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CH3: 4 minor changes #125

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merged 4 commits into from
Aug 22, 2013
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CH3: 4 minor changes #125

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3137f6a
the example shows 500 samples, not 300
haraldschilly Aug 22, 2013
834022c
lw variable is used everywhere except here
haraldschilly Aug 22, 2013
bd45dd4
lowercase x and y are defined (same object), but only x is used. subs…
haraldschilly Aug 22, 2013
0ac9ca9
superfluous parenthesis
haraldschilly Aug 22, 2013
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12 changes: 6 additions & 6 deletions Chapter3_MCMC/IntroMCMC.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -47,12 +47,12 @@
"from mpl_toolkits.mplot3d import Axes3D\n",
"\n",
"fig = plt.figure()\n",
"x = y = np.linspace(0,5, 100 )\n",
"x = y = np.linspace(0, 5, 100 )\n",
"X, Y = np.meshgrid(x, y)\n",
"\n",
"subplot(121)\n",
"uni_x = stats.uniform.pdf( x,loc=0, scale = 5)\n",
"uni_y = stats.uniform.pdf(x, loc=0, scale = 5)\n",
"uni_x = stats.uniform.pdf(x, loc=0, scale = 5)\n",
"uni_y = stats.uniform.pdf(y, loc=0, scale = 5)\n",
"M = np.dot( uni_x[:,None], uni_y[None,:] )\n",
"im = plt.imshow(M, interpolation='none', origin='lower',\n",
" cmap=cm.jet, vmax=1, vmin = -.15, extent=(0,5,0,5))\n",
Expand Down Expand Up @@ -557,7 +557,7 @@
"subplot(313)\n",
"p_trace = mcmc.trace(\"p\")[:]\n",
"plot( p_trace, label = \"$p$: frequency of assignment to cluster 0\",\n",
" color = \"#467821\", lw = 1)\n",
" color = \"#467821\", lw = lw)\n",
"plt.xlabel( \"Steps\" )\n",
"plt.ylim(0,1)\n",
"plt.legend();"
Expand Down Expand Up @@ -809,7 +809,7 @@
"\n",
"During MCMC, we are returned vectors representing samples from the unknown posteriors. Elements of different vectors cannot be used together, as this would break the above logic: perhaps a sample has returned that cluster 1 has a small standard deviation, hence all the other variables in that sample would incorporate that and be adjusted accordingly. It is easy to avoid this problem though, just make sure you are indexing traces correctly. \n",
"\n",
"Another small example to illustrate the point. Suppose two variables, $x$ and $y$, are related by $x+y=10$. We model $x$ as a Normal random variable with mean 4 and explore 300 samples. "
"Another small example to illustrate the point. Suppose two variables, $x$ and $y$, are related by $x+y=10$. We model $x$ as a Normal random variable with mean 4 and explore 500 samples. "
]
},
{
Expand Down Expand Up @@ -878,7 +878,7 @@
"In our case, $A$ represents $L_x = 1$ and $X$ is the evidence we have: we observe that $x = 175$. For a particular sample set of parameters for our posterior distribution, $( \\mu_0, \\sigma_0, \\mu_1, \\sigma_1, p)$, we are interested in asking \"Is the probability that $x$ is in cluster 1 *greater* than the probability it is in cluster 0?\", where the probability is dependent on the chosen parameters.\n",
"\n",
"\\begin{align}\n",
"& P(L_x = 1| x = 175 ) \\gt P( (L_x = 0| x = 175 ) \\\\\\\\[5pt]\n",
"& P(L_x = 1| x = 175 ) \\gt P(L_x = 0| x = 175 ) \\\\\\\\[5pt]\n",
"& \\frac{ P( x=175 | L_x = 1 )P( L_x = 1 ) }{P(x = 175) } \\gt \\frac{ P( x=175 | L_x = 0 )P( L_x = 0 )}{P(x = 175) }\n",
"\\end{align}\n",
"\n",
Expand Down