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--- data_mining:neural_network:model_combination [2017/04/01 12:45] – phreazer
+++ data_mining:neural_network:model_combination [2017/08/19 22:12] (current) – [Approximating full Bayesian learning in a NN] phreazer
@@ Line 22: / Line 22: @@
 ====== Mixtures of Experts ======
+Suitable for large data sets.
+Look at input data, to decide on which model to rely.
+=> Specialization
+Spectrum of models
+Local models like nearest neighbors
+Global models like polynomyal regression (each parameter depends on all the data)
+Use models of intermediate complexity. How to fit data to differen regimes?
+Need to cluster training cases into subsets, one for each local model. Each cluster to have a relationship between input and output that can be well-modeled by one local model.
+Error function that encourages cooperation
+* Compare the **average** of all the predictors with the target and train to **reduce discrepancy**.
+$E = (t-<y_i>_i)^2$
+Error function that encourages specialization
+Compare each predictor seperatly with the target. Manager to determine the probability of picking each expert.
+$E = (p_i t-<y_i>_i)^2$
+====== Full Bayesian Learning ======
+Instead of finding best single setting of the parameters (as in Maximum Likelihood /MAP), compute **full posterior distribution** over **all possible parameter** settings.
+Advantage: Complicated models can be used, even if only few data is available.
+If you don't have much data, you should use a simple model
+  * True, but only if you assume that fitting a model measn choosing a **single best setting** of the **parameters** (Ml learning w that maximizes $P(\text{Data}|w)$).
+If you use full posterior distribution, overfitting disappears: You will get very vague predictions, because many different parameter settings have significant posterior probability (Learn $P(w|\text{Data})$).
+Example: learn lots of polynomial (distribution), average.
+====== Approximating full Bayesian learning in a NN ======
+  * NN with few parameters. Put grid over parameter space and evaluate $p(W|D)$ at each grid-point. (xpensive, but no local optimum issues).
+  * After evaluating each grid point, we use all of them to make predictions on test data.
+    * Expensive, but works much better than ML learning, when posteriror is vague or multimodal (data is scarce).
+Monte Carlo method
+Idea: Might be good enough to sample weight vectors according to their posterior probabilities.
+$p(y_{\text{test}} | \text{input}_\text{test}, D) = \sum_i p(W_i|D) p(y_{\text{test}} | \text{input}_\text{test}, W_i)$
+Sample weight vectors $p(W_i|D)$.
+In Backpropagation, we keep moving weights in the direction that decreases the costs.
+With sampling: Add some gaussion noise to weight vector, after each update.
+Markov Chain Monte Carlo:
+If we use just the right amount of noise, and if we let thei weight vector wander around for long enough before we take a sample, we will get an ubiased sample form the true posterior over weight vectors.
+More complicated and effective methods than MCMC method: Don't need to wander the space long.
+If we compute gradient of cost function on a **random mini-batch**, we will get an unbiased estimate with sampling noise.
+====== Dropout ======
+See [[data_mining:neural_network:regularization|Regularization]]