From c05e99854780bfb596407380b955bc042b12c66a Mon Sep 17 00:00:00 2001 From: Robert Saalbach Date: Tue, 28 Oct 2025 23:52:11 +0100 Subject: [PATCH] more notes for ml --- .obsidian/workspace.json | 36 +++--- Pasted image 20251027151936.png | Bin 0 -> 35234 bytes Pasted image 20251027152814.png | Bin 0 -> 57647 bytes Pasted image 20251027154610.png | Bin 0 -> 33909 bytes Pasted image 20251027155909.png | Bin 0 -> 129577 bytes Pasted image 20251027155945.png | Bin 0 -> 102862 bytes Pasted image 20251027160014.png | Bin 0 -> 112304 bytes Pasted image 20251027160532.png | Bin 0 -> 33082 bytes Pasted image 20251027160553.png | Bin 0 -> 44379 bytes Pasted image 20251027160603.png | Bin 0 -> 42292 bytes Pasted image 20251027160619.png | Bin 0 -> 55671 bytes Pasted image 20251027160630.png | Bin 0 -> 53132 bytes Pasted image 20251027173751.png | Bin 0 -> 47133 bytes Pasted image 20251027175634.png | Bin 0 -> 62583 bytes University/Machine Learning/Full Notes.md | 142 ++++++++++++++++++++++ 15 files changed, 161 insertions(+), 17 deletions(-) create mode 100644 Pasted image 20251027151936.png create mode 100644 Pasted image 20251027152814.png create mode 100644 Pasted image 20251027154610.png create mode 100644 Pasted image 20251027155909.png create mode 100644 Pasted image 20251027155945.png create mode 100644 Pasted image 20251027160014.png create mode 100644 Pasted image 20251027160532.png create mode 100644 Pasted image 20251027160553.png create mode 100644 Pasted image 20251027160603.png create mode 100644 Pasted image 20251027160619.png create mode 100644 Pasted image 20251027160630.png create mode 100644 Pasted image 20251027173751.png create mode 100644 Pasted image 20251027175634.png diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 4db4229..b831289 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -4,16 +4,16 @@ "type": "split", "children": [ { - "id": "073fe34e944da9c4", + "id": "c6dea15c43e47f34", "type": "tabs", 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zt_s;`dis->S`gUWInQFcI`lHKQ>>l1TRZCxMq1@>FaAVhn$B({ut4 z7K0p6;|Hp^7Ck)NK6|Qm_$<)n3_1}T6i#nq=a)0tUH;z8-~R8G$`=dUmDag~7E^sP zQ~e@1al?%0oBL|3rNyMr{`&gb_vxvri*?mN3!N&bfLaVML5H)cdQCB?{{HT3jcvrn zq@!m*S1@piXb6->f;@L}_7(VwF3p8thjw((W;qzhI;yTOBg(*k@>8C_nRmGKv?2oo O1B0ilpUXO@geCydO-=Ry literal 0 HcmV?d00001 diff --git a/University/Machine Learning/Full Notes.md b/University/Machine Learning/Full Notes.md index 2651bf0..9967887 100644 --- a/University/Machine Learning/Full Notes.md +++ b/University/Machine Learning/Full Notes.md @@ -240,3 +240,145 @@ i.e. $\lambda_{21}\space p(y_2|x)$ and $\lambda_{12}\space p(y_1|x)$ (missclassification error loss times the prob its in that class (posterior prob.)) + +# Parametric Densities + +For the output of a model we would find, for each object in the feature space: $p(y|x)$ +In practice we approx: $\hat{p}(y|x)$ +or we fit a function. + +Difference between $p(x)$ and $P(x)$ (first is probability density, second is probability mass, the first is continuous, the second discrete) + +In Bayes' rule how do you get $p(x)$? +You can compute it: (assuming two classes here): +$p(x)=p(x|y_1)p(y_1)+p(x|y_2)p(y_2)$ +![[Pasted image 20251027151936.png]] + +up to now we assumed we know $p(y|x)$ or $p(x|y), p(y)$ +but realistically we only get a sample - so we have to approx. + +For this, we need models of multiple categories: +- Discriminative and Generative +- Parametric and Nonparametric + +## Generative Models +$p(y|x)\propto p(y)p(x|y)$ + +When we know the prior and conditional densities we know everything about the data for classification +the density has to be estimated and given examples from different classes, 'standard' density estimation is sufficient +It is possible to 'generate' (sample) from the classes + +## Discriminative Models +$\hat{p}(y|x)$ +When we don't know the class conditional probs and prios, directly estimate posterior? +- hard problem: given measurements e.g. height, how to estimate $p(\text{woman}|\text{height})$ +- Strong assumptions or sloppy approx + +## Parametric Modeling & Estimation +Density Estimation and related topics: +- Simple Nonparametric approach +- curse of dimensionality +- parametric models +- sphering +- properties of gaussian +- mixture modeling + +## Histogram based Density Estimation +![[Pasted image 20251027152814.png]] + +the problem here though, is accuracy, you may need a lot of repetitions, like in this case. + +for 1 dimensional data apparently +- 1000 objects needed +For each bin we estimate on value: 50 bins, 50 parameters + +for $M$-dimensional data $\pm 1000^M$ objects needed. this becomes basically unworkable if $M>2$ + +This is part of the ***curse of dimensionality*** +intuitively, using more features should give us more information, and make prediction easier +But we hve to estimate the densities, and the number of parameters increases with the number of features +to estimate these well you need more objects +Consequence: There is an optimal number of features to use + +![[Pasted image 20251027154610.png]] + +> Parametric: need only a few parameters and assume a simple global model e.g. Gaussian +> Non-Parametric: depends on training data, simple local model such as uniform or Gassuian + +Normal Distribution === Gaussian Distribution +Standard normal distro: $\mu=0, \sigma^2=1$ +95% of data between $[u-2\sigma, u+2\sigma]$ (1 dimensional) + +1D formula: +$$p(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$ +p(x) is the density don't forget that + +Why gauss? +special, central limit for large i.i.d. random vars will be gaussian +approx occurs in real life +e.g. sum of eyes on 10,000 dice throws +also has few params +easy to estimate parems when using max. likelihood + +### Multivariate Gaussians +$M$-dimensional density: +$$p(x)=\frac{1}{\sqrt{(2\pi)^M\det(\Sigma)}}\exp\left(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)\right)$$ +Which is also written as (crazy) +$$N(x|\mu,\Sigma)$$ +![[Pasted image 20251027155909.png]] + +![[Pasted image 20251027155945.png]] + +![[Pasted image 20251027160014.png]] +(from top-down view) + +## Max. likelihood estimates + +What are the max. likelihood estimators for the mean and the covariance matrix? (the parameters we want to estimate) +$$ +\begin{align} +\hat{\mu}=\frac{1}{n}\sum^n_{i=1}x_i \\ +\hat{\Sigma}=\frac{1}{n}\sum^n_{i=1}(x_i-\hat{\mu})(x_i-\hat{\mu})^T +\end{align} +$$ +seeing as estimate the mean for the second forumla using the first, the second estimator is biased. + +to make it unbiased, we can make one simple change +$$ +\hat{\Sigma}=\frac{1}{n-1}\sum^n_{i=1}(x_i-\hat{\mu})(x_i-\hat{\mu})^T +$$ +#### example as on slides: +![[Pasted image 20251027160553.png]] +![[Pasted image 20251027160603.png]] +![[Pasted image 20251027160619.png]]![[Pasted image 20251027160630.png]] +(x is typically on bold on these slides, because after all it is a vector if there are multiple features, and I guess technically even if there is only one feature) + +REMIND YOURSELF: "T" operation (and matrix stuff in general such as finding inverse) +![[Pasted image 20251027173751.png]] +to solve that you would need to get some 'c' for which $0\times c = 1$, which doesn't work for obvious reasons + +Number of objects is insufficient to find the inverse, two object in a 2 dimensional feature space is a _degenerate_ Gaussian distribution + +### Parametric Estimation +Now for $M$-dimensional data: +$\mu$: is a vector with $M$ elements +$\Sigma$: is a matrix with $0.5M\space (M+1)$ elements + +number of parems increases quadratically with m, and you might still need a lot of data + +I am not sure what this means but apparently "Any projection of a high-dimensional gaussian is itself again Gaussian", so I guess reducing it to one feature? but that makes sense (or a lower feature count rather) + +### Estimating class priors +Given a training set, how can you estimate $\hat{p}(y)$? +The classes are discrete, $\hat{p}(y)$ is a true probability, and often are known or assumed. If not, we need to learn them + +Max. likelihood estimator for priors turns out to be counting: +$\hat{p}(y_1)=\frac{N_1}{N}$ and $\hat{p}(y_2)=\frac{N_2}{N}$ +(you don't need unconditional probablity for finding which is larger remember) ![[Pasted image 20251027151936.png]] + +### How to define the classifier based on the estimates: +#### two-class case +Discriminant: $f(x)=\log p(y_1|x)-\log p(y_2|x)$ +and then this wonderful jumble, which is just simply plugging everything in: +![[Pasted image 20251027175634.png]] + + -- 2.54.0