Python Tutorial : Optimal parameters

Key Takeaways

after completing the prequel to this course you are now beginning to think probabilistically outcomes of measurements follow probability distributions defined by the story of how the data came to be when we looked at Michelson speed of light and air measurements we assumed that the results were normally distributed we verified that both by looking at the PDF and the CDF which was more effective because there is no binning bias to compute and plot the CDF we needed our old friends numpy and matplotlib pipeline so the first step was to import them with their traditional aliases to compute the theoretical CDF by sampling we passed two parameters into NP at random dot normal the mean and standard deviation the values we chose for these parameters were in fact the mean and standard deviation we calculated directly from the data the result was that the theoretical CDF overlaid beautifully with the empirical CDF how did we know that the mean and standard deviation calculated from the data were the appropriate values for the normal parameters we could have chosen others what if the standard deviation differs by 50% the CDF's no longer match or if the mean varies by just point O 1% so if we believe that the process that generates our data gives normally distributed results the set of parameters that brings the model in this case the normal distribution and closest agreement with the data uses the mean and standard deviation computed directly from the data these are the optimal parameters remember though the parameters are only optimal for the model you choose for your data when your model is wrong the optimal parameters are really not meaningful finding the optimal parameters is not always as easy as just computing the mean and standard deviation from the data we will encounter this later in this chapter when we do linear regressions and we rely on built-in numpy functions to find the optimal parameters for us I pause here to note that there are great tools in the Python ecosystem for doing statistical inference including by optimization psy PI dot stats and stats models being two good examples in this course however we focus on Hacker statistics because the technique is like a Swiss Army knife the same simple principle is applicable to a wide variety of statistical problems now it's time for you to do some exercises to demonstrate how choosing optimal parameters results in best agreement between the theoretical model distribution and your data