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Homework 1 Reminder Due date: 21.11.2011 (till 23:59) Submission: – – – [email protected] Write the names of students in your team Send only one e-mail Midterm 1 Reminder Date: 24.11.211 14:30 – 16:20 Place: 1029, 2009 Hidden Markov Models Hidden Markow Models: – – – A hidden Markov model (HMM) is a statistical model, in which the system being modeled is assumed to be a Markov process (Memoryless process: its future and past are independent ), with hidden states. Hidden Markov Models Hidden Markow Models: – – – Has a set of states each of which has limited number of transitions and emissions, Each transition between states has an assisgned probability, Each model strarts from start state and ends in end state, Hidden Markov Models Hidden Markow Models parameters: – – – – A set of finite number of states, Si, The transition probability from state Si to Sj, aij, The emission probability density of a symbol ω in state Si Hidden Markov Models Hidden Markow Models parameters: – Firstly discuss: Morkov Models, Markov Assumption Hidden Markov Models Markow Models and Assumption (cont.): – To understand HMMs: Talk about weather, Assume there are three types of weather: – Sunny, – Rainy, – Foggy. Assume weather does not change during the day (if it is sunny it will sunny all the day) Hidden Markov Models Markow Models and Assumption (cont.): Weather prediction is about the what would be the weather tomorrow, – Based on the observations on the past. Hidden Markov Models Markow Models and Assumption (cont.): Weather at day n is qn {sunny , rainy , foggy} – qn depends on the known weathers of the past days (qn-1, qn-2,…) Hidden Markov Models Markow Models and Assumption (cont.): We want to find that: – means given the past weathers what is the probability of any possible weather of today. Hidden Markov Models Markow Models and Assumption (cont.): For example: if we knew the weather for last three days was: the probability that tomorrow would be P(q4 = | q3 = , q2 = , q1 = is: ) Hidden Markov Models Markow Models and Assumption (cont.): – For example: this probability could be infered from the statistics of past observations the problem: the larger n is, the more observations we must collect. – for example: if n = 6 we need 3(6-1) = 243 past observations. Hidden Markov Models Markow Models and Assumption (cont.): – Therefore, make a simplifying assumption Markov assumption: For sequence: the weather of tomorrow only depends on today (first order Markov model) Hidden Markov Models Markow Models and Assumption (cont.): Examples: The probabilities table: Hidden Markov Models Markow Models and Assumption (cont.): Examples: HMM: Hidden Markov Models Markow Models and Assumption (cont.): Examples: Given that day the weather is sunny, what is the probability that tomorrow is sunny and the next day is rainy ? Markov assumption Hidden Markov Models Markow Models and Assumption (cont.): Examples: If the weather yesterday was rainy and today is foggy what is the probability that tomorrow it will be sunny? Hidden Markov Models Markow Models and Assumption (cont.): – Examples: If the weather yesterday was rainy and today is foggy what is the probability that tomorrow it will be sunny? Markov assumption Hidden Markov Models Hidden Markov Models (HMMs): – What is HMM: Suppose that you are locked in a room for several days, you try to predict the weather outside, The only piece of evidence you have is whether the person who comes into the room bringing your daily meal is carrying an umbrella or not. Hidden Markov Models Hidden Markov Models (HMMs): – What is HMM (cont.): assume probabilities as seen in the table: Hidden Markov Models Hidden Markov Models (HMMs): – What is HMM (cont.): Now the actual weather is hidden from you. You can not directly see what is the weather. Hidden Markov Models Hidden Markov Models (HMMs): – What is HMM (cont.): Finding the probability of a certain weather is based on the observations xi: qn {sunny , rainy , foggy} Hidden Markov Models Hidden Markov Models (HMMs): – What is HMM (cont.): Using Bayes rule: For n days: Hidden Markov Models Hidden Markov Models (HMMs): – What is HMM (cont.): We can omit With Markov assumptions: So: Hidden Markov Models Hidden Markov Models (HMMs): – Examples: Suppose the day you were locked in it was sunny. The next day, the caretaker carried an umbrella into the room. You would like to know, what the weather was like on this second day. Hidden Markov Models Hidden Markov Models (HMMs): – Examples: Calculate 3 probabilities: Hidden Markov Models Hidden Markov Models (HMMs): – Examples: Consider the event with highest value. It is most likely to happen. Hidden Markov Models Hidden Markov Models (HMMs): – Another Examples: Suppose you do not know how the weather was when your were locked in. The following three days the caretaker always comes without an umbrella. Calculate the likelihood for the weather on these three days to have been Hidden Markov Models Hidden Markov Models (HMMs): – Another Examples: As you do not know how the weather is on the first day, you assume the 3 weather situations are equiprobable on this day and the prior probability for sun on day one is therefore Hidden Markov Models Hidden Markov Models (HMMs): – Another Examples: Assumption: Hidden Markov Models Hidden Markov Models: – Another Examples: 3 3 Discrete Markov Processes (Markov Chains) 3 4 First-Order Markov Models 3 5 First-Order Markov Models 3 6 First-Order Markov Models 3 7 First-Order Markov Model Examples 3 8 First-Order Markov Model Examples 3 9 First-Order Markov Model Examples 4 0 First-Order Markov Models 4 1 First-Order Markov Models 4 2 First-Order Markov Models 4 3 First-Order Markov Models 4 4 First-Order HMM Examples 4 5 First-Order HMM Examples 4 6 First-Order HMM Examples 4 7 First-Order HMM Examples 4 8 Three Fundamental Problems for HMMs 4 9 HMM Evaluation Problem 5 0 HMM Evaluation Problem 5 1 HMM Evaluation Problem 5 2 HMM Evaluation Problem 5 3 HMM Evaluation Problem 5 4 HMM Decoding Problem 5 5 HMM Decoding Problem 5 6 HMM Decoding Problem 5 7 HMM Learning Problem 5 8 HMM Learning Problem 5 9 HMM Learning Problem 6 0 HMM Learning Problem References R.O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, New York: John Wiley, 2001. Selim Aksoy, “Pattern Recognition Course Materials”, Bilkent University, 2011.