Similarly, in inference, one should write down every possibility imaginable, and delete the ones that are inconsistent with all of our experiences data. However, this is impossible in practice, so for pragmatic reasons we summarise some of the data by a constraint on the probabilities we should use on a smaller hypothesis space , in much the same way that, in physics, we reduce the whole rest of the universe to just a single damping coefficient, or a driving force term.
Once we have them, we need to process them to update our probabilities, and ME is the right tool for this job. It is just the opposite — we are really trying to preserve as much prior information as possible! This leaves one remaining loose end — that pesky biased die problem, or the analogous one discussed by MacKay. Which answer is correct? Note that not all of these sequences of tosses are equally likely. If we condition on a 1 for the first toss, this raises our probability for the 2nd toss being a 1 as well. This is relevant prior information that should, and does, affect the result.
This is the source of the disagreement between MaxEnt and Bayes. It is this case where MaxEnt is appropriate because we really do possess no information other than the specified average value. This concludes my narrative of my journey from confusion to some level of understanding of this issue. At the moment, I am working on some ideas related to ME that can help clear up some difficulties in conventional Bayesian Inference.
Ramachandran, that gave me the confidence to reveal my heretical thoughts on this matter. Anyone who has actually used the maxent code for image reconstruction can see the influence of the prior information or enforced lack thereof. These days I use the maxent principle to justify various pragmatically simple priors, but always feel a little weedy while doing so… Strange how wedded people are to the ideal of an uniformative prior — I think it must be because we never really understand our experimental setups, and at least in astronomy are often blundering around in the dark.
Then as an after-thought I mentioned that exponential family distributions are clearly pretty significant. I like your analogy with physical systems. Any arbitrary predictor that gives a proper conditional probability distribution given some observations is trivially compatible with a Bayesian interpretation. Just pick any joint prior distribution over observations and new outcomes that has that conditional.
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So any simple updating procedure, not just MaxEnt, could be seen as a compact description of the effect of some, potentially complicated, beliefs. I would treat these with care. This should be a signal that too many assumptions have been made. Three examples: 1 Example This leads to an apparent paradox, although the book spuriously blames Bayesian inference as the cause with no real explanation. The reason the procedure helps is that it adds flexibility to the models. If this had been realized, it could have been motivation to improve the models directly instead.
Using MaxEnt to build a prior for bayesian inference
I hope your MaxEnt journey does give you insight into your problem and leads to better models of it. Statistics is one of those rare fields of study which is far more interesting in its fundamentals than in its applications …. I disagree with your broad point, Luke: granting the false dichotomy between fundamentals and applications, I challenge defy you to name three disciplines in which applications are a key attraction.
The second part is with math: four axioms that make entropy a unique function are recapped. The one of the four axioms is the most important to my taste.. The fourth part concludes. I have a dice with 6 sides from 1 to 6.
The only thing I know about the dice is that the mean of outcomes is 3. What are the probabilities which I have to assign to the sides of the dice? The mean will be:. Of course, there is an infinite number of choices which satisfies the mean 3. The dice can be biased and have higher probabilities for some sides while still having a mean of 3.
But among all possibilities, the uniform assignment feels intuitively justified. Let me modify the task a bit and consider that the only thing I know is that the mean is 4. Well, there is again an infinite number of possibilities of assignment. Probably, the most natural answer is that you can not tell, as you have not enough information.
But what is the difference between 3. Why do I have a feeling and a natural answer on one value of the mean 3. Let us first write this condition formally.
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I can assign probabilities of sides of a dice to be p1,…p6. I know that:. How to choose a certain assignment of probabilities from many which satisfy these equations constraints? That can be stated as an optimization task. I can maximize some function of all probabilities:.
But what does this function mean and how should I choose it? The only information that I know is that the mean value is 4. I have already used 4. Here is a criterion! I should optimize function F in a way that I will not plug any additional information about the dice in our estimation. Which function then do I have to choose?
That is the question which is answered in Jaynes work. I have to maximize entropy:. Any other function would imply additional information. Jaynes explained in his work:. The maximum entropy distribution may be asserted for the positive reason that it is uniquely determined as the one which is maximally noncommittal with regard to missing information.
There are some interesting properties about entropy.
For example, if I have 6-sided dice with the mean 3. The only components of this function that depend on are the polynomail constraints of form. As such, these constraints are the only components at risk to force the function towards infinity, provided that Therefore, because the corresponding to can any can be positive or negative, the function will be able to be defined so long for all , or for all.
Finally, we can consider the conditions for which these criteria for will be satisfied. In short, the only way to guarantee that remain either positive for negative will be if the dominant component of the polynomial is of an EVEN order for all s. If the dominant component is odd, then will either move from negative infinity to positive infinity or, if negated, from positive infinity to negative infinity as x moves across the domain, which means that no finite and nonzero could be chosen to maintain the criteria outlined above.
Derivation of maximum entropy probability distribution with no other constraints uniform distribution Satisfy constraint Putting Together 2. Derivation of maximum entropy probability distribution for given fixed mean and variance gaussian distribution Satisfy first constraint Satisfy second constraint Putting together 3. Derivation of maximum entropy probability distribution of half-bounded random variable with fixed mean exponential distribution Satisfying first constraint Satisfying the second constraint Putting together 4.
Maximum entropy of random variable over range with set of constraints with and is of polynomial order Introduction In this post, I derive the uniform, gaussian, exponential, and another funky probability distribution from the first principles of information theory.
Throwing dice with maximum entropy principle
Lagrange Multipliers Given the above, we can use the maximum entropy principle to derive the best probability distribution for a given use. The above can then be extended to additional variables and constraints as: and solving or, equivalently, solving In this case, since we are deriving probability distributions, the integral of the pdf must sum to one, and as such, every derivation will include the constraint.
With all that, we can begin: 1. Derivation of maximum entropy probability distribution with no other constraints uniform distribution First, we solve for the case where the only constraint is that the distribution is a pdf, which we will see is the uniform distribution.
Taking the derivative with respect ot and setting to zero, , which in turn must satisfy Note: To check if this is a minimum which would maximize entropy given the way the equation was set up , we also need to see if the second derivative with respect to is positive here or not, which it clearly always is: Satisfy constraint Putting Together Plugging the constraint into the pdf , we have:. Of course, this is only defined in the range between and , however, so the final function is: 2.
Derivation of maximum entropy probability distribution for given fixed mean and variance gaussian distribution Now, for the case when we have a specified mean and variance, which we will see is the gaussian distribution. To maximize entropy, we want to minimize the following function: , where the first constraint is the definition of pdf and the second is the definition of the variance which also gives us the mean for free.
Related Where do we go from maximum entropy
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