how can I convert a negative log likelihood to likelihood between 0 and 1 ?   I use HMMs package in R and  I keep getting strange results of the log likelihood for example, -48569 ! I need to understand these values so I can pick the best model. any advice will be appreciated.  thanks
The  claims that likelihood and probability are distinct concepts.     In non-technical parlance, "likelihood" is usually a synonym for "probability," but in statistical usage there is a clear distinction in perspective: the number that is the probability of some observed outcomes given a set of parameter values is regarded as the likelihood of the set of parameter values given the observed outcomes.    Can someone give a more down-to-earth description of what this means?  In addition, some examples of how "probability" and "likelihood" disagree would be nice.
Let $x = (a, b), y = (c, d) \in \mathbb{Z}^2$. What is the condition on $a, b, c, d$ so that ${x, y}$ is a basis?   My answer: $ad\neq bc$ and $gcd(a, c) = gcd(b, d) = 1$.  The first condition ensures that they aren't the same vector; the second ensures that we can actually "get" all of the integer values/lattice points.  Is this correct?  Thanks. How do we prove $n^n \mid m^m \Rightarrow n \mid m$? I'm not sure I've got this right. When proving $a^n \mid b^n \Rightarrow a \mid b$, can we do this indirectly? In short,  "Suppose $a$ does not divide $b$, this implies that $a^n$ does not divide $b^n$. But $a^n \mid b^n$, hence $a$ divides $b$."  How about $n^n \mid m^m \Rightarrow n \mid m$? Can we do this the same way? Is there an IDictionary implementation that, on missing key, returns the default value instead of throwing? How to negate a specific symbol? I want to negate the symbol whose code is \models (this would be specific to Formal Logic). How can I do that? I tried a number of combination, but didn't find how this could be done. I tried \neqmodels, \nemodels, \notmodels, \nmodels  but all of them are wrong.  Is there a general way to negate symbols? Octal number literals: When? Why? Ever? I have never used octal numbers in my code nor come across any code that used it (hexadecimal and bit twiddling notwithstanding).  I started programming in C/C++ about 1994 so maybe I'm too young for this? Does older code use octal? C includes support for these by prepending a 0, but where is the code that uses these base 8 number literals? Sum of First $n$ Squares Equals $\frac{n(n+1)(2n+1)}{6}$ I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals:  $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$  I really have no idea why this statement is true. Can someone please explain why this is true and if possible show how to arrive at one given the other? Proof that $1729$ is the smallest taxicab number I need another LP-E5 type battery for my Canon 450D. Should I buy an original Canon battery or one of the many generic brands available? There are loads of cheapos on , for example.  I should add that the new battery will be added to my existing Canon battery in a battery grip.  Does anyone have any experience (good or bad) of using generic batteries? How do you get Mac OS Finder to group folders together when sorting? I have Mac OS 10.7.4 and I'd like Finder to use "Is Folder?" as its first sort criteria, or in other words, group folders before files regardless of the sorting method.  Windows and Linux both do this by default. I'm trying to write a program in C (on Linux) that loops until the user presses a key, but shouldn't require a keypress to continue each loop.  Is there a simple way to do this? I figure I could possibly do it with select() but that seems like a lot of work.  Alternatively, is there a way to catch a ctrl-c keypress to do cleanup before the program closes instead of non-blocking io? For as long as I can remember I have avoided using switch statement fall-through. Actually, I can't remember it ever entering my consciousness as a possible way to do things as it was drilled into my head early on that it was nothing more than a bug in the switch statement. However, today I ran across some code that uses it by design, which got me immediately wondering what everyone in the community thinks about switch statement fall-through. Is it something that a programming language should explicitly not allow (like C# does, though it supplies a workaround) or is it a feature of any language that is powerful enough to leave in the programmer's hands? Edit: I wasn't specific enough to what I meant by fall-through. I use this type a lot:     switch(m_loadAnimSubCt){         case 0:         case 1:             // Do something             break;         case 2:         case 3:         case 4:             // Do something             break;    }  However, I'm concerned about something like this.    switch(m_loadAnimSubCt){         case 0:         case 1:             // Do something, but fall through to the other cases             // after doing it.          case 2:         case 3:         case 4:             // Do something else.             break;    }  This way whenever the case is 0, 1 it will do everything in the switch statement. I've seen this by design and I just don't know if I agree that switch statements should be used this way. I think the first code example is very useful and safe. The second seems kind of dangerous. In my object record type i have created formula field. Now i want my formula field to work as below.  If Basement__c = NA and Property__r.Basement__c = NA OR  Basement__c = Crawl and Property__r.Basement__c = Crawl OR  Basement__c = Slab and Property__r.Basement__c = Slab Then it will return Type__c(formula field) = "MAtch"  And (Another rule) Basement__c DOES NOT contain any of the following "N/A,Crawl,Slab,None and Property__r.Basement__c Does not contain any of this then formula field will return "HAs basement"  How can i achieve this? Please help. Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $ Let $f:\Bbb R\to (0,\infty)$ be a differentiable function such that for some constant $k$ we have, $$|f'(x)-f'(y)|\le k|x-y|$$   for all $x,y \in\Bbb R.$ Then prove that, $$(f'(x))^2&lt;2 kf(x).$$   Due to lack of arguments I could not proof this inequality. But rather I proved that is true for the particular function,   $$f(x) =\cos^2(x)+1\implies f'(x) = -\sin(2x).$$ And we readily have,$$|\sin(2x)-\sin(2y)| = \left|\int_{2x}^{2y}\cos t dt\right|\le 2|x-y|$$ as well   $$(f'(x))^2 = 4 \sin^2 x\cos^2x &lt; 4(\cos^2x +1) = 4f(x).$$  It is also true for the function $x\mapsto \sin^2x +1.$   From these example I don't see how to prove the general case. Any hint will be welcome. Fluent and Query Expression — Is there any benefit(s) of one over other? LINQ is one of the greatest improvements to .NET since generics and it saves me tons of time, and lines of code.  However, the fluent syntax seems to come much more natural to me than the query expression syntax.  var title = entries.Where(e =&gt; e.Approved)     .OrderBy(e =&gt; e.Rating).Select(e =&gt; e.Title)     .FirstOrDefault();  var query = (from e in entries              where e.Approved              orderby e.Rating              select e.Title).FirstOrDefault();   Is there any difference between the two or is there any particular benefit of one over other? Question is :  Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:   $X$ is not connected. $X$ is not compact $X$ is not homeomorphic to a subset of $\mathbb{R}$ None of the above.   I guess first two options are false.  We do have possibility that product of two connected spaces is connected.  So, $X\times X$ is connected if $X$ is connected. So I guess there is no problem.  We do have possibility that product of two compact spaces is compact.  So, $X\times X$ is compact if $X$ is compact. So I guess there is no problem.  I understand that this is not the proof to exclude first two options but I guess the chance is more for them to be false.  So, only thing I have problem with is third option.  I could do nothing for that third option..  I would be thankful if some one can help me out to clear this.  Thank you :) How do I add the kernel PPA? Showing that $X^2$ and $X^3$ are irreducible but not prime in $K[X^2,X^3]$ Show that $X^2$ and $X^3$ are irreducible but not prime in $K[X^2,X^3]$.   My reasoning is as follows:  Since the only ways $X^2$ and $X^3$ can be factored in $K[X^2,X^3]$ are $X^2 = X^2 * 1$ and $X^3=X^3*1$ it follows that $X^2$ and $X^3$ are irreducible.  To show that they are not prime in $K[X^2,X^3]$ I argue that $X^2\mid X^3*X^3$ but $X^2\nmid X^3$ and $X^3\mid X^4*X^2$ but $X^3\nmid X^4$ or $X^3\nmid X^2$.  Is that correct? How do I recover from the Heartbleed bug in OpenSSL? Is there any significant difference between Nikon and Canon? So I need to find the Convergence Radius and Convergence Domain of the complex power series: $\sum\limits_{n=0}^\infty \frac{1}{2^{n}} z^{2^n}$  I've tried looking at the individual parts of the series and rewriting it somehow but couldn't find a way.  It can probably be solved, after a step or two, by the formula $R=\frac{1}{\displaystyle\limsup_{n\rightarrow \infty}\sqrt[n]{|a_n|}}$ .  Thanks.