Write solution of Data Science & Statistics assignment using Python

Write solutions to my 18 questions of Data Science & Statistics assignment (a few questions requires Python)

(Find the complete assignment in the attachment below)

For EXAMPLE:

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:

MLE=argmaxw p(y∣w,X)(2)

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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