Week-8 (11/03) Baye’s theorem.

The Bayes’ theorem is a statistical and probability theory that may be used to analyze data in Python. It allows you to adjust probability or draw conclusions about events based on fresh facts or data. Bayes’ theorem is frequently employed in statistical contexts for Bayesian inference, which is a versatile and powerful framework for statistical modeling and parameter estimation.

Bayes’ theorem can be expressed as

Where
– (P(A|B)) is the conditional probability of event A given that event B has occurred.
– (P(B|A)) is the conditional probability of event B given that event A has occurred.
– (P(A)) is the prior probability of event A, which is your initial belief in the probability of A before considering any new evidence.
– (P(B)) is the prior probability of event B, which is your initial belief in the probability of B before considering any new evidence.

In the context of Python data analysis, Bayes’ theorem can be utilized for a variety of tasks, including:
– Bayes’ theorem can be used to update your views about the parameters of a statistical model depending on observed data. This is especially handy if you know the settings ahead of time.
– Hypothesis Testing: Bayesian hypothesis testing uses the posterior probability to analyze the evidence for or against a hypothesis.
– Predictive Modeling: Bayesian approaches, such as Bayesian regression or Bayesian networks, can be utilized to develop predictive models that yield more robust predictions and uncertainty estimates.
The code generates two graphs to visualize the results of the Bayesian parameter estimation using the Metropolis-Hastings sampler.
Mean Estimation:

• The first graph depicts the estimated values of the normal distribution’s mean during the Markov Chain Monte Carlo (MCMC) sampling procedure.
• The sample number or iteration is represented on the x-axis, demonstrating how the parameter estimations change over time.
• The calculated mean values are represented on the y-axis.
• In this graph, you can see how the sampler investigates various mean values to locate the region with the highest posterior probability, which corresponds to the most likely mean value given the data.

Standard Deviation Estimation:

• The second graph depicts the estimated values of the normal distribution’s standard deviation over the course of the MCMC sampling procedure.
• The x-axis shows the sample number or iteration, while the y-axis represents the estimated standard deviation values, as in the first graph.
• This graph depicts how the sampler investigates various standard deviation values in order to locate the region with the highest posterior probability, showing the most likely value of the standard deviation given the data.

  These graphs depict the parameter estimation process as well as the MCMC sampler’s convergence. The sampled values should stabilize as the number of samples rises, producing more accurate estimations of the parameters.
  This is done using the dummy data for learning purposes
  I’ll implement this in my project