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Question 1

In the context of bayesian regression, how do we define a linear model (for simplicity). (hint: expansion of a normal distribution that we assume the model data is generated from)

Answer 1

see photo

Question 2

How do we use bayes theorem in bayesian regression

Answer 2

Calculate the posterior distribution of weights given observed data by the distribution of data given weights (likelihood) multiplied by the prior distribution of weights divided by distribution of data p(t), but we often forgo the last one

Question 3

how do we define a gaussian prior for the parameters?

Answer 3

Define a normal distribution parameterised by the mean of the weights and a covariance matrix which just makes sure the variables are not statistically independent

Question 4

After calculating the posterior by a simplified bayesian, we get the result e^F(w,t, beta), which we know is quadratic. How do we expand a quadratic of that form generally

Answer 4

see image

Question 5

How do we find the explicit expressions for the covariance and mean (new ones) given a general expansion for the F function quadratic and the expanded version of the bayesian update

Answer 5

Look at the quadratic terms of each function and compare the like terms. I.e. which terms must correspond to one another. See image