Write solutions to my 18 questions of Data Science & Statistics assignment (a few questions requires Python)
(Find the complete assignment in the attachment below)
Bayesian Regression/markov chain
1 GENERAL REGRESSION PROBLEM
In the Bayesian viewpoint, we formulate linear regression using
probability distributions rather than point estimates. The response, y,
is not estimated as a single value, but is assumed to be drawn from a
probability distribution. The model for Bayesian Linear Regression
with the response sampled from a normal distribution is:
The output, y is generated from a normal (Gaussian) Distribution characterized by a mean and
variance. The mean for linear regression is the transpose of the weight matrix multiplied by the
predictor matrix. The variance is the square of the standard deviation σ (multiplied by the
Identity matrix because this is a multi-dimensional formulation of the model).
The aim of Bayesian Linear Regression is not to find the single “best” value of the model
parameters, but rather to determine the posterior distribution for the model parameters. Not only
is the response generated from a probability distribution, but the model parameters are
assumed to come from a distribution as well. The posterior probability of the model parameters
is conditional upon the training inputs and outputs:
Here, P(β|y, X) is the posterior probability distribution of the model parameters given the inputs
and outputs. This is equal to the likelihood of the data, P(y|β, X), multiplied by the prior
probability of the parameters and divided by a normalization constant. This is a simple
expression of Bayes Theorem, the fundamental underpinning of Bayesian Inference:
2. COMPUTE THE PO STERIOR OF W
1. Write the likelihood of w.
When we try to find how likely is for an output y to belong to a model defined by data X, weights
w and model parameters σ (if any), or maximize the likelihood p(y∣w,X,σ2), we perform a
Maximum Likelihood Estimator (MLE). Maximizing the likelihood means maximizing the
probability that models the training data, given the model parameters, as:
Let us interpret what the probability density p(x∣θ) is modeling for a fixed value of θ. It is a
distribution that models the uncertainty of the data for a given parameter setting. In a
complementary view, if we consider the data to be fixed (because it has been observed), and
we vary the parameters θ, what does the MLE tell us? It tells us how likely a particular setting of
θ is for the observations x. Based on this second view, the maximum likelihood estimator gives
us the most likely parameter θ for the set of data.
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