1,106
edits
Changes
no edit summary
==Bayesian Approaches==
==Bayesian Approaches==
[[Bayesian statistics]] is a method of applying [[Bayes equation]] to data analysis. One of the biggest difference between Bayesian approaches and frequentist approaches is that Bayesians attempt to determine the probability that a given hypothesis is true given the data, while frequentist attempt to define the probability of getting the data given that a particular hypothesis is true.
Bayesian approaches are becoming more and more popular in science because what most people are interested in is the probability of proposed hypothesis no the probability of the data. However, Bayesian methods have come under fire from many frequentist proponents. There is actually a very heated debate in statistical circles about the respective validity of both methods. The primary complaint leveled at Bayesian statistics is that it must use a [[prior probability]] of a hypothesis in its analysis. Often this prior is not known out right and assigned seemingly arbitrary values based on particular distributions such as the [[uniform distribution]] or [[beta distribution]].
===Descriptive statistics===
===Descriptive statistics===
Bayesian methods all use Bayes equation, this applies for both descriptive and inferential statistics. To find such things as the mean and standard deviation first a prior probability for all means and standard deviations must be assigned. In practice this usually means assigning uniform probabilities to values equally spaced between what we think is the minimum and maximum values for the statistic we are interested in (the number of values depends on the grid density, which is proportional to accuracy and inversely proportional to computation time). Then a likelihood of each value is then calculated based on the data and then Bayes equation is used to assign a [[posterior probability]] for each value. These posterior probabilities can be plotted as a [[probability density function]] (PDF) to see the various probabilites for the value given the data, or often simply the value with the highest posterior probability is simply chosen.
===Inferential statistics===
===Inferential statistics===
Inferential statistics in Bayesian methods looks much the same as descriptive statistics since both use the Bayes equation and the same basic approach. To compare to means you would calculate the PDF for each data set then subtract them from each other to figure out the probability that they differ.
In order to compare hypothesis [[Bayesian model selection]] is often used. This is when each hypothesis you want to test is assigned a prior probability, and then the likelihood of the data given each hypothesis being test is calculated. You can then us Bayes equation to determine the relative probabilities that each hypothesis is correct. This method is almost always testing relative probabilites since to calculate and absolute probability would require knowing every possible hypothesis. Usually this is not possible, but sometimes the subset is finite enough it can be tested.
Because of the large number of calculations needed for model selection Bayesian approaches have only became practical and popular with the advent of computers. But even with the most modern computers available many Bayesian models remain computational intractable. Recent developments in applying [[Monte Carlo method | Markov chain Monte Carlo methods]] to these problems have lead to promising results.
==Non-parametric and Bootstrapping methods==
==Non-parametric and Bootstrapping methods==