Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This paper gives a somewhat gentle introduction to Bayesian inference: http://www.miketipping.com/papers/met-mlbayes.pdf

I got to section 2.3 without much problems but got stuck in understanding that section onwards. It starts by presenting a probabilistic regression framework where the likelihood of all data is given as:

$$ p(t|x,w,\sigma^2) = \prod_{n}p\left(t_n|x_n,w,\sigma^2\right) $$ where $t_n=y(x_n;w)+\epsilon_n$ is the 'target' value. Next, given a set of parameters $w$ and a hyperparameter $\alpha$, the prior is given as: $$ p(w|\alpha)=\prod_{m}\left(\frac{\alpha}{2\pi}\right)^{1/2}\exp\left({-\frac{\alpha}{2}w_m^2}\right) $$

I can then compute the posterior $p\left(w|t,\alpha,\sigma^2\right)$. What I don't understand is the following:

  • In the first equation above, how should I interpret the product over the $N$ pairs of data $(t_n,x_n)$? Lets say I get two initial measurements from the real world, is $p\left(t|x,w,\sigma^2\right)$ supposed to give me a single real-valued probability? And how do I account for $w$ since it is not known yet?
  • As far as I got it, $w$ is supposed to be a vector of size $M$ where $w_i$ contains the $i$th estimated value. Now, how can a prior for $w$ have a reference to its own vector elements if I don't know them yet? Shouldn't a prior be an independent distribution such as a Gaussian or Beta? Also, shouldn't a prior be independent of hyperparameters?
  • Figure 4, on the article's page 8 has a plot from the prior and from the posteriors of an example using the $y=\sin(x)$ function with added Gaussian variance 0.2. How could I plot something similar in, say, Octave/Matlab or R?

I don't have a strong background in statistics so forgive me if this is too basic. Any help is appreciated.

Thanks in advance!

share|improve this question
    
To answer your second question, the prior has $w$ as a variable because it is function of $w$. It maps all possible values of $w$ to a probability. Furthermore, it is a Gaussian. See footnote #3 in that paper...I think you're getting confused by the subtle distinction b/t a probability density and a likelihood function. –  jerad Dec 2 '12 at 22:56
    
@jerad Ok, would that answer the first question a little as well? Since tn and xn are known, is the first equation also a function of w? Thanks! –  jokerbrb Dec 3 '12 at 9:57
    
The first equation is a distribution over $t$ conditional on some $x,w,\sigma^2$. –  jerad Dec 3 '12 at 20:48
    
Thanks. This is part of my confusion. Take a look at this videolecture for instance: videolectures.net/mlss09uk_bishop_ibi, and jump to minute 10:33 (Bayesian inference). There he says that p(x-hat|theta) is a function over theta given the new observed values x-hat. –  jokerbrb Dec 4 '12 at 15:06
    
Yes, well as explained in the wikipedia article on likelihood functions, it is merely a matter of perspective. I think anytime you see $p(\cdot)$ you should try to visualize a plot with probability on the Y-axis and parameters on the X-axis. You can either evaluate that function for a parameter value and return a probability, or you can view it as a function of the variables, ie. the whole plot. –  jerad Dec 4 '12 at 15:22

1 Answer 1

First question:

The product is the joint probability of the sample, often also called the likelihood (see the footnote on page 5). Yes, it gives you a single probability. It is simply the individual probabilities multiplied together, since they are assumed independent. This equation is sort of like an intermediate step. From there on, they drop $x$ from the notation. Then they end up with equation (11), where this first equation is combined with a prior and the normalizing constant. This is sort of the essence of Bayesian inference: we don't know the parameter $w$, but we know that the data depends on it. Using Bayes' theorem, we can thus get a posterior distribution by having a prior distribution.

Second question:

The vector $\mathbf{w}=(w_1, w_2, \dots, w_M)$ does not contain estimates. It contains the random variables $w_1, w_2, \dots, w_M$, i.e. the parameters. Not sure how/where they reference themselves?

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.