Overview of Machine Learning introducing the field and some of its - - PowerPoint PPT Presentation
Overview of Machine Learning introducing the field and some of its key concepts Thomas Sch on Division of Systems and Control Department of Information Technology Uppsala University. Email: thomas.schon@it.uu.se, www:
Overview of Machine Learning
“introducing the field and some of its key concepts” Thomas Sch¨
Division of Systems and Control Department of Information Technology Uppsala University. Email: thomas.schon@it.uu.se, www: user.it.uu.se/~thosc112
Overview of Machine Learning, Autonomous systems, WASP PhD course Thomas Sch¨
What is machine learning all about? ”Machine learning is about learning, reasoning and acting based on data.”
“It is one of today’s most rapidly growing technical fields, lying at the intersection of computer science and statistics, and at the core
Ghahramani, Z. Probabilistic machine learning and artificial intelligence. Nature 521:452-459, 2015. Jordan, M. I. and Mitchell, T. M. Machine Learning: Trends, perspectives and prospects. Science, 349(6245):255-260, 2015. 1 / 48 Overview of Machine Learning, Autonomous systems, WASP PhD course Thomas Sch¨
A probabilistic approach
Machine learning is about methods allowing computers/machines automatically make use of data to solve tasks. Data on its own is typically useless, it is only when we can extract knowledge from the data that it becomes useful. Representation of the data: A model with unknown (a.k.a. latent
Key concept: Uncertainty. Key ingredient: Data. Probability theory and statistics provide the theory and practice that is needed for representing and manipulating uncertainty about data, models and predictions. Learn the unknown variables from the data.
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The data – model relationship
The first step in the extraction of knowledge from data often amounts to finding the unknown parameters in the model using the data that we have available. To do this the learning system needs links between the latent and the observed data. The links are made via assumptions and taken together these assumptions constitute the model. A mathematical model is a compact representation (set of assumptions) of the data that in precise mathematical form captures the key properties of the underlying system.
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Mathematical models in machine learning
To enable reasoning about uncertainty we make use of extensive use of statistics and probability theory in building our models. We often work with very flexible models and methods, such as for example Gaussian processes, neural networks (deep learning). Simpler models, like the linear regression remains of key
components within model complex models. “All models are wrong but some are useful.” Uncertainty plays a fundamental role since any reasonable model will be uncertain when making predictions of unobserved data.
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Mathematical models – representations
The performance of an algorithms typically depends on which representation that is used for the data. Learned representations often provide better solutions than hand-designed representations. When solving a problem – start by thinking about which model/representation to use!
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Representation learning
Input Hand- designed program Output Input Hand- designed features Mapping from features Output Input Features Mapping from features Output Input Simple features Mapping from features Output Additional layers of more abstract features Rule-based systems Classic machine learning Representation learning Deep learning
Figure 1.5
From http://www.deeplearningbook.org/
Problem: How can we learn good representations of data?
the problem by introducing representations that are expressed in terms of other, simpler representations. International Conference on Learning Representations http://www.iclr.cc/
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The two basic rules from probability theory
Let x and y be continuous random variables. Let p(·) denote a general probability density function.
p(x) =
p(x, y) = p(x | y)p(y). Combine them into Bayes’ rule: p(y | x) = p(x | y)p(y) p(x) = p(x | y)p(y)
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Key objects – Learning a model
D - measured data. z - unknown model variables. The full probabilistic model is given by p(D, z) = p(D | z)
data distribution
p(z)
Inference amounts to computing the posterior distribution p(z | D) =
data distribution
p(D | z)
prior
p(D)
model evidence
Soon we will make this much more concrete.
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The model – inference relationship
The problem of inferring (estimating) a model based on data leads to computational challenges, both
e.g. the HD integrals arising during marg. (averaging over all possible parameter values z): p(D) =
e.g. when extracting point estimates, for example by maximizing the posterior or the likelihood
z
p(D | z) Typically impossible to compute exactly, use approximate methods
sequential MC (SMC).
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Example of 3 of the 4 cornerstones
The three cornerstones: 1. Data, 2. Model, and 3. Inference. Aim: Compute the position and
segments of a person moving around indoors (motion capture). Sensors (data) used:
A situation where we need to find latent variables based on
We need a model to extract knowledge from the observed data.
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Example data-model-inference
Illustrate the use of three different models:
Add ultrawideband (UWB) measurements for absolute position.
Manon Kok, Jeroen D. Hol and Thomas B. Sch¨
capture using inertial sensors. In Proceedings of the 19th World Congress of the International Federation of Automatic Control (IFAC), Cape Town, South Africa, August 2014. Manon Kok, Jeroen D. Hol and Thomas B. Sch¨
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Example – ambient magnetic field map
The Earth’s magnetic field sets a background for the ambient magnetic field. Deviations make the field vary from point to point. Aim: Build a map (i.e., a model) of the magnetic environment based on measurements from magnetometers. Solution: Customized Gaussian process that obeys Maxwell’s equations.
www.youtube.com/watch?v=enlMiUqPVJo
Arno Solin, Manon Kok, Niklas Wahlstr¨
arkk¨
the ambient magnetic field by Gaussian processes. arXiv:1509.04634, 2015. 12 / 48 Overview of Machine Learning, Autonomous systems, WASP PhD course Thomas Sch¨
Example – waveNet
A generative model capable to reading a written text using an artificial voice, beating all existing techniques. Using enough samples of a persons voice this can be used to synthesize new a written text using this particular voice. Application example: Audiobooks?
https://deepmind.com/blog/wavenet-generative-model-raw-audio/
van den Oord, A., Dieleman, S., Zen, H., Simonyan, K., Vinyals, O., Graves, A., Kalchbrenner, N., Senior, A. and Kavukcuoglu, K. WaveNet: a generative model for raw audio. arXiv:1609.03499v2, September, 2016. 13 / 48 Overview of Machine Learning, Autonomous systems, WASP PhD course Thomas Sch¨
The nature of Machine Learning
It is sometimes (often...) easier to solve a problem by starting from examples of input-output data than trying to manually program it. In ML we start from the data. We need models that are flexible enough to capture the properties of the data that are necessary to achieve a certain task. There are basically two ways of building flexible models:
compared with the data set. (parametric, ex. deep learning)
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The scientific field of Machine Learning
There are many related terms, e.g. Pattern Recognition, Statistical Modelling, Data Mining, Adaptive Control, Data Analytics, Data Science, Artificial Intelligence, and Machine Learning. Learning is clearly multidisciplinary, view from different fields:
Signal processing, system identification, adaptive and optimal control, computer vision/image processing, information theory, robotics, . . .
Artificial Intelligence, information retrieval, . . .
Learning theory, data mining, learning and inference from data, . . .
perception, mathematical psychology, computational linguistics, . . .
Decision theory, game theory, operational research, . . .
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Different scientific fields, same mathematics
Machine learning develops methods allowing computers to improve their performance at certain tasks based on observed data. Find and understand hidden structures and regularities in data.
Provides you with a timely and highly sought after skill-set. The importance of these skills is very likely to increase in the future.
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Field of machine learning
Top conferences on general machine learning
International Conference on Machine Learning (ICML)
(AISTATS) and Uncertainty in Artificial Intelligence (UAI) Top journals on general machine learning
For new (and non-peer reviewed) material see arXiv.org arxiv.org/list/stat.ML/recent
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My assignment
Aim of these lectures: To give an introduction to statistical Machine Learning (SML) by:
I will also make use of our own and others research in the area to exemplify the concepts that I am introducing.
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Outline
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Key building block – probability distributions
models.
exponential, Laplace, Wishart, Gamma, Student-t and inverse-Wishart, just to mention a few.
Given the computational tools we have today it is often rewarding to resist the linear Gaussian convenience!!
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Probabilistic linear regression
Linear regression models the relationship between a continuous
yn = θ1xn,1 + θ2xn,2 + · · · + θdxn,d + εn, εn ∼ N(0, β−1) = θTxn
f(xn,θ)
+
noise
n = 1, . . . , N. We have N input-output pairs available. Model for n = 1, . . . , N yn = θTxn + εn, εn ∼ N(0, β−1), where β is a known constant. Let the input xn be modeled as a know deterministic variable.
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Probabilistic linear regression
Recall that the full probabilistic model is given by p(Y , θ) = p(Y | θ)
data distribution
p(θ)
where Y = (y1, y2, . . . , yN)T. p(Y | θ) =
N
p(yn | θ) =
N
N(yn | θTxn, β−1). For simplicity we let the prior be p(θ) = N(θ | 0, α−1Id). Hence, for this example, the full probabilistic model is p(Y , θ) =
N
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Probabilistic linear regression
The posterior distribution is now given by p(θ | Y ) = p(Y | θ)p(θ) p(Y ) = · · · = N(θ | mN, SN), where mN = βSNXTY, SN = (αId + βXTX)−1, Y = y1 y2 . . . yN X = xT
1
xT
2
. . . xT
N
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Probabilistic linear reg. – example (I/VI)
Consider the problem of fitting a straight line to noisy measurements. Let the model be (yn ∈ R, xn ∈ R) yn = θ0 + θ1xn
+εn, n = 1, . . . , N. where εn ∼ N(0, 0.22), β = 1 0.22 = 25. The example lives in two dimensions, allowing us to plot the distributions in illustrating the inference.
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Probabilistic linear reg. – example (II/VI)
Let the true values for θ be θ⋆ =
0.5 T , plotted using a filled white circle below. Generate synthetic measurements by yn = θ⋆
0 + θ⋆ 1xn + εn,
εn ∼ N(0, 0.22), where xn ∼ U(−1, 1). Furthermore, let the prior be p(θ) = N
where α = 2.
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Probabilistic linear reg. – example (III/VI)
Plot of the situation before any data arrives. Prior, p(θ) = N
2I
−0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 x y
Example of a few realizations from the prior.
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Probabilistic linear reg. – example (IV/VI)
Plot of the situation after one measurement has arrived.
−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 x yData distribution (plotted as a function of θ) p(y1 | θ) = N(y1 | θ0 + θ1x1, β−1) Posterior/prior, p(θ | y1) = N (θ | m1, S1) , m1 = βS1XTY, S1 = (αI + βXTX)−1. Example of a few realizations from the posterior and the first measurement (black circle).
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Probabilistic linear reg. – example (V/VI)
Plot of the situation after two measurements have arrived.
−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 x yData distribution (plotted as a function of θ) p(y2 | θ) = N(y2 | θ0 + θ1x2, β−1) Posterior/prior, p(θ | Y ) = N (θ | m2, S2) , m2 = βS2XTY, S2 = (αI + βXTX)−1. Example of a few realizations from the posterior and the measurements (black circles).
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Probabilistic linear reg. – example (VI/VI)
Plot of the situation after 30 measurements have arrived.
−1 −0.5 0.5 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 x yData distribution (plotted as a function of θ) p(y30 | θ) = N(y30 | θ0 + θ1x30, β−1) Posterior/prior, p(θ | Y ) = N (θ | m30, S30) m30 = βS30XTY, S30 = (αI + βXTX)−1. Example of a few realizations from the posterior and the measurements (black circles).
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Kernelized linear regression
We can predict the output y⋆ for a previously unseen input x⋆ (a “test” input) according to ˆ y⋆ = f(x⋆, mN) = mT
Nx⋆,
which we can rewrite ˆ y⋆ = xT
⋆ mN = βxT ⋆ SNXTY = N
βxT
⋆ SNxn
yn. Hence, the predictive mean can be written ˆ y⋆ =
N
k(x⋆, xn)yn where k(x, x′) = βxTSNx′ = ψ(x)Tψ(x′) (with ψ(x) = β1/2S1/2
N x) is called the equivalent kernel.
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Kernelized linear regression
The exercise on the previous slide suggests an alternative approach to regression, where we instead of postulating a parametric model y = θTx + ε directly make use of a kernel. Provides a non-parametric alternative to linear regression. General property of kernels k(x, x′) = ψ(x)Tψ(x′), (an inner product of the input). Kernel trick: In any algorithm where the input data enters only in the form of an inner product, we can replace this inner product with any kernel! The Gaussian process is one construction that provides a non- parametric alternative to regression via the direct use of a kernel.
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Learning the hyperparameters from data
yn = θTxn + εn, εn ∼ N(0, β−1), θ ∼ N(0, α−1Id). Important question: How do we decide on the suitable values for the hyperparameters η = (α, β)? Pragmatic idea: Estimate the hyperparameters η from the data by selecting them such that they maximize the marginal likelihood function, p(Y | η) =
Travels under many names and besides empirical Bayes this is also referred to as type 2 maximum likelihood, generalized maximum likelihood, and evidence approximation.
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Summary – Probabilistic linear regression
Linear regression models the relationship between a continuous
yn = θTxn + εn, εn ∼ N(0, β−1), θ ∼ N(0, α−1I). We derived the expression for the posterior distribution p(θ | Y ) = p(Y | θ)p(θ) p(Y ) Kernelized linear regression via the use of the kernel trick. Showed how to estimate the hyperparameters from data. We can show that the MAP point estimate with the data distribution (likelihood) p(Y | θ) ∝ N
n=1(yn − θTxn)2 together
with a Gaussian prior leads to ridge regression and with a Laplacian prior it leads to the LASSO.
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Probabilistic mod. – Want to know more?
Good introduction
Ghahramani, Z. Probabilistic machine learning and artificial intelligence. Nature 521:452-459, 2015.
A clear tutorial of probabilistic modelling
Bishop, C. M. Model-based machine learning. Philosophical Transactions of the Royal Society A, 371, 20120222 (2013).
Textbook style introductions
Bishop, C. M. Pattern Recognition and Machine Learning, Springer, 2006. Murphy, K. P. Machine learning – a probabilistic perspective, MIT Press, 2012. Koller, D. and Friedman, N. Probabilistic Graphical Models: Principles and Techniques. MIT Press, 2009.
Lecture on Probabilistic modelling
https://www.microsoft.com/en-us/research/video/ posner-lecture-probabilistic-machine-learning-foundations-and-frontiers/ 34 / 48 Overview of Machine Learning, Autonomous systems, WASP PhD course Thomas Sch¨
Outline
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Motivating the name deep learning
Let the computer learn from experience and understand the situation in terms of a hierarchy of concepts, where each concepts is defined in terms of its relation to simpler concepts. If we draw a graph showing these concepts of top of each other, the graph is deep, hence the name deep learning. Key aspect: We avoid the need for a human to formally specify all the knowledge that the computer needs. Deep learning achieves a highly flexible model by representing the world in terms of a sequential hierarchy of concepts, where more abstract representations are computed in terms of less abstract ones.
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Why deep? - Image classification example
Image classification (Input: pixels of an image. Output: object identity.) 1 megapixel (B/W) → 21 000 000 possible images! A deep neural network can solve this with a few million parameters! Each hidden layer extracts increasingly abstract features.
Zeiler, M. D. and Fergus, R. Visualizing and understanding convolutional networks. In European Conference on Computer Vision (ECCV), Z¨ urich, Switzerland, September, 2014. 37 / 48 Overview of Machine Learning, Autonomous systems, WASP PhD course Thomas Sch¨
An example – deep autoencoder
Unsupervised learning procedure for dimensionality reduction. Notation:
yk,1 yk,2 yk,3 yk,n zk,1 zk,m ˆ yk,1 ˆ yk,2 ˆ yk,3 ˆ yk,n Input layer Hidden layer ”bottleneck” Output layer
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An example – deep autoencoder
Unsupervised learning procedure for dimensionality reduction. Notation:
yk,1 yk,2 yk,3 yk,n zk,1 zk,m ˆ yk,1 ˆ yk,2 ˆ yk,3 ˆ yk,n Input layer Hidden layer ”bottleneck” Output layer
Model components:
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An example – deep autoencoder
Unsupervised learning procedure for dimensionality reduction. Notation:
yk,1 yk,2 yk,3 yk,n zk,1 zk,m ˆ yk,1 ˆ yk,2 ˆ yk,3 ˆ yk,n Input layer Hidden layer ”bottleneck” Output layer
Model components:
yR
k = g(zk; θD)
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An example – deep autoencoder
Unsupervised learning procedure for dimensionality reduction. Notation:
yk,1 yk,2 yk,3 yk,n zk,1 zk,m ˆ yk,1 ˆ yk,2 ˆ yk,3 ˆ yk,n Input layer Hidden layer ”bottleneck” Output layer
Model components:
yR
k = g(zk; θD)
Reconstruction error: VR(θE, θD) = N
k=1 yk−
yR
k (θE, θD)2
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Constructing an NN for regression
A neural network (NN) is a hierarchical nonlinear function y = gθ(x) from an input variable x to an output variable y parameterized by θ. Linear regression models the relationship between a continuous
y =
n
xiθi + θ0 + ε = xTθ + ε, where θ is the parameters composed by the “weights” θi and the
θ =
θ1 θ2 · · · θn T , x =
x1 x2 · · · xn T .
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Generalized linear regression
We can generalize this by introducing nonlinear transformations of the predictor uTθ, y = f(uTθ). Let us consider an example of a feed-forward NN, indicating that the information flows from the input to the output layer.
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NN for regression – an example
a(1)
j
=
n
θ(1)
ji xi + θ(1) j0 ,
j = 1, . . . , m1.
zj = f
j
j = 1, . . . , m1.
yk =
my
θ(2)
kj zj + θ(2) k0 ,
k = 1, . . . , my.
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NN for regression – an example
ˆ yk(θ) =
m1
θ(2)
kj f
n
θ(1)
ji xi + θ(1) j0
k0
. . . f f f . . .f f . . . x1 x2 xn z1 ˆ y1 ˆ ymy θ(1)
11
θ(2)
11
Inputs Hidden layer Output layer
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Deep neural networks
We can think of the neural network as a sequential construction of several generalized linear regressions. Each layer in a multi-layer NN is modelled as z(l+1) = f
starting with the input z(0) = x. (The nonlinearity operates element-wise.) The scalar nonlinear function f(·) is what makes the neural network nonlinear. Common functions are f(z) = 1/(1 + e−z), f(z) = tanh(z) and f(z) = max(0, z). The so-called rectified linear unit (ReLU) f(z) = max(0, z) is heavily used for deep architectures.
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Deep neural networks
Deep learning methods allow a machine to make use of raw data to automatically discover the representations (abstractions) that are necessary to solve a particular task. It is accomplished by using multiple levels of representation. Each level transforms the representation at the previous level into a new and more abstract representation, z(l+1) = f
starting from the input (raw data) z(0) = x. Key aspect: The layers are not designed by human engineers, they are generated from (typically lots of) data using a learning procedure and lots of computations.
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Training an NN
The final layer z(L) of the network is used for making a prediction ˆ y(θ) = z(L) and we train the network by employing:
y(θ), y) and a regularizer J(θ).
V (θ) =
N
L (ˆ yn(θ), yn) + λJ(θ). There is great software support available, use it! For example: TensorFlow (and playground.tensorflow.org), Theano, Torch.
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Some comments - Why now?
Neural networks have been around for more than fifty years. Why have they become so popular now (again)? To solve really interesting problems you need:
These three factors have not been fulfilled to a satisfactory level until the last 5-10 years.
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Summary – Deep learning
A neural network (NN) is a hierarchical nonlinear function y = gθ(x) from an input variable x to an output variable y parameterized by θ. We can think of an NN as a sequential construction of several generalized linear regressions. Deep learning refers to learning NNs with several hidden layers. Allows for data-driven models that automatically learn representations (features) of data with multiple layers of abstraction. A deep NN is very parameter efficient when modelling high-dimensional, complex data.
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Deep learning – Want to know more?
Good introduction
LeCun, Y., Bengio, Y., and Hinton, G. (2015) Deep learning, Nature, 521(7553), 436–444.
Timely introduction (target audience include: software engineers who do not have a machine learning or statistics background)
http://www.deeplearningbook.org
Interesting discussion about why it works so well
Lin, H. W. and Tegmark, M. (2016) Why does deep and cheap learning work so well?, arXiv
Deep learning summer school 2016
https://sites.google.com/site/deeplearningsummerschool2016/
Geoffrey Hinton’s Coursera course
https://www.coursera.org/learn/neural-networks/home/welcome
NIPS and ICML conferences and workshops!
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Appendix
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Probabilistic linear regression – summary
We have N input-output pairs available. Model for n = 1, . . . , N yn = θTxn + εn, εn ∼ N(0, β−1), θ ∼ N(0, α−1Id). We have shown that the posterior distribution is p(θ | Y ) = N(θ | mN, SN), where mN = βSNXTY, SN = (αId + βXTX)−1, Y = y1 y2 . . . yN X = xT
1
xT
2
. . . xT
N
What is the maximum aposteriori (MAP) point estimate? ˆ θMAP = arg max
θ
p(θ | Y )
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On the relationship to maximum likelihood
The MAP solution is given by ˆ θMAP = β(αI + βXTX)−1XTY. (1) Do you recognize this solution? What if α = 0? This is the maximum likelihood solution, i.e. the solution to ˆ θML = arg min
θ
||Y − Xθ||2
2
Keeping α = 0 we can show that the solution to ˆ θRR = arg min
θ
||Y − Xθ||2
2 + α
β ||θ||2
2
is given by (1). Commonly referred to as ridge regression. Hence, ridge regression is equivalent to computing the MAP estimate using a Gaussian prior for θ.
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On the relationship to LASSO
Analogously we can show an equivalence between the use of a Laplacian prior and ℓ1-norm regularization, ˆ θLASSO = arg min
θ
||Y − Xθ||2
2 + λ||θ||1
Commonly referred to as the LASSO. Ridge regression and the LASSO are two examples of regularized (or penalized) maximum likelihood.
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Probabilistic modeling of linear SSMs
A linear Gaussian state space model (SSM) consists of a Markov process {xt}t≥1 that is indirectly observed via a linear, Gaussian measurement process {yt}t≥1, xt+1 = Axt + But + vt, vt ∼ N(0, Q) yt = Cxt + Dut + et, et ∼ N(0, R) x1 ∼ µη(x1), θ ∼ π(θ), where θ = {A, B, C, D, Q, R, η}. The full probabilistic model is given by p(x1:T , θ, y1:T ) = p(y1:T | x1:T , θ)
p(x1:T , θ)
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