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Probabilistic Reasoning for Modeling Unreliable Data . Ron Tal York University. Agenda. Modeling Uncertainty Bayesian Reasoning M-Estimation Maximum Likelihood Common Pitfall More Advanced Models . Modeling Uncertainty. Why is it necessary?

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## Probabilistic Reasoning for Modeling Unreliable Data

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**Probabilistic Reasoning for Modeling Unreliable Data**Ron Tal York University**Agenda**• Modeling Uncertainty • Bayesian Reasoning • M-Estimation • Maximum Likelihood • Common Pitfall • More Advanced Models**Modeling Uncertainty**• Why is it necessary? • The only certainty in this world is uncertainty • Often we cannot afford or are not capable of explicitly enumerating all variables absolutely • Sometimes uncertainty is caused by a limit of the reliability of the technology • Making decisions with unreliable data**Modeling Uncertainty (cont.)**• Three competing paradigms: • Non-monotonous Calculus • Fuzzy Logic • Probability Theory • Since we cannot construct a deterministic solution to many problems, we model sources of uncertainty as probability distributions**Bayesian Reasoning**• At the core of probabilistic frameworks is Bayesian Inference • Let’s define a few concepts: • - The probability of witnessing evidence E given a hypothesis H • - The probability of hypothesis H given the evidence E • - Probability of Hprior to observing E • -**Bayesian Reasoning: Bayes’ theorem**• States that: • Our life becomes simpler Expressed in terms of our model We usually know! We don’t always care! What we want to maximize**Bayesian Reasoning: Bayes’ theorem**• If we prefer, it can also be written as The joint probability**M-Estimation**Bayesian Inference gives us a powerful tool to choose the hypothesis that models the data A simple example is the set of parameters of a line of best fit through noisy data Statistical tools to achieve this are called M-Estimators The most popular choice is a special case called “Maximum Likelihood Estimator”**Maximum Likelihood**Recall Bayes’ theorem: The denominator is merely a normalization constant Maximum Likelihood can be applied if we assume the model prior is known**Maximum Likelihood (cont.)**• When model prior is constant: • Thus, we can fit model parameters by maximizing the likelihood**Maximum Likelihood (cont.)**• To determine parameters of a model, we maximize the negative log likelihood: • This let’s us avoid playing with products**Maximum Likelihood (cont.)**• For Gaussian distribution this is especially convenient:**Maximum Likelihood**• Becoming: Least Squares Constant**Common Pitfall**• We love Gaussian Distributions • We love Least-Squares • However, using Least-Squares without the process of probabilistic reasoning is a common rookie mistake**Better Modeling**• Many statistical tools are available for when the Gaussian assumption fails • Assumptions can include • Good Data is Gaussian, Outliers are present • pdf can be represented as a mixture of causes • No parametric model is best suited for the job**Robust Statistics**In Robust M-Estimators it is assumed that the data is locally Gaussian but outliers make traditional Least-Squares unsuitable Essentially, we give ‘bad’ data more credibility than it deserves Robust formulation ‘weighs’ the data with a Robust Influence Function**Robust Statistics (cont.)**E.g. Tukey’s Biweight:**Mixture Models**• Data can be represented as caused by one of several possible causes • Essentially a weighted sum of distributions • GMM is extremely powerful • EM Clustering is the ideal estimator for that**Non-parametric**Actual observed data is used in place of a fitted model Usually a histogram To find the ML fit between new observed data and the histogram we can minimize the Bhattachariyya Distance:**Non-parametric**Very simple to use Sometimes most accurate Very inefficient for problems with high dimensionality

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