Probabilistic Graphical Models - Day 2

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Joint Distributions | Continue |
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Description: # Joint Distributions Consider two discrete random variables X and Y. The function given by f (x, y) = P(X = x, Y = y) for each pair of values (x, y) within the range of X is called the joint probability distribution of X and Y.

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Description: # Joint Distributions Consider two discrete random variables X and Y. The function given by f (x, y) = P(X = x, Y = y) for each pair of values (x, y) within the range of X is called the joint probability distribution of X and Y.

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Independent and Identical Random Variables | Continue |
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Description: # Independent and Identical Random Variables When observations drawn from a random sample are independent of each other, but have the same probability distribution, then the observations are said to be independent and identical random variables. Lets say from a population of transactions, we drew a random sample and we try to observe if the transaction is a fradulent transaction or not. Assuming that the probability of a transaction to be 'p', we can say each transaction from the random sample, has the same probability distribution but the transactions themselves are independent of each other. Hence, these are said to be independent and identical random variables. An interesting point to note is that the random variables are said to be independent only when the joint probability distribution of the variables is a product of the individual marginal probability distributions. (For further reading refer to the following link: https://inst.eecs.berkeley.edu/~cs70/sp13/notes/n17.pdf)

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Description: # Independent and Identical Random Variables When observations drawn from a random sample are independent of each other, but have the same probability distribution, then the observations are said to be independent and identical random variables. Lets say from a population of transactions, we drew a random sample and we try to observe if the transaction is a fradulent transaction or not. Assuming that the probability of a transaction to be 'p', we can say each transaction from the random sample, has the same probability distribution but the transactions themselves are independent of each other. Hence, these are said to be independent and identical random variables. An interesting point to note is that the random variables are said to be independent only when the joint probability distribution of the variables is a product of the individual marginal probability distributions. (For further reading refer to the following link: https://inst.eecs.berkeley.edu/~cs70/sp13/notes/n17.pdf)

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Bayesian Networks | Continue |
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Fraud Modeling Example with pgmpy | Continue |
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Description: # Fraud Modeling Example with pgmpy pgmpy is one of the popular packages to do Bayesian Network modeling. We shall continue to use the fraud modeling example to visualize our network. pgmpy is good for simpler problems, to visualize the indepencies and CPDs. It doesn't work very well for large dimensional problems. There are other toolkits which are available such as: * WINMINE by Microsoft: https://www.microsoft.com/en-us/research/project/winmine-toolkit/

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Description: # Fraud Modeling Example with pgmpy pgmpy is one of the popular packages to do Bayesian Network modeling. We shall continue to use the fraud modeling example to visualize our network. pgmpy is good for simpler problems, to visualize the indepencies and CPDs. It doesn't work very well for large dimensional problems. There are other toolkits which are available such as: * WINMINE by Microsoft: https://www.microsoft.com/en-us/research/project/winmine-toolkit/

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Features of a Bayesian Network | Continue |
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Description: # Features of a Bayesian Network So far we have seen that: * A Bayesian Network is a joint probability distribution of a set of random variables.

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Description: # Features of a Bayesian Network So far we have seen that: * A Bayesian Network is a joint probability distribution of a set of random variables.

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Credit Approval Model using a Bayesian Network | Continue |
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Description: # Credit Approval Model using a Bayesian Network Let us look at a credit approval process example. Please note that the model/process shown here does not closely follow any real life approval process. This model is a completely generated from scratch solely for the purpose of practice and easy explanation. There are two factors, Outstanding Loan (OL) and Payment History (PH) which are independent of each other and influence another factor Credit Rating (CR). Credit Rating and Income Level (IL) are in turn two independent factors which influence Interest Rate (IR) of a credit line that would be extended to a customer. Depending upon CR and IL, a customer may receive a credit/loan at a premium rate, par rate or discounted interest rate.

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Description: # Credit Approval Model using a Bayesian Network Let us look at a credit approval process example. Please note that the model/process shown here does not closely follow any real life approval process. This model is a completely generated from scratch solely for the purpose of practice and easy explanation. There are two factors, Outstanding Loan (OL) and Payment History (PH) which are independent of each other and influence another factor Credit Rating (CR). Credit Rating and Income Level (IL) are in turn two independent factors which influence Interest Rate (IR) of a credit line that would be extended to a customer. Depending upon CR and IL, a customer may receive a credit/loan at a premium rate, par rate or discounted interest rate.

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Gibbs Sampling | Continue |
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Description: # Gibbs Sampling Bayesian inference generates full posterior probability distribution over a set of random variables. Gibbs Sampling algorithm is based on Monte Carlo Markov Chain (MCMC) technique. The underlying logic of MCMC sampling is that we can estimate any desired expectation by ergodic averages [Gibbs]. ## Gibbs Sampling in pgmpy

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Description: # Gibbs Sampling Bayesian inference generates full posterior probability distribution over a set of random variables. Gibbs Sampling algorithm is based on Monte Carlo Markov Chain (MCMC) technique. The underlying logic of MCMC sampling is that we can estimate any desired expectation by ergodic averages [Gibbs]. ## Gibbs Sampling in pgmpy

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Gaussian Mixture Models (GMMs) | Continue |
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Description: # Gaussian Mixture Models (GMMs) ## The Three Archers - Not So 'normal' Data In an archery training session, three archers are told to shoot at the target. Assume that the archers are shooting the same arrows and later at the end of the competition they need to count their scores. What would be the best estimate of their scores? Assume that inner yellow has the highest score and the score lowers as the circle your arrow lands in moves away from the center.

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Description: # Gaussian Mixture Models (GMMs) ## The Three Archers - Not So 'normal' Data In an archery training session, three archers are told to shoot at the target. Assume that the archers are shooting the same arrows and later at the end of the competition they need to count their scores. What would be the best estimate of their scores? Assume that inner yellow has the highest score and the score lowers as the circle your arrow lands in moves away from the center.

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Jensen's Inequality that Guarantees Convergence of the EM Algorithm | Continue |
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Description: # Jensen's Inequality that Guarantees Convergence of the EM Algorithm Jensen's Inequality states that given g, a strictly convex function, and X a random variable, then $$ E[g(X)] ≥ g(E[X]) $$

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Description: # Jensen's Inequality that Guarantees Convergence of the EM Algorithm Jensen's Inequality states that given g, a strictly convex function, and X a random variable, then $$ E[g(X)] ≥ g(E[X]) $$

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