I have taken a handful of statistics/ data science-oriented courses. And to this day, I feel like I grasp the underlying concept but I do not comprehend the in-depth understanding. I am asking this because it's been more of a formulaic relationship, and since it is so applicable and important, I was wondering if anybody had any readings or mental breakthroughs to what Bayes Theorem is!
I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation: $$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$ where $A$ represents a hypothesis or belief and $B$ represents data or evidence. I've understood the concept of the posterior - it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator? After reviewing a couple of resources I came across this quote:     The likelihood is the weight of event $B$ given by the occurrence of $A$ ... $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred.   The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two?
How to resolve Google "Indexed, though blocked by robots.txt" Cardinality of $\mathbb{R}^\mathbb{R}$ by diagonal argument I want to prove that the set of all real functions $\mathbb{R}^\mathbb{R}$ has a higher cardinality than the real numbers $\mathbb R$, by Cantor's diagonal argument.  I'm having difficulties with approaching this problem.  What I'm looking for is a hint in the right direction.  I've seen an example where it is shown that the power set $2^S$ of a countable set $S$ is uncountable, by the diagonal argument. I have a DEM layer from ASTER satellite and I want to get the slope information from it.  I use terrain analysis > slope, but the output is so wrong. Why is that?  Here is the DEM:     And here is the "Slope" that is totally wrong. It should stretch the values from 0 - 89.9097 but it does not. Defining font color inside highlight fails Since I wanted a better readability of highlights of a would be black &amp; white-printed document, I needed to change both the highlighting color (gray) AND the font color (white).  Depending which command you implement inside the other will compile or not: \hl{\textcolor{white}{Fail}} does not work, compiling returns Package xcolor Error: Undefined color '{white}'.  Here's the mwe:  \documentclass{article} \usepackage{xcolor} \usepackage{color} \usepackage{soul} % Enables highlighting \sethlcolor{gray}  \begin{document}      \textcolor{white}{\hl{Success}}     %% Error:     %% Package xcolor Error: Undefined color `{white}'.     \hl{\textcolor{white}{Fail}}  \end{document}   So, in the end, the problem is sorta solved but I thought you might have an answer as to what happens. Cannot reach public WIFI login page in Ubuntu 18.04 Captive Portal Issue PHP regular expressions: No ending delimiter '^' found in Most efficient way to append arrays in C#? I am pulling data out of an old-school ActiveX in the form of arrays of doubles. I don't initially know the final number of samples I will actually retrieve.  What is the most efficient way to concatenate these arrays together in C# as I pull them  out of the system? Scalar Field Theories The Lagrangian density for a single real scalar field theory is \begin{equation}\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\end{equation} I have often seen this written \begin{equation}\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\end{equation} which appears to differ in the kinetic term.  Specifically \begin{equation}(\partial_{\mu}\phi)^{2}=(\partial_{\mu}\phi)(\partial^{\mu}\phi)\neq\partial_{\mu}\phi\partial^{\mu}\phi=(\partial_{\mu}\phi)(\partial^{\mu}\phi)+\phi\partial_{\mu}\partial^{\mu}\phi\end{equation} where the product rule has been used in the last equality. Is this just sloppy notation (which is acceptable because only first derivatives in the field are considered anyway), or is my maths very bad? I'm not a git master, but I have been working with it for some time now, with several different projects.  In each project, I always git clone [repository] and from that point, can always git pull, so long as I don't have outstanding changes, of course.  Recently, I had to revert to a previous branch, and did so with git checkout 4f82a29.  When I was again ready to pull, I found that I had to set my branch back to master.  Now, I can not pull using a straight git pull but instead, have to specify git pull origin master, which is annoying, and indicates to me that I don't fully understand what is going on.  What has changed which does not allow me to do a straight git pull without specifying origin master, and how to I change it back?  UPDATE:  -bash-3.1$ cat config [core]     repositoryformatversion = 0     filemode = true     bare = false     logallrefupdates = true [branch "master"] [remote "origin"]     url = git@github.com:user/project.git     fetch = refs/heads/*:refs/remotes/origin/*   UPDATE 2: To be clear, I understand that my original method may have been incorrect, but I need to fix this repo so that I can simply use git pull again.  Currently, git pull results in:  -bash-3.1$ git pull You asked me to pull without telling me which branch you want to merge with, and 'branch.master.merge' in your configuration file does not tell me either.  Please name which branch you want to merge on the command line and try again (e.g. 'git pull  '). See git-pull(1) for details on the refspec.  If you often merge with the same branch, you may want to configure the following variables in your configuration file:      branch.master.remote =      branch.master.merge =      remote..url =      remote..fetch =   See git-config(1) for details.   I can tell git pull which branch to merge, and it works correctly, but git pull does not work as it did originally before my git checkout. Troubleshooting "program does not contain a static 'Main' method" when it clearly does...? Can a Spring form command be a Map? Is the Moonbeam spell amazing, or are we doing it wrong? The description of the moonbeam spell (PHB pg. 261) says:  When a creature enters the spell's area for the first time on a turn or starts its turn there, it is engulfed in ghostly flames that cause searing pain... On each of your turns after you cast this spell, you can use an action to move the beam 60 feet in any direction.  We read this to mean a creature takes damage from moonbeam when any of the following conditions are met:  It begins its turn inside the moonbeam It enters the moonbeam on its turn The creature finds itself inside the beam on another character's turn, no matter how it got there  The creature will not take damage in the following instances:  A creature moves from one part of the beam to another (obviously it had to take damage, but it won't take more damage) A creature is in the beam after having been inside the beam already on that turn.  Following these assumptions, our druid cast a moonbeam and did damage to two enemies, Bob and Frank. On their turn, they took damage again, killing Bob. Frank left the beam. On the druid's next turn, she moved the beam in a circular motion, not exceeding 60 total feet of movement, and in the process the beam came in contact with a dozen enemies and ended on Frank. All dozen enemies received damage, as they were inside the beam at some point during the turn. On Frank's next turn, he again took damage for the fourth time and died. On each of the druid's turns, she moved the moonbeam around like this until the fight ended (it did not exceed the time limit and she made all concentration checks). Is this a legitimate use of moonbeam? It seems awfully powerful for a level 2 spell. I read a story--the first part of one, at any rate--years ago, and haven't tracked it down since. It started with a man been sent to a planet under attack to pose as a monk--he arrived in the middle of an assault, but still had to do penance for being late. So he wore a hair shirt, that created an agonising reaction. He was captured by the attackers (who took him for the man he was posing as), who then stood off the planet in their starships and watched as a 'hell burner' weapon tumbled out of deep space and torched the planet. The attackers then explained that they weren't responsible for the hell burner, but every time they arrived at a planet, a hell burner turned up afterwards, at shorter and shorter intervals.  As the story developed, there was some kind of time travelling dimension, and a character turned out to have his organs reversed. I may have got some details wrong, but this was the shape of it.  Anyone know this story, and the author? Main Reference + Appendix with "Other Sources" I need your help with references. I have 1 main reference section, and I want to include a second list of references as an appendix. For the main reference I use:  \newpage  \bibliographystyle{plainnat}  \bibliography{library}   The problem is that I don't know how to include the second list using a different bib file, and then calling the function \nocite{*} to show every sources inside the file.   I have the appendix part workout. I need your help to design the function to call the second bibliography, and that does't interfere with the main reference:   \newpage  \appendix  \section{Other Sources} \label{app:ReferencesB}   In addition, I will prefer that the word References is not shown after Other Sources.  Thank you so much!! There is no fruit jet pack. I can't get it. How do I get it? I looked at the jetpacks many times. There are none. "Require Installation Of Untrusted Packages" while upgrading with the update manager It didn't fixed the error. So, how do I fix my GUI update?  For every software I have to install it from command line because I got same error when install software from Ubuntu Software Center. How to shuffle all songs in iOS 8.4 music app In the old version of the music app you could shuffle all songs when songs were selected:      This feature seems to be absent in iOS 8.4 and the new music app:      What are the options for shuffling all my songs in the new version of the music app under iOS 8.4? -> 683  -> 678  ???  (Doesn't really bother me too much but figured I'd report the bug...) This is an exercise in some textbooks.     Let $E$ be an algebraic extension of $F$. Suppose $R$ is ring that contains $F$ and is contained in $E$. Prove that $R$ is a field.   The trouble is really with the inverse of $r$, where $r\in R$. How to prove that $r^{-1}\in R$, in apparent lack of a characterization of $R$.  It occurred to me to use the smallest field containing $R$ ($R$ is easily shown to be an integral domain), that's the field of quotients, and proving that it's $R$ itself, but I don't really know how to proceed.  A not-too-weak, not-too-strong hint will be much appreciated.   Beware $ $ Readers seeking only hints should  beware that there is now a complete answer. Problems with pie charts There seems to be in increasing discussion about pie charts.   The main arguments against it seem to be:   Area is perceived with less power than length. Pie charts have very low data-point-to-pixel ratio   However, I think they can be somehow useful when portraying proportions. I agree to use a table in most cases but when you are writing a business report and you've just included hundreds of tables why not having a pie chart?  I'm curious about what the community thinks about this topic. Further references are welcome.  I include a couple of links:         In order to conclude this question I decided to build an example of pie-chart vs waffle-chart.