MoreLikelihood - Revision history

Carver: /* Likelihood for randn as a function of Mean and Standard Deviation */

2009-02-04T00:12:06Z

‎Likelihood for randn as a function of Mean and Standard Deviation

Carver: /* Likelihood for randn as a function of Mean and Standard Deviation */

2009-02-04T00:10:36Z

‎Likelihood for randn as a function of Mean and Standard Deviation

Carver: /* Likelihood for randn as a function of Mean and Standard Deviation */

2009-02-03T23:35:41Z

‎Likelihood for randn as a function of Mean and Standard Deviation

Carver: /* Two Random Variables, Independence */

2009-02-03T22:49:07Z

‎Two Random Variables, Independence

Carver: /* Two Random Variables, Independence */

2009-02-03T22:47:30Z

‎Two Random Variables, Independence

Carver: /* Likelihood for randn as a function of Mean and Standard Deviation */

2009-02-03T21:17:27Z

‎Likelihood for randn as a function of Mean and Standard Deviation

Carver: /* Two Random Variables, Independence */

2009-02-03T20:12:42Z

‎Two Random Variables, Independence

Carver: /* Likelihood for randn as a function of Mean and Standard Deviation */

2009-02-03T20:11:58Z

‎Likelihood for randn as a function of Mean and Standard Deviation

Carver: /* Homework from Lab B */

2009-02-03T20:03:57Z

‎Homework from Lab B

Carver: /* Likelihood for randn as a function of Mean and Standard Deviation */

2009-02-03T19:17:39Z

‎Likelihood for randn as a function of Mean and Standard Deviation

@@ Line 556: / Line 556: @@
 [[Image:Randsurf.jpg|center|thumb|300px|Click image for full size surface plot]]
-[[Image:Randcontour.jpg|center|thumb|300px|Click image for full size contour plot.  ''Added later: The asterisk is the true (mean, std); the peak of the function is the estimate of (mean,std)]]
+[[Image:Randcontour.jpg|center|thumb|300px|Click image for full size contour plot.  ''Added later: The asterisk is the true'' (std, mean); ''the peak of the function is the estimate of'' (std, mean)]]
 * '''Homework C.*:''' Bonus Problem (one point).  Be creative!  See [[Filename_Convention|filename conventions]].

@@ Line 556: / Line 556: @@
 [[Image:Randsurf.jpg|center|thumb|300px|Click image for full size surface plot]]
-[[Image:Randcontour.jpg|center|thumb|300px|Click image for full size contour plot]]
+[[Image:Randcontour.jpg|center|thumb|300px|Click image for full size contour plot.  ''Added later: The asterisk is the true (mean, std); the peak of the function is the estimate of (mean,std)]]
 * '''Homework C.*:''' Bonus Problem (one point).  Be creative!  See [[Filename_Convention|filename conventions]].

@@ Line 542: / Line 542: @@
 [[Image:Emeanmis.jpg|center|thumb|300px|Click image for full size surface plot]]
-* '''Homework C.4:'''  What if the data were generated by the "rand" command instead of the "randnms" command we created, based on randn.  How well can you estimate the mean and standard deviation?  See [[Filename_Convention|filename conventions]].
+* '''Homework C.4:'''  What if the data were generated by the "rand" command instead of the "randnms" command we created, based on randn.  How well can you estimate the mean and standard deviation?  See [[Filename_Convention|filename conventions]].  ''Clarification added later:  This question concerns the plots below.  I generated data with the command rand, but calculated the likelihood using a model that incorrectly assumes the data are from randmns.''
 I replaced the command

@@ Line 231: / Line 231: @@
 V1 and N are independent and V1 and V2 are not.
-'''Mathematical Definition of Independence:''' Two continuous random variables X1 and X2 are independent if and only their joint density is the product of their marginal densities: p(X1,X2) = p(X1)*p(X2)
+'''Mathematical Definition of Independence:''' Two continuous random variables X1 and X2 are independent if and only if their joint density is the product of their marginal densities: p(X1,X2) = p(X1)*p(X2)
 * '''Homework C.1:''' What is special about the joint density of independent random variables?  How can you tell from the second 3D histogram that V1 and V2 are not independent?  For this problem, think more about the definition of independence and use it to justify your answer.  For more intuition, you could try plotting 3D histograms of other independent random variables [RV1, RV2] where the random variables are functions of different calls to rand or randn (as toward the end of Lab B).  This extra work could be the basis of your bonus problem, if you want.  See [[Filename_Convention|filename conventions]].

← Older revision		Revision as of 22:47, 3 February 2009
Line 230:		Line 230:

	V1 and N are independent and V1 and V2 are not.		V1 and N are independent and V1 and V2 are not.
		+
		+	'''Mathematical Definition of Independence:''' Two continuous random variables X1 and X2 are independent if and only their joint density is the product of their marginal densities: p(X1,X2) = p(X1)*p(X2)

	* '''Homework C.1:''' What is special about the joint density of independent random variables? How can you tell from the second 3D histogram that V1 and V2 are not independent? For this problem, think more about the definition of independence and use it to justify your answer. For more intuition, you could try plotting 3D histograms of other independent random variables [RV1, RV2] where the random variables are functions of different calls to rand or randn (as toward the end of Lab B). This extra work could be the basis of your bonus problem, if you want. See [[Filename_Convention\|filename conventions]].		* '''Homework C.1:''' What is special about the joint density of independent random variables? How can you tell from the second 3D histogram that V1 and V2 are not independent? For this problem, think more about the definition of independence and use it to justify your answer. For more intuition, you could try plotting 3D histograms of other independent random variables [RV1, RV2] where the random variables are functions of different calls to rand or randn (as toward the end of Lab B). This extra work could be the basis of your bonus problem, if you want. See [[Filename_Convention\|filename conventions]].

← Older revision		Revision as of 21:17, 3 February 2009
Line 559:		Line 559:

	If you finish lab early, you might want to look at the textbook software, which should be on the lab computers now. Next Thursday's lab will use this software. If it does not work, tell me.		If you finish lab early, you might want to look at the textbook software, which should be on the lab computers now. Next Thursday's lab will use this software. If it does not work, tell me.
		+
		+	== Coming Soon ==
		+
		+	Here is the present plan:
		+
		+	'''Thursday, Jan 5:''' Introduction to Neuron's in a Action, textbook.
		+
		+	'''Tuesday, Jan 10:''' Model Selection and Discussion of Projects.
		+
		+	'''Beyond:''' More textbook tutorials, and eventually a lab that discussion Bayesian filtering, how likelihood is calculated for the Hodgkin-Huxley model.

← Older revision		Revision as of 20:11, 3 February 2009
Line 559:		Line 559:

	* '''Homework C.*:''' Bonus Problem (one point). Be creative! See [[Filename_Convention\|filename conventions]].		* '''Homework C.*:''' Bonus Problem (one point). Be creative! See [[Filename_Convention\|filename conventions]].
		+
		+	If you finish lab early, you might want to look at the textbook software, which should be on the lab computers now. Next Thursday's lab will use this software. If it does not work, tell me.

@@ Line 138: / Line 138: @@
   hist(F,100)
-* '''Homework B.5:''' Consider the uniform random variable A = rand and the random variable F = sigmoid(A,k,h).  Where does the PDF of F have peaks; where does it have troughs.  Hint: For different k,h, be sure to plot both the function over the range of rand (0 to 1) and the histogram.
+* '''Homework B.5 (from Lab B):''' Consider the uniform random variable A = rand and the random variable F = sigmoid(A,k,h).  Where does the PDF of F have peaks; where does it have troughs.  Hint: For different k,h, be sure to plot both the function over the range of rand (0 to 1) and the histogram.
 Many students did not execute the commands in order.  If I had anticipated this I would have used different names (A and B) instead of just A.  Your plots for problem 5 should have looked like this:

@@ Line 552: / Line 552: @@
   data = rand(n,1);
-Then I spent considerable time tweaking the ranges to get the plot to look reasonable.  Just look at the results below:
+Then I spent considerable time tweaking the ranges (of the assumed means and stds) to get the plot to look reasonable.  Just look at the results below:
 [[Image:Randsurf.jpg|center|thumb|300px|Click image for full size surface plot]]