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We write this function Q in the form Q(hllh) to indicate that it is defined in part by the assumption that the current hypothesis h is equal to 8 In its general form, the EM algorithm repeats the following two steps until convergence:

RESEARCH INTERESTS The nature of concept formation and theory construction in political science with particular emphasis on utopian impulses in political philosophy

t argmax Q (hf h) h'

Listing 5 7. drawRect - (void)drawRect:(NSRect)rect { [super drawRect:rect]; [[glView openGLContext] makeCurrentContext]; // clear the background this sets a pale blue-ish background glClearColor(0.75, 0.75, 0.95, 0.0); glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); // set the viewpoint glMatrixMode(GL_MODELVIEW); glLoadIdentity(); // the arguments to glRotatef determine the axes of rotation // (in this case, x and y) glRotatef( rotation, 1.0f, 1.0f, 0.0f ); // put the light in place GLfloat lightPosition [] = {1.6, 1, 3, 0.0}; glLightfv(GL_LIGHT0, GL_POSITION, lightPosition); if (!displayList) { displayList = glGenLists(1); glNewList(displayList, GL_COMPILE_AND_EXECUTE); // do the drawing this draws a teapot at the center of the viewport glTranslatef(0, 0, 0); glutSolidTeapot(0.5); glEndList(); } else { glCallList(displayList); } glFinish(); }

When the function Q is continuous, the EM algorithm converges to a stationary point of the likelihood function P ( Y ( h l ) When this likelihood function has a single maximum, EM will converge to this global maximum likelihood estimate for h' Otherwise, it is guaranteed only to converge to a local maximum In this respect, EM shares some of the same limitations as other optimization methods such as gradient descent, line search, and conjugate gradient discussed in 4

To illustrate the general EM algorithm, let us use it to derive the algorithm given in Section 6121 for estimating the means of a mixture of k Normal distributions As discussed above, the k-means problem is to estimate the parameters 0 = ( P I p k ) that define the means of the k Normal distributions We are given the observed The hidden variables Z = { ( z i l , , z i k ) }in this case indicate data X = { ( x i ) } which of the k Normal distributions was used to generate xi To apply EM we must derive an expression for Q ( h ( h f )that applies to our k-means problem First, let us derive an expression for 1np(Y(h1) Note the ) probability p(yi (h') of a single instance yi = ( x i ,Z i l , ~ i k of the full data can be written

First the graphics context is established. This is in effect the environment in which the drawing happens. Then the method clears the view and sets the background to a pale blue color. It then sets up a variety of OpenGL properties governing the placement and lighting for the diagram. Finally it draws the teapot. Note that if you go exploring in glut.h you will find method signatures for a whole range of shapes, including spheres, toruses, cubes, and so on. You might like to try adding drawing steps for other shapes either in addition or as alternatives to the teapot. Now add the code for the animation. This is much shorter it simply sets the incremental rotation for the object and refreshes the display. If this were a simple example, all of the drawing code might be in animateOneFrame, and in that case you wouldn t need to add the setNeedsDisplay command (see Listing 5 8).

To verify this note that only one of the zij can have the value 1, and all others must be 0 Therefore, this expression gives the probability distribution for xi generated by the selected Normal distribution Given this probability for a single instance p(yi(hl), the logarithm of the probability In P(Y(hl) for all m instances in the data is

lnP(Ylhf) = l n n p ( , l h l )

Listing 5 8. animateOneFrame - (void)animateOneFrame { rotation += 0.2f; [self setNeedsDisplay:YES]; }

COURSEWORK Political Science Introductory I Comparative European Politics American Political System National Institutions and the Policy Process Parties in the Political Process European Security and Integration National Security Family in Political Thought Living Politicians: The Electoral Process Geography Human Geography Political Geography Geography of Development: Asia Environmental Geography Environmental Issues Environmental Hazards Environmental Impact Assessment

Finally we must take the expected value of this In P(Ylhl) over the probability distribution governing Y or, equivalently, over the distribution governing the unobserved components zij of Y Note the above expression for In P(Ylhl) is a linear function of these zij In general, for any function f (z) that is a linear function of z, the following equality holds E[f (z)l = f (Ek1) This general fact about linear functions allows us to write