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‎doc/pub/week12/ipynb/ipynb-week12-src.tar.gz‎

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‎doc/pub/week12/ipynb/week12.ipynb‎

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‎doc/pub/week12/pdf/week12.pdf‎

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‎doc/src/week12/_minted/2F94C7C8030AA1854BB403778B0F2084.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{c+c1}{\PYGZsh{} Set up the conditional probability distribution for each dimension}
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\PYG{c+c1}{\PYGZsh{} For example, I can sample p(a | b) using sample\PYGZus{}for\PYGZus{}dim[0].}
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\PYG{n}{univariate\PYGZus{}conditionals} \PYG{o}{=} \PYG{p}{[}
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\PYG{n}{get\PYGZus{}conditional\PYGZus{}dist}\PYG{p}{(}\PYG{n}{joint\PYGZus{}mu}\PYG{p}{,} \PYG{n}{joint\PYGZus{}cov}\PYG{p}{,} \PYG{n}{d}\PYG{p}{)}
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\PYG{k}{for} \PYG{n}{d} \PYG{o+ow}{in} \PYG{n+nb}{range}\PYG{p}{(}\PYG{n}{D}\PYG{p}{)}
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\PYG{p}{]}
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\end{MintedVerbatim}

‎doc/src/week12/_minted/43267A0395131F82B3FEF654B556A0B9.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{k}{def}\PYG{+w}{ }\PYG{n+nf}{get\PYGZus{}conditional\PYGZus{}dist}\PYG{p}{(}\PYG{n}{joint\PYGZus{}mu}\PYG{p}{,} \PYG{n}{joint\PYGZus{}cov}\PYG{p}{,} \PYG{n}{var\PYGZus{}index}\PYG{p}{)}\PYG{p}{:}
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\PYG{+w}{ }\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}Returns the conditional distribution given the joint distribution and which variable}
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\PYG{l+s+sd}{ the conditional probability should use.}
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\PYG{l+s+sd}{ Right now this only works for 2\PYGZhy{}variable joint distributions.}
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\PYG{l+s+sd}{ joint\PYGZus{}mu: joint distribution\PYGZsq{}s mu}
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\PYG{l+s+sd}{ joint\PYGZus{}cov: joint distribution\PYGZsq{}s covariance}
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\PYG{l+s+sd}{ var\PYGZus{}index: index of the variable in the joint distribution. Everything else will be}
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\PYG{l+s+sd}{ conditioned on. For example, if the joint distribution p(a, b, c) has mu [mu\PYGZus{}a, mu\PYGZus{}b, mu\PYGZus{}c],}
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\PYG{l+s+sd}{ to get p(c | a, b), use var\PYGZus{}index = 2.}
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\PYG{l+s+sd}{ returns:}
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\PYG{l+s+sd}{ a function that can sample from the univariate conditional distribution}
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\PYG{l+s+sd}{ \PYGZsq{}\PYGZsq{}\PYGZsq{}}
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\PYG{k}{assert} \PYG{n}{joint\PYGZus{}mu}\PYG{o}{.}\PYG{n}{shape}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{]} \PYG{o}{==} \PYG{l+m+mi}{2}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{Sorry, this function only works for 2\PYGZhy{}dimensional joint distributions right now}\PYG{l+s+s1}{\PYGZsq{}}
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\PYG{n}{a} \PYG{o}{=} \PYG{n}{joint\PYGZus{}mu}\PYG{p}{[}\PYG{n}{var\PYGZus{}index}\PYG{p}{]}
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\PYG{n}{b} \PYG{o}{=} \PYG{n}{joint\PYGZus{}mu}\PYG{p}{[}\PYG{o}{\PYGZti{}}\PYG{n}{var\PYGZus{}index}\PYG{p}{]}
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\PYG{n}{A} \PYG{o}{=} \PYG{n}{joint\PYGZus{}cov}\PYG{p}{[}\PYG{n}{var\PYGZus{}index}\PYG{p}{,} \PYG{n}{var\PYGZus{}index}\PYG{p}{]}
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\PYG{n}{B} \PYG{o}{=} \PYG{n}{joint\PYGZus{}cov}\PYG{p}{[}\PYG{o}{\PYGZti{}}\PYG{n}{var\PYGZus{}index}\PYG{p}{,} \PYG{o}{\PYGZti{}}\PYG{n}{var\PYGZus{}index}\PYG{p}{]}
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\PYG{n}{C} \PYG{o}{=} \PYG{n}{joint\PYGZus{}cov}\PYG{p}{[}\PYG{n}{var\PYGZus{}index}\PYG{p}{,} \PYG{o}{\PYGZti{}}\PYG{n}{var\PYGZus{}index}\PYG{p}{]}
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\PYG{c+c1}{\PYGZsh{} we\PYGZsq{}re dealing with one dimension so}
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\PYG{n}{B\PYGZus{}inv} \PYG{o}{=} \PYG{l+m+mi}{1}\PYG{o}{/}\PYG{n}{B}
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\PYG{c+c1}{\PYGZsh{} Return a function that can sample given a value of g}
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\PYG{k}{def}\PYG{+w}{ }\PYG{n+nf}{dist}\PYG{p}{(}\PYG{n}{g}\PYG{p}{)}\PYG{p}{:}
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\PYG{c+c1}{\PYGZsh{} a + C*B\PYGZca{}\PYGZob{}\PYGZhy{}1\PYGZcb{}(g \PYGZhy{} b)}
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\PYG{n}{mu} \PYG{o}{=} \PYG{n}{a} \PYG{o}{+} \PYG{n}{C} \PYG{o}{*} \PYG{n}{B\PYGZus{}inv} \PYG{o}{*} \PYG{p}{(}\PYG{n}{g} \PYG{o}{\PYGZhy{}} \PYG{n}{b}\PYG{p}{)}
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\PYG{c+c1}{\PYGZsh{} A \PYGZhy{} C * B\PYGZca{}\PYGZob{}\PYGZhy{}1\PYGZcb{} * C\PYGZca{}T}
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\PYG{n}{cov} \PYG{o}{=} \PYG{n}{A} \PYG{o}{\PYGZhy{}} \PYG{n}{B\PYGZus{}inv} \PYG{o}{*} \PYG{n}{C} \PYG{o}{*} \PYG{n}{C}
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\PYG{k}{return} \PYG{n}{np}\PYG{o}{.}\PYG{n}{sqrt}\PYG{p}{(}\PYG{n}{cov}\PYG{p}{)} \PYG{o}{*} \PYG{n}{np}\PYG{o}{.}\PYG{n}{random}\PYG{o}{.}\PYG{n}{randn}\PYG{p}{(}\PYG{l+m+mi}{1}\PYG{p}{)} \PYG{o}{+} \PYG{n}{mu}
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\PYG{k}{return} \PYG{n}{dist}
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\end{MintedVerbatim}

‎doc/src/week12/_minted/6E30EEDCA6E4F0B63088422FCFB9AE8A.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{n}{N} \PYG{o}{=} \PYG{l+m+mi}{10000}
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\PYG{n}{L} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{linalg}\PYG{o}{.}\PYG{n}{cholesky}\PYG{p}{(}\PYG{n}{joint\PYGZus{}cov}\PYG{p}{)}
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\PYG{n}{samples\PYGZus{}from\PYGZus{}true\PYGZus{}distribution} \PYG{o}{=} \PYG{n}{L} \PYG{o}{@} \PYG{n}{np}\PYG{o}{.}\PYG{n}{random}\PYG{o}{.}\PYG{n}{randn}\PYG{p}{(}\PYG{n}{D}\PYG{p}{,} \PYG{n}{N}\PYG{p}{)} \PYG{o}{+} \PYG{n}{joint\PYGZus{}mu}
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\PYG{n}{plt}\PYG{o}{.}\PYG{n}{plot}\PYG{p}{(}\PYG{o}{*}\PYG{n}{samples\PYGZus{}from\PYGZus{}true\PYGZus{}distribution}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{.}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{,} \PYG{n}{alpha}\PYG{o}{=}\PYG{l+m+mf}{0.1}\PYG{p}{)}
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\PYG{n}{plt}\PYG{o}{.}\PYG{n}{axis}\PYG{p}{(}\PYG{p}{[}\PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{4}\PYG{p}{,} \PYG{l+m+mi}{4}\PYG{p}{,} \PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{4}\PYG{p}{,} \PYG{l+m+mi}{4}\PYG{p}{]}\PYG{p}{)}
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\PYG{n}{plt}\PYG{o}{.}\PYG{n}{show}\PYG{p}{(}\PYG{p}{)}
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\end{MintedVerbatim}

‎doc/src/week12/_minted/A3E61CC5AEB9A07789FE9BD2AD9F8010.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{n}{samples} \PYG{o}{=} \PYG{n}{gibbs\PYGZus{}sample}\PYG{p}{(}\PYG{n}{univariate\PYGZus{}conditionals}\PYG{p}{,} \PYG{n}{sample\PYGZus{}count}\PYG{o}{=}\PYG{l+m+mi}{100}\PYG{p}{)}
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\PYG{n}{fig}\PYG{p}{,} \PYG{n}{ax} \PYG{o}{=} \PYG{n}{plt}\PYG{o}{.}\PYG{n}{subplots}\PYG{p}{(}\PYG{p}{)}
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\PYG{n}{ax}\PYG{o}{.}\PYG{n}{plot}\PYG{p}{(}\PYG{o}{*}\PYG{n}{samples\PYGZus{}from\PYGZus{}true\PYGZus{}distribution}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{.}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{,} \PYG{n}{alpha}\PYG{o}{=}\PYG{l+m+mf}{0.1}\PYG{p}{)}
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\PYG{n}{ax}\PYG{o}{.}\PYG{n}{plot}\PYG{p}{(}\PYG{o}{*}\PYG{n}{samples}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{k}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{)}
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\PYG{n}{ax}\PYG{o}{.}\PYG{n}{plot}\PYG{p}{(}\PYG{o}{*}\PYG{n}{samples}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{.r}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{)}
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\PYG{n}{ax}\PYG{o}{.}\PYG{n}{axis}\PYG{p}{(}\PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{square}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{)}
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\PYG{n}{plt}\PYG{o}{.}\PYG{n}{show}\PYG{p}{(}\PYG{p}{)}
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\end{MintedVerbatim}

‎doc/src/week12/_minted/D2DFF7AAB78247C1D4D2B10B41895007.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{k+kn}{import}\PYG{+w}{ }\PYG{n+nn}{numpy}\PYG{+w}{ }\PYG{k}{as}\PYG{+w}{ }\PYG{n+nn}{np}
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\PYG{k+kn}{import}\PYG{+w}{ }\PYG{n+nn}{matplotlib}\PYG{n+nn}{.}\PYG{n+nn}{pyplot}\PYG{+w}{ }\PYG{k}{as}\PYG{+w}{ }\PYG{n+nn}{plt}
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\PYG{c+c1}{\PYGZsh{} two dimensions}
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\PYG{n}{D} \PYG{o}{=} \PYG{l+m+mi}{2}
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\PYG{c+c1}{\PYGZsh{} set up the means (standard normal distribution}
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\PYG{n}{a\PYGZus{}mu} \PYG{o}{=} \PYG{l+m+mi}{0}
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\PYG{n}{b\PYGZus{}mu} \PYG{o}{=} \PYG{l+m+mi}{0}
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\PYG{c+c1}{\PYGZsh{} and the variances and covariances}
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\PYG{n}{a\PYGZus{}sigma} \PYG{o}{=} \PYG{l+m+mi}{1}
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\PYG{n}{b\PYGZus{}sigma} \PYG{o}{=} \PYG{l+m+mi}{1}
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\PYG{n}{a\PYGZus{}b\PYGZus{}cov} \PYG{o}{=} \PYG{l+m+mf}{0.5}
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\PYG{n}{joint\PYGZus{}cov} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{vstack}\PYG{p}{(}\PYG{p}{(}\PYG{p}{(}\PYG{n}{a\PYGZus{}sigma}\PYG{p}{,} \PYG{n}{a\PYGZus{}b\PYGZus{}cov}\PYG{p}{)}\PYG{p}{,} \PYG{p}{(}\PYG{n}{a\PYGZus{}b\PYGZus{}cov}\PYG{p}{,} \PYG{n}{b\PYGZus{}sigma}\PYG{p}{)}\PYG{p}{)}\PYG{p}{)}
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\PYG{n}{joint\PYGZus{}mu} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{vstack}\PYG{p}{(}\PYG{p}{(}\PYG{n}{a\PYGZus{}mu}\PYG{p}{,} \PYG{n}{b\PYGZus{}mu}\PYG{p}{)}\PYG{p}{)}
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\end{MintedVerbatim}

‎doc/src/week12/_minted/E102C28695B012E3145F33C4906CBE49.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{k}{def}\PYG{+w}{ }\PYG{n+nf}{gibbs\PYGZus{}sample}\PYG{p}{(}\PYG{n}{univariate\PYGZus{}conditionals}\PYG{p}{,} \PYG{n}{sample\PYGZus{}count}\PYG{p}{)}\PYG{p}{:}
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\PYG{+w}{ }\PYG{l+s+sd}{\PYGZsq{}\PYGZsq{}\PYGZsq{}Does Gibbs sampling given the distribution\PYGZsq{}s univariate conditionals.}
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\PYG{l+s+sd}{ Returns a D x N matrix}
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\PYG{l+s+sd}{ \PYGZsq{}\PYGZsq{}\PYGZsq{}}
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\PYG{n}{D} \PYG{o}{=} \PYG{n+nb}{len}\PYG{p}{(}\PYG{n}{univariate\PYGZus{}conditionals}\PYG{p}{)}
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\PYG{k}{assert} \PYG{n}{D} \PYG{o}{==} \PYG{l+m+mi}{2}\PYG{p}{,} \PYG{l+s+s2}{\PYGZdq{}}\PYG{l+s+s2}{Sorry, this only supports 2 dimensions right now}\PYG{l+s+s2}{\PYGZdq{}}
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\PYG{c+c1}{\PYGZsh{} initializes an empty matrix for the samples}
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\PYG{n}{samples} \PYG{o}{=} \PYG{n}{np}\PYG{o}{.}\PYG{n}{zeros}\PYG{p}{(}\PYG{p}{(}\PYG{n}{D}\PYG{p}{,} \PYG{n}{sample\PYGZus{}count}\PYG{p}{)}\PYG{p}{)}
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\PYG{c+c1}{\PYGZsh{} initialize the first sample to some arbitrary value}
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\PYG{n}{samples}\PYG{p}{[}\PYG{p}{:}\PYG{p}{,} \PYG{l+m+mi}{0}\PYG{p}{]} \PYG{o}{=} \PYG{p}{[}\PYG{l+m+mi}{3}\PYG{p}{,} \PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{3}\PYG{p}{]}
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\PYG{k}{for} \PYG{n}{i} \PYG{o+ow}{in} \PYG{n+nb}{range}\PYG{p}{(}\PYG{l+m+mi}{1}\PYG{p}{,} \PYG{n}{sample\PYGZus{}count}\PYG{p}{)}\PYG{p}{:}
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\PYG{c+c1}{\PYGZsh{} first set this sample equal to the previous sample}
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\PYG{n}{samples}\PYG{p}{[}\PYG{p}{:}\PYG{p}{,} \PYG{n}{i}\PYG{p}{]} \PYG{o}{=} \PYG{n}{samples}\PYG{p}{[}\PYG{p}{:}\PYG{p}{,} \PYG{n}{i} \PYG{o}{\PYGZhy{}} \PYG{l+m+mi}{1}\PYG{p}{]}
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\PYG{c+c1}{\PYGZsh{} now update the dimension whose turn it is using the conditional distribution}
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\PYG{c+c1}{\PYGZsh{} pass in all dimension from the previous sample except this dimension}
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\PYG{n}{d} \PYG{o}{=} \PYG{n}{i} \PYG{o}{\PYGZpc{}} \PYG{n}{D}
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\PYG{n}{samples}\PYG{p}{[}\PYG{n}{d}\PYG{p}{,} \PYG{n}{i}\PYG{p}{]} \PYG{o}{=} \PYG{n}{univariate\PYGZus{}conditionals}\PYG{p}{[}\PYG{n}{d}\PYG{p}{]}\PYG{p}{(}\PYG{n}{samples}\PYG{p}{[}\PYG{o}{\PYGZti{}}\PYG{n}{d}\PYG{p}{,} \PYG{n}{i} \PYG{o}{\PYGZhy{}} \PYG{l+m+mi}{1}\PYG{p}{]}\PYG{p}{)}
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\PYG{k}{return} \PYG{n}{samples}
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\end{MintedVerbatim}

‎doc/src/week12/_minted/E42139F55CD9AA76EC146D3B93DD6A09.highlight.minted‎

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\begin{MintedVerbatim}[commandchars=\\\{\},codes={\catcode`\$=3\catcode`\^=7\catcode`\_=8\relax}]
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\PYG{n}{samples} \PYG{o}{=} \PYG{n}{gibbs\PYGZus{}sample}\PYG{p}{(}\PYG{n}{univariate\PYGZus{}conditionals}\PYG{p}{,} \PYG{n}{sample\PYGZus{}count}\PYG{o}{=}\PYG{n}{N}\PYG{p}{)}
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\PYG{n}{fig}\PYG{p}{,} \PYG{n}{axs} \PYG{o}{=} \PYG{n}{plt}\PYG{o}{.}\PYG{n}{subplots}\PYG{p}{(}\PYG{l+m+mi}{1}\PYG{p}{,} \PYG{l+m+mi}{3}\PYG{p}{,} \PYG{n}{figsize}\PYG{o}{=}\PYG{p}{(}\PYG{l+m+mi}{12}\PYG{p}{,} \PYG{l+m+mi}{4}\PYG{p}{)}\PYG{p}{,} \PYG{n}{sharex}\PYG{o}{=}\PYG{k+kc}{True}\PYG{p}{,} \PYG{n}{sharey}\PYG{o}{=}\PYG{k+kc}{True}\PYG{p}{)}
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\PYG{c+c1}{\PYGZsh{} set all the axes}
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\PYG{n}{axs}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{o}{.}\PYG{n}{axis}\PYG{p}{(}\PYG{p}{[}\PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{4}\PYG{p}{,} \PYG{l+m+mi}{4}\PYG{p}{,} \PYG{o}{\PYGZhy{}}\PYG{l+m+mi}{4}\PYG{p}{,} \PYG{l+m+mi}{4}\PYG{p}{]}\PYG{p}{)}
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\PYG{n}{axs}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{o}{.}\PYG{n}{plot}\PYG{p}{(}\PYG{o}{*}\PYG{n}{samples\PYGZus{}from\PYGZus{}true\PYGZus{}distribution}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{.}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{,} \PYG{n}{alpha}\PYG{o}{=}\PYG{l+m+mf}{0.1}\PYG{p}{)}
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\PYG{n}{axs}\PYG{p}{[}\PYG{l+m+mi}{0}\PYG{p}{]}\PYG{o}{.}\PYG{n}{set\PYGZus{}title}\PYG{p}{(}\PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{original p(a, b)}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{)}
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\PYG{n}{axs}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{]}\PYG{o}{.}\PYG{n}{plot}\PYG{p}{(}\PYG{o}{*}\PYG{n}{samples}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{k}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{,} \PYG{n}{alpha}\PYG{o}{=}\PYG{l+m+mf}{0.8}\PYG{p}{)}
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\PYG{n}{axs}\PYG{p}{[}\PYG{l+m+mi}{1}\PYG{p}{]}\PYG{o}{.}\PYG{n}{set\PYGZus{}title}\PYG{p}{(}\PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{gibbs sampling path}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{)}
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\PYG{n}{axs}\PYG{p}{[}\PYG{l+m+mi}{2}\PYG{p}{]}\PYG{o}{.}\PYG{n}{plot}\PYG{p}{(}\PYG{o}{*}\PYG{n}{samples}\PYG{p}{,} \PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{.g}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{,} \PYG{n}{alpha}\PYG{o}{=}\PYG{l+m+mf}{0.1}\PYG{p}{)}
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\PYG{n}{axs}\PYG{p}{[}\PYG{l+m+mi}{2}\PYG{p}{]}\PYG{o}{.}\PYG{n}{set\PYGZus{}title}\PYG{p}{(}\PYG{l+s+s1}{\PYGZsq{}}\PYG{l+s+s1}{gibbs samples}\PYG{l+s+s1}{\PYGZsq{}}\PYG{p}{)}
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\PYG{n}{plt}\PYG{o}{.}\PYG{n}{show}\PYG{p}{(}\PYG{p}{)}
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\end{MintedVerbatim}

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