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--- cs401r_w2016:lab3 [2015/12/23 20:04]
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+++ cs401r_w2016:lab3 [2015/12/23 21:16]
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-===Objective:===
+====Objective:====
 To understand how to sample from different distributions, and to
@@ Line 6: / Line 6: @@
 a small library of random variable types.
-===Deliverable:===
+====Deliverable:====
 You should turn in an ipython notebook that implements and tests a
@@ Line 18: / Line 18: @@
 {{:cs401r_w2016:lab3.png?nolink|}}
+For multidimensional variables, your visualization should convey information in a natural way; you can either use 3d surfaces, or 2d contour plots:
-===Description:===
+{{:cs401r_w2016:lab3_2d.png?nolink|}}
-You must implement:
+====Description:====
-* The following one dimensional, continuous valued distributions.  For
+You must implement seven random variable objects.  For each type, you should be able to sample from that distribution, and compute the log-likelihood of a particular value.  All of your classes should inherit from a base random variable object that supports the following methods:
-  these, you should also plot the PDF of the random variable on the
-  same plot; the curves should match.
-''
+<code python>
-Beta (alpha=1, beta=3)
-Poisson (lambda=7)
+class RandomVariable:
-Univariate Gaussian (mean=2, variance=3)
+    def __init__( self ):
-''
+        self.state = None
+        pass
+    def get( self ):
+        return self.state
+    def sample( self ):
+        pass
+    def log_likelihood( self ):
+        pass
+    def propose( self ):
+        pass
+</code>
+You don't need to implement the ''get'' or ''propose'' methods yet.  For example, your univariate Gaussian class might look like this:
+<code python>
+class Gaussian( RandomVariable ):
+    def __init__( self, mu, sigma ):
+        self.mu = mu
+        self.sigma = sigma
+        self.state = 0
+    def sample( self ):
+        return self.mu + self.sigma * numpy.Random.randn()
+    def log_likelihood( self, X, mu, sigma ):
+        return -numpy.log( sigma*numpy.sqrt(2*pi) ) - (X-mu)**2/(sigma**2)
+</code>
+**Given that framework, you should implement:**
+* The following one dimensional, continuous valued distributions.  To visualize
+these, you should plot a histogram of sampled values, and also plot the PDF of the random variable on the
+same axis; they should (roughly) match. //Note: it is **not** sufficient to let seaborn estimate the PDF using its built-in KDE estimator; you need to plot the true PDF.  In other words, you can't just use seaborn.kdeplot!//
+  * ''Beta (a=1, b=3)''
+  * ''Poisson (lambda=7)''
+  * ''Univariate Gaussian (mean=2, variance=3)''
 * The following discrete distributions.  For these, plot predicted and
-  empirical histograms side-by-side:
+empirical histograms side-by-side:
+  * ''Bernoulli (p=0.7)'' (hint: you may need a uniform random number)
+  * ''Multinomial (pvals=[0.1, 0.2, 0.7])''
-''
+* The following multidimensional distributions. For these, use a contour or surface plot to visualize the empirical distribution of samples vs. the PDF:
-Bernoulli (p=0.7)
+  * ''Multivariate Gaussian ( mean=[2.0,3.0], cov=[[1.0,0.9],[0.9,1.0]] )''
-Multinomial (theta=[0.1, 0.2, 0.7])
+  * ''Dirichlet ( alpha=[ 0.1, 0.2, 0.7 ] )''
-''
-* The following multidimensional distributions.
+**Important notes:**
-* Two-dimensional Gaussian
+**You //may// use [[http://docs.scipy.org/doc/numpy-1.10.0/reference/routines.random.html|numpy.random]] to sample from the appropriate distributions.**
-* 3-dimensional Dirichlet
-===Hints:===
+**You may //not// use any existing code to calculate the log-likelihoods.**  But you can, of course, use any online resources or the book to find the appropriate definition of each PDF.
+====Hints:====
 The following functions may be useful to you:
 <code python>
+numpy.random
+matplotlib.pyplot.contour
+seaborn.kdeplot
+seaborn.jointplot
 hist( data, bins=50, normed=True )
@@ Line 61: / Line 113: @@
 </code>

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