Publications by Jake
Markov Models
Markov Models Jake 31/10/2022 Dynamic Bayesian Networks Graphical models with a repeating structure that grows with time Time/space dynamic Utilise the markov property Future states are independent of past given the current state Markov Chains Consider a discrete variable \(X\) with states Transitions between states are nondeterministi...
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MPE/MAP Queries
MPE and MAP Queries Jake 02/11/2022 MPE and MAP Queries MAP: Maximum a posteriori hypothesis Given map variables \(\mathbf{M}\subseteq\mathbf{X}\) and evidence \(\mathbf{e}\), our goal is to find instantiation \(\mathbf{m}\) for which \(P(\mathbf{m|e})\) is maximised MPE: Maximum a psoteriori explanation Most probable probable variable ins...
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Markov Networks
MarkovNetworks Jake 02/11/2022 Markov Networks Markov networks are undirected graphical models Often used to model symmetric dependencies where each state can be influenced by the state of its neighbours A variable is independent of all other variables given its neighbours \[ X_1\perp X_3,X_4|X_2\] Markov networks encode independence assump...
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Bayes Factors
Bayes Factors Jake 28/11/2022 Bayesian Hypothesis Testing Considering a test statistic \(T = T(x_1,...,x_n)\) we can calculate a posterior of a hypothesis given observed data \(T\). For \(H_0\): \[ P(H_0|T) = \frac{P(T|H_0)P(H_0)}{P(T|H_0)P(H_0)+P(T|H_1)P(H_1)}\] * To avoid computing normalisation constant, can compute the posterior odds ratio...
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hierarcha;
Heirarchial Models Jake 28/11/2022 Principles Relaxes the assumption of fixed parameters of parameter priors. \[ \theta\sim N(a,b),\quad\text{where }a,b\text{ are unknown}\] Multi-level structure Exchangability of model parameters between units Modelling Dependent Parameters Three approaches to modelling a problem with data measured as \(x_...
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MCMC
MarkovChainMonteCarlo Jake 11/11/2022 Markov Chain Monte Carlo (MCMC) Given a distribution \(\pi(X), X\in E\) known up to a normalisation constant. Construct Markov Chain with state space \(E\) and stationary distribution \(\pi(X)\) Simulate Markov Chain sample paths until convergence Once converged, chain iterations can be treated as a sample ...
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Mixture Models
Mixture_Models Jake 29/11/2022 Mixture Models Mixture models naturally arise when measurements of individuals within a population can be considered to arise from different distributions For example, male and female For \(y = (y_1,...,y_m)\), the \(M\) component mixture distribution is: \(f_m(y_i|\theta_m)\) is the distribution of \(y_i\) f...
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jointree
Jointree Jake 06/12/2022 Jointree Jointree is a variation of variable elimination that utilises factor elimination rather than variable elimination Jointree algorithm is particularly useful when we want to compute posterior marginals for each of \(n\) variables Jointree has a complexity of \(O(n\exp(w))\) while running VE \(n\) times would lea...
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Graph Decomposition
Graph Decomposition Jake 06/12/2022 Node Elimination Elimination order for a graph \(G\) is the total ordering \(\pi\) of nodes of \(G\), where \(\pi(i)\) is the \(i\)th node in the ordering The result of eliminating node \(X\) from graph \(G\) is another graph obtained from \(G\) by: Adding an edge between every pair of non-adjacent neighbor...
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Gaussian Models
Gaussian Models Jake 06/12/2022 Graphical Models with Continuous Variables Everything from discrete probabilities holds for continuous distributions However cannot use tables, have to use probability distribution functions Gaussian Bayesian Networks All variables are continuous and modelled by Gaussian Densities Often good approximation fo...
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