My son and I watched this entire video together. You did a great job teaching. My son understood this, I made sure to ask him questions to ensure he surly did. He's nearly 13 now and loves data-science. Great lecture and you have an excellent voice for teaching. Thanks again for this video.
The best explanation of Metropolis-Hastings algorithm. Thank you for this, no university course nor research paper will provide this much of understandable clarification.
Really nicely done - a very clear and concise explanation. I got a better sense of MCMC and M-H from these 8 minutes than several hours of reading on the topic beforehand. Thank you.
Note for revision. Why "Markov Chain"? Draw a theta (which will be accepted or rejected based on Hastings algo), then drawn another theta based on the new theta. The new theta is dependent on the old theta, so will be each successive theta, hence Markov Chain.
My question is why? Why do we generate a new distribution from an already existing distribution? Why is each new generated value based on the previous value? Coin tosses are independent, are they not? Why would we do this to distributions of independent probabilities?
You wouldn’t do this for simple situations where it’s possible to sample independently. This is just a simplified example. MCMC algorithms are designed for use in multidimensional models where it isn’t feasible or efficient to calculate p(y), and thus a proper absolute posterior probability of theta. In such situations, when p(y) is intractable, dependent MCMC sampling is needed because it allows for the exploration of posterior parameter space through the use of relative frequencies of posterior samples rather than absolute probabilities
It depends on the parameter type. If it’s an unconstrained parameter, then a symmetrical is fine, usually the normal curve is used. If it’s a constrained parameter (e.g. sigma), then the Hastings variation of the Metropolis algorithm is used because it allows for non symmetric proposal distributions. The key to the algorithm and how it works to map out the posterior is the acceptance ratio, which can be affected by the size of the steps proposed (sigma in the normal curve proposal distribution). Auto-tuning allows the proposal distribution to be adjusted periodically to make sure the acceptance ratio is within the target threshold.
All this is excellent for physical, electrical / electronic systems, and the like. However, I have seen people try to use it for human based systems (like predicting the stock market, yeah, right). For that, it would serve best to know if a person likes Shakespeare, Frost, or limericks.
03:35 A new theta is drawn from the Proposal Distribution, which has a normal distribution in this example. How to decide/select the proposal distribution function?
Thanks! This explains MCMC perfectly. However it does not seem to explain why Markov chain is required? Why not just use Monte Carlo alone for drawing from the distribution?
Can someone please explain to me how step one is calculated (posterior theta new / posterior theta old) because if you are sampling from a binomial distribution the output would be 0 or 1. and sampling from beta (1,1) is just sampling from a uniform distribution. for example what does beta(1,1,0.088) and binomial(10,4,0.88) each equal individially? Thanks in advance
With the first example in the video: R code: prior1=dbeta(shape1=1,shape2=1,x=0.517) likelihood1=dbinom(x=4,size=10,prob=0.517) prior2=dbeta(shape1=1,shape2=1,x=0.380) likelihood2=dbinom(x=4,size=10,prob=0.380) r=(prior2*likelihood2)/(prior1*likelihood1) r 1.31 Each part equals: prior1=1 prior2=1 likelihood1=0.19 likelihood2=0.25
thanx for a beautiful video that has made me form an image of what mcmc is all about. God bless you. do kalman filters demo ,eg, for time dpseries predictions,using stata.
It seems, usage of Beta(1,1,0.286), Beta(1,1,0.380), Binomial (10,4,0.286) and Binomial(10,4,0.380) at the time frame 4:16 has no meaning: the first two are constants and equal to each other, and the latter two describe PDF to express outside [0,1] region, as shown in the plot.
Sir..so MCMC is not seprarete algorithm but uses Meterpolis hasting algorithm.. i thought MCMC was a seperate procedure that helps us to simulate data to help us identify posterior distribution. Also i want to learn more.. how can i do so?
This is how MH accepts candidate/proposed values where the posterior ratio of proposed/current is less than 1. When it’s less than 1, a uniform distribution is used to draw a random number. If that random number is below the acceptance ratio, we accept the candidate value. It’s just a random number generator to make sure we accept our values at that probability
Wait so if we use MCMC to walk around the parameter space why do we even need a prior distribution? Is it only used to initialize the starting value of the MCMC?
Thank you for this great tutorial! I have a question though: What happens if the mcmc sampler gives theta>1. You won't be able to calculate the likelihood or the prior. Do you discard this theta in this case? Shouldn't one use a uniform distribution u(0,1) to make sure we are getting a theta in [0, 1]?
You’ll need to read up on the distinction between Metropolis and Metropolis Hastings. The MH was created for this very reason, to address proposal distributions for constrained parameters, like sigma. The regular Metropolis algorithm only works for non-constrained parameters where this isn’t an issue. Basically, MH adjusts for this by allowing non symmetric proposal distributions in that instance. Beyond the scope of this video, I’d recommend watching Ben Lamberts series on Bayesian Statisitcs. He has a video specific to MH and this question
The target distribution is just the posterior, which is proportional to the prior * likelihood. A beta prior is a conjugate prior for the beta binomial posterior (target) distribution. Not all priors are conjugate priors. Regarding the proposal distribution, a normal distribution is often used b/c it is a symmetric distribution, but it’s not specifically mandated to be a normal
