What happens in a Bayesian update of a normal prior distribution?
The following data and calculation shows what happens in this example. In this example an increase of the mean of 6 % is the influence of the 8 new data points. However, the variance is now considerably reduced, or in terms of the standard deviation: from a 7.838 down to 6.232, which is ~80% of the prior st.dev.
How is the Gaussian distribution used in statistics?
The Gaussian or normal distribution is one of the most widely used in statistics. Estimating its parameters using Bayesian inference and conjugate priors is also widely used. The use of conjugate priors allows all the results to be derived in closed form.
How to calculate an update of a normal prior distribution?
It has a mean equal to the posterior mean and a standard deviation which we derive from the posterior variance by multiplying with the regression sample size and taking the square root. Example of an update The following example of estimating the API gravity (or oil density) may help to see the above formulas at work.
Is the Bayesian update mechanism similar to an expert system?
In that sense the baysian update mechanism is similar to what many “expert systems” are providing. The formulas involved are shown here without giving the derivation (Jacob, 2008, Winkler, 1972). They are valid under the simplifying assumption that we know the “process” variance.
How is the mean of a normal prior distribution calculated?
It combines in the simulation model the world-wide variation of a variable and the relevant local data to give the most realistic value and its uncertainty. The mean of this distribution is the same as the posterior mean, but the variance of this mean is a weighted combination of process and posterior variance.
Why is the variance of a normal prior distribution different for each depth?
This distribution has a varying mean and variance along the depth, with the mean following the regression line and the variance dependent on the residual standard deviation and the distance to the mean depth of the data used for the regression. So, each target depth has a (possibly slightly) different distribution.
Which is more complete update of probability or normal distribution?
I can recommend an explanation of the update process for a probability and for a normal distribution given by Jacobs (2008), which is more complete than what I have given here and explains the derivation of the formulas.
How does bayesian inference work for normal mean?
This illustrates how the prior, likelihood, and posterior behave for inference for a normal mean ( μ) from normal-distributed data, with a conjugate prior on μ. Specifically the prior on μ is N ( μ 0, τ 0 2) [dotted line] and the data is sampled from a normal distribution N ( μ, σ 2 ), which gives the likelihood [black line].