How can you update a prior distribution in Bayesian statistics?

2 What is Bayesian updating?

Bayesian updating is the process of revising your prior distribution based on new data or evidence, using a mathematical formula called Bayes' theorem. Bayes' theorem states that the posterior distribution, which is the updated distribution after observing the data, is proportional to the product of the prior distribution and the likelihood function, which is the probability of the data given the parameter or hypothesis. In other words, Bayesian updating combines your prior beliefs and the data to produce a new distribution that reflects your updated beliefs.

3 How to perform Bayesian updating?

To perform Bayesian updating, you must start by selecting a prior distribution that reflects your initial beliefs or assumptions about the parameter or hypothesis of interest. This could be based on historical data, expert opinions, or theoretical considerations. Alternatively, you can opt for a non-informative prior that assigns equal probabilities to all possible values. Then, you need to collect new data or evidence that is relevant to the parameter or hypothesis of interest. This should be reliable, representative, and independent data. After that, calculate the likelihood function, which is the probability of the data given the parameter or hypothesis of interest. Then, use Bayes' theorem to update your prior distribution with the likelihood function and obtain the posterior distribution. You can do this analytically or numerically. Finally, interpret and communicate the results of your posterior distribution by summarizing its mean, median, mode, variance, confidence intervals, or credible intervals. You can also use graphical tools to visualize it and its uncertainty.

4 What are some examples of Bayesian updating?

Bayesian updating can be applied to a variety of R&D problems and scenarios, such as testing the performance of a new product or service, optimizing a design or process, and estimating a market or demand. For example, you can use a beta prior and a binomial likelihood to update your beliefs about the proportion of satisfied customers after launching an app. Additionally, you can use a normal prior and a normal likelihood to update your beliefs about the optimal parameter value for a machine or system after testing different settings. Lastly, you can use a Poisson prior and a Poisson likelihood to update your beliefs about the average number of sales or orders for a product or service after observing the actual sales or orders. By using Bayesian updating, you can gain valuable insights into the performance, quality, customer satisfaction, optimal parameter values, and market demand.

5 What are some tips for Bayesian updating?

Bayesian updating is a powerful tool for R&D, but it also requires careful considerations and best practices. It is important to select an appropriate and reasonable prior distribution, collect enough data or evidence that is reliable and representative, use a model or assumption that is suitable and realistic, apply Bayes' theorem correctly and efficiently, and interpret and communicate the results of the posterior distribution clearly and effectively. Additionally, you should perform a sensitivity analysis to check how different priors affect the posterior distribution, a data quality assessment to check the validity and reliability of the data, a model validation or comparison to check the fit and accuracy of the model, and use software tools or packages such as R, Python, Stan, or JAGS to perform Bayesian updating. Furthermore, statistical or graphical methods can be used to summarize and visualize the posterior distribution and its uncertainty.

6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?