Publications by Jake
Modelling Bayesian
Modelling_Bayesian Jake 29/09/2022 Bayesian Networks can be queried for probabilities Probability of Evidence A simple query is the probability of some variable instantiation Consider evidence \(\mathbf{E}\) and network variables \(\mathbf{W}\) \[ P(\mathbf{e}),\quad\mathbf{E}\in\mathbf{W}\] For example: \[ P(X=True, D=True)\] Prior and ...
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Propositional Logic
Propositional Logic Jake 22/09/2022 Propositional Syntax Propositional variables can be sentences or sentences of sentences Typicall binary, however can be extended to be multi-valued \[ P_1,...,P_n\] Logical Connectives There are three primitive logical connectives: \[ \land\text{ - Logical conjunction (and)} \] \[ \lor\text{ - Logical d...
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Bayesian Networks
Bayesian Networks Jake 27/09/2022 Bayesian Networks are a modelling tool to specify joint probability distributions Can be used to decrease the computational complexity of a joint probability distribution through independence modelling. Graphs Bayesian Networks employ directed acyclic graphs (DAG) Nodes represent variables Edges can be use...
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Probability Calculus
Probability_Calculus Jake 24/09/2022 Worlds and Degrees of Belief We can assign a ‘degree of belief’ (probability) to each world The probability of a sentence, \(\alpha\), is the sum of probabilities where \(\alpha\) is true. \[ P(\alpha)=\sum_{w_i\models\alpha}P(w_i)\] Can use this idea to create joint probability distribution tables ...
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BayesNet Classification
Classification_BayesNet Jake 05/10/2022 Classification with Bayesian Networks We can divide a network variable set into class (query) variables, \(C\), and attributes (evidence variables), \(\mathbf{e}\). Attributes are typically all variables except the class variables. Given a set of evidence variables, we want to know the most likely clas...
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Variable Elimination
Variable Elimination Jake 06/10/2022 Variable elimination successively removes variables from a Bayesian Network while maintaining its ability to answer queries of interest MPE,MAP, probability of evidence, prior and posterior marginals Main insight is that we can use variable elimination to sum out variables without constructing joint probab...
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bayes_inf_intro
Intro_Bayesian_Inference Jake 01/10/2022 Bayes Theorem \[ P(A|B) = \frac{P(A,B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)}\] Bayes Theorem and Distributions \[ P(\theta|x) = \frac{P(x|\theta)P(\theta)}{P(x)}\] \[ \pi(\theta|x) = \frac{L(x|\theta)\pi(\theta)}{\pi(x)}\propto L(x|\theta)\pi(\theta)\] Definitions Note our main idea is to define parameters ...
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Further_Priors
Further Priors Jake 01/10/2022 Conjugate Priors Conjugate priors permit the posterior distribution to stay in the same family as the prior. Same distribution but altered parameters Only easy case for conjugate pairs is the exponential family: \[ f(x|\theta) = h(x)g(\theta)\exp(t(x)c(x)) \] \[ \pi(\theta | x)\propto \pi(\theta)g(\theta)^n\ex...
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Multivariate_Bayes
Multivariate Bayesian Models Jake 01/10/2022 Mixture Priors We may require more flexibility in our priors Mixture of distributions can add flexibility Particularly useful to mix conjugate distributions Consider conjugate priors \(\pi_1(\theta),...,\pi_k(\theta)\), leading to posteriors \(\pi_1(\theta),...,\pi_k(\theta|x)\) We consider the m...
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Loss Functions and Asymptotics
Loss Functions and Asymptotics Jake 13/10/2022 Loss Functions For a decision \(d\in\mathcal{D}\), a loss function defines the penalty of decision \(d\): \[ L(\theta,d)\] We want to minimise the loss function \(d^*=arg\min_dL(\theta,d)\), however we consider \(\theta\sim\pi(\theta|x)\), therefore we want: \[ d^*=arg\min_d\mathbb{E}_\pi[L(\the...
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