Forms of uncertainty:

Epistemic uncertainty - translates to 'knowledge' in Greek, is the uncertainty derived from lack of knowledge of information which can be reduced with further data. Aleatoric uncertainty - comes from the word "alea" which means "the roll of the dice" and is defined as internal, unexplainable randomness

Heat equation https://tutorial.math.lamar.edu/classes/de/theheatequation.aspx

Bibliography:

https://homepages.inf.ed.ac.uk/rbf/HIPR2/gsmooth.htm

https://pdfs.semanticscholar.org/ad19/0bc250bb16202a032a5a2ba50fd1085c3c79.pdf

https://stackoverflow.com/questions/28754252/smoothing-a-noisy-image-then-sharpening

Carl Rasmussen GPs http://gaussianprocess.org/gpml/chapters/RW.pdf "There are several ways to interpret Gaussian Process regression models:

1. Weight space
2. Function space - a probability density directly over the function output space.

Predictive distribution, P(y*|x*, y, x) = \int P(y*|f,x*) P(f|y, x) df)

A Gaussian Process (prior) is a prior distribution on some unknown function, \mu(x).

Kriging

Kernels are can be added together to capture multiple factors.

Fantastic introduction resource video on GPs: https://www.youtube.com/watch?v=UBDgSHPxVME

https://towardsdatascience.com/gaussian-process-regression-from-first-principles-833f4aa5f842 https://www.cs.cmu.edu/~epxing/Class/10708-20/scribe/lec21_scribe.pdf

Gaussian Processes have propertry where by as you move away from data uncertainty decreases. Such increase in uncertainty propagates both along the x-axis and the y-axis. Bibliography:

http://cs229.stanford.edu/section/cs229-gaussian_processes.pdf

Bayesian optimisation was pivotal in the implementation of AlphaGo achieving its success (paper on Bayesian optimisation in AlphaGo).

Allows for non-linear trends to be fit. I.e. impact of watering lawn via both sprinklers and rain. Don't do one and the lawn dies, do one the lawn lives and do both the lawn dies. Another, example is that of the XOR gate which can be modelled using an interaction effect.

An interaction effect is modelled by combining parameter multiplicatively: Y = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2

Kalman filter demystified: from intuition to probabilistic graphical model to real case in financial markets Kalman filter video

1. Laplace's demon, full determinism in physical world is not possible.
2. Machine learning, data + model = prediction. Too much of ML has had pure focus on prediction more work needs to be causal ML to endeavour to understand system = more transferability.
3. Analogy of Stochastic Vs Determinism, likened to hurricane Vs Navier-Stokes equation. Be mindful of your model’s granularity.
4. Physical Vs Simulation Vs emulations. UK's BOM uses too much low level physics, not enough high level emulations /ML.
5. OR and Machine Learning hasn't come together as of yet. There’s a lot of low hanging fruit to be had at the intersection of these fields.
6. Separation of concern architecture – again has focused on prediction at the cost of explanability, needs to incorporate causal lens.

Video: https://youtu.be/FuJgGeKMIJM

Lecture series: https://mlatcl.github.io/mlphysical/