and see how often different values come up. Problem: I only allow you to play the game once Bayesian probability o o o o Have prior belief of distribution for X (or anything for that matter) Prior belief is a subjective probability By extension, all probabilities are subjective Allows us to answer question when we have no/limited data E.g., probability a coin you ve never flipped lands on heads As we get more data, prior belief is swamped by data

Subjectivity of Probability p(x) 0 10 20 40 60 80 100 X

The Envelope, Please Bayesian: have prior distribution over X, P(X) Let Y = $ in envelope you selected Let Z = $ in other envelope Open your envelope to determine Y If Y > E[Z Y], keep your envelope, otherwise switch o No inconsistency! Opening envelope provides data to compute P(X Y) and thereby compute E[Z Y] Of course, there s the issue of how you determined your prior distribution over X o Bayesian: Doesn t matter how you determined prior, but you must have one (whatever it is) o Imagine if envelope you opened contained $20.01

Envelope Summary: Probabilities are beliefs Incorporating prior beliefs is useful

Priors for Parameter Estimation?

Flash Back: Bayes Theorem Bayes Theorem (q = model parameters, D = data): Posterior Likelihood Prior Likelihood: you ve seen this before (in context of MLE) o Probability of data given probability model (parameter q) Prior: before seeing any data, what is belief about model o I.e., what is distribution over parameters q Posterior: after seeing data, what is belief about model o P(q D) = P(D q) P(q) P(D) After data D observed, have posterior distribution p(q D) over parameters q conditioned on data. Use this to predict new data.

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sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> Maximum Likelihood Estimation ˆ MLE = argmax = argmax f(x (1) = x (1),...,X (n) = x (n) ) X log f(x (i) = x (i) ) i Maximum A Posteriori ˆ MAP = argmax f( = X (1) = x (1),...,X (n) = x (n) )

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sha1_base64="ocuu1hhk/az3jg80lccu+mgitt4=">aaacj3icbvbnaxsxenwmaeu4h3hbyy5dtkens1skcyguqc49uma7adsyrtzrfdzkizqbyhb/m176v3ojncekx/ytypyeejvvosd7m6ozf+dkogrd+2dr1fbrn29ro/v37z983g18+txzpracu8ioy89j7lbjjv2sppa8t8izwofzpd1d+gcxaj00ukozhiczn2izsmhjs6pgt4abpujcp3hjjughljwwuq7hkbgl3gw8qe1hahmydjbfx1xxqnemw+essemiijrzhfao8x8wnqli/dyhuhp9kmxpwhjluiic1wefw5ylkz9g31pnm3tdcnnnhl56zbvuyjtbun3auflmuvkw+8qmu+rwvyx4ktcvkdkclllnbaewq4+sqgezwiqgy2lrkjp5wowvflcqkbdcki+27koi1k/ejl3vrshsrb9/ne8oqzhqbi/ts28sygfshp1ibdzlgv1h/9g1uwn+blfbbxc3kt0kqp4v7bmch0clvkte</latexit> <latexit 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sha1_base64="5typk9xpuuq9e5+lwmrcj7zgilk=">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</latexit> <latexit sha1_base64="5typk9xpuuq9e5+lwmrcj7zgilk=">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</latexit> ˆ MAP = argmax Notation Shorthand MAP, without shorthand f( = X (1) = x (1),...,X (n) = x (n) ) Our shorthand notation is shorthand for the event: = x (i) is shorthand for the event: X (i) = x (i) MAP, now with shorthand ˆ MAP = argmax f( x (1),...,x (n) )

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sha1_base64="hwtjai6zj9txhdv1epw014cbypq=">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</latexit> <latexit sha1_base64="hwtjai6zj9txhdv1epw014cbypq=">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</latexit> MLE vs MAP Data: x (1),...,x (n) <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> Maximum Likelihood Estimation ˆ MLE = argmax = argmax f(x (1),...,x (n) ) X log f(x (i) ) Maximum A Posteriori i ˆ MAP = argmax f( x (1),...,x (n) )

Most important slide of today

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sha1_base64="5typk9xpuuq9e5+lwmrcj7zgilk=">aaacvxicbvbnaxsxenvu0ir1p+kmx15etysnweygqnmpjptss8gfogl4xtmrz9oiwmmrzoonsn8yl9j/0kuh8gelttogehpvnmb08kpjr0nyi4p3dh/t7r88bj15+uz5yfvf0yuztru4feyze5wdqyu1dkmswqvkips5wsv8+unkv7xb66trx2lz4biemzaffecbmrrvngfygc2rojn4z+edhr/9wlnat4ml/0g+mxvaigm1lojbzkpyna3ngegcpg+k7qat3/lfn99ne80xz6ag3ph6rnsn703anasfris/bokwdni2bpp2xxhc1cvqegqcg6vjremwnarq2lsy2mef4hpmoapwtzob+3uqdx8tmckvja1he1+zfzs8lm4tyzx0lkbzd19bkf/trjuvp2mvdvutarezvnskk+gripluwhsklggasdlsysuclagkkbzccon9lz8efyf9nomnx951zk63crywv+w167kuvwdn7bmbscet7i79jkiojr5hv+ldeg/tgkdbz0v2t8whvwhtflpl</latexit> <latexit sha1_base64="5typk9xpuuq9e5+lwmrcj7zgilk=">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</latexit> Maximum A Posteriori data: x (1),...,x (n) <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> ˆ MAP = argmax f( x (1),...,x (n) ) ˆ MAP =argmax likelihood f(x (1),x (2),...,x (n) )g( ) h(x (1),x (2),...x (n) ) posterior prior

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sha1_base64="5typk9xpuuq9e5+lwmrcj7zgilk=">aaacvxicbvbnaxsxenvu0ir1p+kmx15etysnweygqnmpjptss8gfogl4xtmrz9oiwmmrzoonsn8yl9j/0kuh8gelttogehpvnmb08kpjr0nyi4p3dh/t7r88bj15+uz5yfvf0yuztru4feyze5wdqyu1dkmswqvkips5wsv8+unkv7xb66trx2lz4biemzaffecbmrrvngfygc2rojn4z+edhr/9wlnat4ml/0g+mxvaigm1lojbzkpyna3ngegcpg+k7qat3/lfn99ne80xz6ag3ph6rnsn703anasfris/bokwdni2bpp2xxhc1cvqegqcg6vjremwnarq2lsy2mef4hpmoapwtzob+3uqdx8tmckvja1he1+zfzs8lm4tyzx0lkbzd19bkf/trjuvp2mvdvutarezvnskk+gripluwhsklggasdlsysuclagkkbzccon9lz8efyf9nomnx951zk63crywv+w167kuvwdn7bmbscet7i79jkiojr5hv+ldeg/tgkdbz0v2t8whvwhtflpl</latexit> <latexit 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sha1_base64="5typk9xpuuq9e5+lwmrcj7zgilk=">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</latexit> <latexit sha1_base64="5typk9xpuuq9e5+lwmrcj7zgilk=">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</latexit> Maximum A Posteriori data: x (1),...,x (n) <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> ˆ MAP = argmax f( x (1),...,x (n) ) ˆ MAP =argmax =argmax =argmax =argmax g( )f(x (1),x (2),...,x (n) ) h(x (1),x (2),...x (n) ) g( ) Q n i=1 f(x(i) ) h(x (1),x (2),...,x (n) ) g( ) ny f(x (i) ) i=1 log(g( )) +! nx log(f(x (i) )) i=1 Fury K monotonic

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sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> <latexit sha1_base64="gichi1fa8t4wawaicrfsc/9d7vk=">aaacahicbvdlsgmxfm3uv62vqgsxbojfakgugrf04algxmuf+4b2ljk004zmmknyryzdbpwvny4ucetnupnvzlrdaoubwmk5997khi8sxintf1u5ldw19y38zmfre2d3r7h/0njhrchr0lcequmrzqsxrakcboteiphae6ztja8zv/3aloahvinjxnyadcx3osvgph7x6pe+ktuvtip7gxb0fwd3wun7xzjds6fay8szkxkao9evfpkbna6ybcqi1l3hjsbniajobuslvvizinaxgbkuozietlvjdieunxplgp1qmsmbt9xfhqkjtj4enqkmciz0opej/3ndgpxln+eyiofjonvijwwgegdp4afxjikygeko4uavmi6iihrmzgutgro48jjpnducu+bcnpfqv/m48ugynaayctafqqmb1ebnrfgkntereroerbfr3fqyleasec8h+gpr8wewf5uw</latexit> Maximum Likelihood Estimation ˆ MLE = argmax = argmax f(x (1),...,x (n) ) X log f(x (i) ) Maximum A Posteriori i ˆ MAP = argmax = argmax f( x (1),...,x (n) )! nx log(g( )) + log(f(x (i) )) i=1

Gotta get that intuition

P(θ D) For Bernoulli l l Prior: q ~ Beta(a, b); data = {n heads, m tails} Estimate p, aka q This is the beta PDF This is???

P(θ D) For Bernoulli l l Prior: q ~ Beta(a, b); data = {n heads, m tails} Estimate p, aka q This is the beta PDF Product of thetas and (1-theta)s = argmax = argmax h 1 log a 1 (1 ) b 1i +n log f(heads ) +m log f(tails ) log 1 +(a 1) log +(b 1) log(1 )+n log + m log(1 ) = argmax(a 1 + n) log +(b 1 + m) log(1 )

P(θ D) For Bernoulli l l Prior: q ~ Beta(a, b); D = {n heads, m tails} Estimate p, aka q = argmax(a 1 + n) log +(b 1 + m) log(1 ) = n + a 1 n + m + a + b 2 That s the mode of the updated beta

Hyper Parameters a b Hyperparameter a, b are fixed p Prior p Beta(a, b) Data distribution X i Bern(p) X 1 X 2 X n MAP will estimate the most likely value of p for this model

Where d Ya Get Them P(!)? l l l q is the probability a coin turns up heads Model q with 2 different priors: l l P 1 (q) is Beta(3,8) (blue) P 2 (q) is Beta(7,4) (red) They look pretty different! l Now flip 100 coins; get 58 heads and 42 tails l What do posteriors look like?

It s Like Having Twins argmax returns the mode l As long as we collect enough data, posteriors will converge to the true value!

Conjugate Distributions Without Tears Just for review Have coin with unknown probability q of heads Our prior (subjective) belief is that q ~ Beta(a, b) Now flip coin k = n + m times, getting n heads, m tails Posterior density: (q n heads, m tails) ~ Beta(a+n, b+\m) o Beta is conjugate for Bernoulli, Binomial, Geometric, and Negative Binomial a and b are called hyperparameters o Saw (a + b 2) imaginary trials, of those (a 1) are successes For a coin you never flipped before, use Beta(x, x) to denote you think coin likely to be fair o How strongly you feel coin is fair is a function of x

Gonna Need Priors Parameter Bernoulli p Binomial p Poisson Exponential Multinomial p i Normal µ Distribution for Parameter Beta Beta Gamma Gamma Dirichlet Normal Normal 2 Inverse Gamma Don t need to know Inverse Gamma. But it will know you

Good Times with Gamma Gamma(k,!) distribution Conjugate for Poisson Rate o Also conjugate for Exponential, but we won t delve into that Intuitive understanding of hyperparameters: o Saw k total imaginary events during! prior time periods 4.0 4.0 4.0 2.0

Good Times with Gamma Gamma(k,!) distribution Conjugate for Poisson Rate o Also conjugate for Exponential, but we won t delve into that Intuitive understanding of hyperparameters: o Saw k total imaginary events during! prior time periods Updating with observations o After observing n events during next t time periods... o... posterior distribution is Gamma(k + n,! + t) o MAP estimator for Poisson with Gamma prior is (k+n)/(! + t) o Example: Prior for rate is Gamma(10, 5) o Saw 10 events in 5 time periods. Like observing at rate = 2 o Now see 11 events in next 2 time periods à Gamma(21, 7) o MAP rate = 3

Reviving an Old Story Line The Multinomial Distribution Mult(p 1,, p k ) p(x 1,...,x k )= n! x 1!...x k! px 1 1...px k k

Multinomial is Multiple Times the Fun Dirichlet(a 1, a 2,..., a m ) distribution Conjugate for Multinomial o Dirichlet generalizes Beta in same way Multinomial generalizes Bernoulli f(x 1 = x 1,X 2 = x 2,...,X m = x m )=K Intuitive understanding of hyperparameters: o Saw åa i - m imaginary trials, with (a i 1) of outcome i Updating to get the posterior distribution o m i= 1 After observing n 1 + n 2 +... + n m, new trials with n i of outcome i... o... posterior distribution is Dirichlet(a 1 + n 1, a 2 + n 2,..., a m + n m ) my i=1 x a i 1 i

Best Short Film in the Dirichlet Category And now a cool animation of Dirichlet(a, a, a) This is actually log density (but you get the idea ) Thanks Wikipedia!

Example: Estimating Die Parameters

Your Happy Laplace Recall example of 6-sides die rolls: X ~ Multinomial(p 1, p 2, p 3, p 4, p 5, p 6 ) Roll n = 12 times Result: 3 ones, 2 twos, 0 threes, 3 fours, 1 fives, 3 sixes o MLE: p 1 =3/12, p 2 =2/12, p 3 =0/12, p 4 =3/12, p 5 =1/12, p 6 =3/12 Dirichlet prior allows us to pretend we saw each X i outcome k times before. MAP estimate: p = + k i n + mk o Laplace s law of succession : idea above with k = 1 X i + 1 o Laplace estimate: pi = n + m o Laplace: p 1 =4/18, p 2 =3/18, p 3 =1/18, p 4 =4/18, p 5 =2/18, p 6 =4/18 o No longer have 0 probability of rolling a three!

The last estimator has risen

One Shot Learning Single training example: Test set:

Is Peer Grading Accurate Enough? Peer Grading on Coursera HCI. 31,067 peer grades for 3,607 students. Tuned Models of Peer Assessment. C Piech, J Huang, A Ng, D Koller

Is Peer Grading Accurate Enough? 1. Defined random variables for: True grade (s i ) for assignment i Observed (z ij ) score for assign i Bias (b j ) for each grader j Variance (r j ) for each grader j 2. Designed a probabilistic model that defined the distributions for all random variables z j i N (µ = s i + b j, = p r j ) s i N(µ 0, 0) b i N(0, 0 ) = hyperparameter r i InvGamma( 0, 0 ) Tuned Models of Peer Assessment. C Piech, J Huang, A Ng, D Koller

Is Peer Grading Accurate Enough? 1. Defined random variables for: True grade (s i ) for assignment i Observed (z ij ) score for assign i Bias (b j ) for each grader j Variance (r j ) for each grader j 2. Designed a probabilistic model that defined the distributions for all random variables 3. Found variable assignments using MAP estimation given the observed data Inference or Machine Learning Tuned Models of Peer Assessment. C Piech, J Huang, A Ng, D Koller

Improved Accuracy Before: After: Some students were getting very erroneous grades 99% within 10pp -100-80 -60-40 -20 0 20 40 60 80-100 -80-60 -40-20 0 20 40 60 80 Error is based on ground truth assignments. Results are across all assignments (~10,000 submissions) Parent s Club