quick note: here we took the same prior for ${}_1$ and ${}_2.$ One can consider different priors. For

example, if we believe that sometime around week $30$ the pattern changed, then we can define

different values of $$ considering separate averages for $,$ one for $W<mn>3</mn><mn>0</mn>$ and the other for

$W30.$ But, be aware that this will have a minimal effect on our final conclusion.

Our model has $3$ parameters, ${}_1$;${}_2$ and $W_s.$ We are going to find the posterior of these three

parameters to infer that how much we believe in the change in Bob's tweeting pattern and when

is the most likely week for this change?

e$)$ Use numpy.linspace to create two variables with $150$ array in the interval of $25$ and $50.$ These

variables are defined as the model space that we want to search for posterior of ${}_1$ and ${}_2.$ You

have also defined an array in part a which shows the week number. This provides a space for $W_s.$

Consider all these points as a $3-$D mesh$-$grid, how can we find posterior for each point in the $3-$D

space given all the information you have? Use Bayes' theorem to elaborate your method in detail.

f$)$ Write a code to find marginalized$-$posterior for ${}_1,{}_2$ and $W_s.$ Plot posteriors for ${}_1$ and ${}_2$ in

the same figure and create a bar plot for posterior of $W_s$ in a separate figure. Running your code

for this part may take a long time since you compute posterior for every single point in your

model space. However, we will learn more efficient way, Markov chain Monte Carlo $($MCMC$),$

later in the next homework.

g$)$ How is your belief updated about a sudden change in Bob's tweeting habit? Can you estimate

the week when tweeting pattern changed? Use marginalized $2-$D posteriors of ${}_1$ and ${}_2$ to obtain

$P({}_2-{}_1>5).$ This shows the probability that Bob's weekly tweet counts has increased by five

at some point.