Does Maximum Likelihood Estimation always yield the same result as Maximum a Posteriori with uniform prior?
I am studying different point estimate methods and read that when using MAP vs ML estimates, when we use a "uniform prior", the estimates are identical. Can somebody explain what a "uniform" prior is and give some (simple) examples of when the MAP and ML estimators would be the same?
A deck of 52 cards is shuffled, and we deal a bridge of 13 cards. Let $X$ be the number of aces and $Y$ be the number of spades. Show that $X$, $Y$ are uncorrelated.  Here is what I did:  $Cov(X,Y) = E[XY]-E[X]E[Y]$   uncorrelated means $Cov(X,Y) = 0$, hence  $E[XY]=E[X]E[Y]$   $E[X] =  \sum_{k=0}^{k=4} k \frac{\dbinom{4}{k} \dbinom{48}{13-k}}{\dbinom{52}{13}} $   $E[Y] =  \sum_{k=0}^{k=13} k \frac{\dbinom{13}{k} \dbinom{39}{13-k}}{\dbinom{52}{13}} $   Are the summations above correct? and how do I calculate $E[XY]$? Question mark for sentence which starts with a question but doesn't end with one I'm basically looking at sentences which start with a question, and end with an explanation to it. But there may be more instances of a similar structure. Sample sentence :      Can you book a room in advance, because otherwise we'd be in trouble   Should this sentence end with a period or a question mark?  PS: I read the apparently similar question  but the sentence structure there is different from what I'm looking for.  Edit: I just discovered that this question is a duplicate. A resolved question already  exists:   But i do like the explanations here better. I often see questions edited into this form:     Lorem ipsum dolor sit amet blah blah blah paragraph paragraph original question.            UPDATE: The Lorem disappeared and left this message: "new information".   I find this format unaesthetic (since Stack Exchange posts are supposed to be useful in the long term and update boundaries are artifacts of the creation rather than the content) and feel that new information should, whenever possible, be incorporated into the original text “as if it had always been that way” if practical, and pointing out an update should be done mostly in comments.  Is this an appropriate writing style, and if so under what circumstances? If not, should it be removed or discouraged? Grep: unexpected results when searching for words in heading from man page How do I make a Villager Trade a Spawn Egg I am trying to make a Custom Trade on a villager. I need to make him buy 8 gold nuggets and sell a Cow Spawn Egg.  basically you pay 8 nuggets for a Cow Spawner.  The ID of the Cow Spawn Egg is 92 but I don't know where to put it in the command /summon Villager ~ ~2 ~ {Invulnerable:1,CustomName:Spawner_Vendor,Profession:2,Career:1,Offers:{Recipes:[{buy:{id:gold_nugget,Count:8},sell:{id:spawn_egg,Count:1}}]}} Getting a standard form quadratic from a set of points ($3$ points) I came home from school today, pulled out my homework, now I'm stumped.  I don't want the answer, I just want to know how to do it.  Here is the question that I'm reading:  Determine a quadratic function in standard form for the set of points $(x_1,y_2)(x_2,y_2)(x_3,y_3)$  Where the $x$'s and $y$'s represent the points.  Here are the points I am working with (remember, I don't want the answer, I just want to know how to do it, there are lots of questions on this so I want to be able to know how to do it).     $(-1, 2), (0, 1), (1, -4)$   Thanks! Using getopts to process long and short command line options I wish to have long and short forms of command line options invoked using my shell script.  I know that getopts can be used, but like in Perl, I have not been able to do the same with shell.  Any ideas on how this can be done, so that I can use options like:  ./shell.sh --copyfile abc.pl /tmp/ ./shell.sh -c abc.pl /tmp/   In the above, both the commands mean the same thing to my shell, but using getopts, I have not been able to implement these? Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$ Can “whose” refer to an inanimate object? Why is polyploidy lethal for some organisms while for others is not? This is similar to  but I want the iterate tool to run to collect all the values and only then continue to the merge. Any ideas? Currently it just runs the whole process for each file (which works but takes much longer than if the merge is done only once) or I have to manually break the collect-merge link, run the iterate, reconnect the collect-merge link and then continue running the model which is a pain as it is a lot of towers that need processing and this collect values-finish getting all values and then continue will be required in other models as well.    I tried the "Select Data" tool as well but this doesn't allow for wildcard searches like the Iterate does -my Iterate has the "_Graded*" as the search param as the gdb included many other files related to each tower chunk.  My STOP is set to "True" (tried false as well). I also tried to connect the Stop to the Iterate instead of Collect values but same result.  === SCRIPT ADDED  I just created a python tool for this and it collects all the values but it passes them back to the model as a long list -how can I get it to output in a proper format?      import sys, os, arcpy  GDB=arcpy.GetParameterAsText(0) Search=arcpy.GetParameterAsText(1) Type=arcpy.GetParameterAsText(2)  ##GDB=r'P:\2012\044_DM_Radio_Propogation\Working\Propogation\Propogation_28355.gdb' ##Search='_m' ##Type='Polygon'  path=os.path.dirname(GDB) arcpy.env.workspace = GDB List=[] for item in arcpy.ListFeatureClasses('*'+Search+'*',Type):     List.append(path+"\\"+item)     #arcpy.SetParameterAsText(3, item)  arcpy.SetParameterAsText(3, List) arcpy.AddMessage("List: "+str(List)) Stack Exchange doesn't allow me to answer protected questions because I gave away earned rep in bounties I noticed that I can't answer protected questions. I don't want to actually, but it says:        It was , but I also didn't have an ability to answer other protected questions.  I earned 270 reputation points + 100 reputation point bonus. Then I  350 reputation points as a bounty. Then I earned 59 reputation points. Today I started another bounty (50 reputation points), etc – I offered 4 bounties with total 500 rep.  Despite the fact I started from the association bonus, I must have an ability to answer protected questions, as I earned reputation points here. But offered them as bounties.    I now have enough reputation points, but I would like it to be fixed, because I like offering bounties very much.  I would like system to check for my bounties when it checks if I can answer protected question. Proof if $a \vec v = 0$ then $a = 0$ or $\vec v = 0$ I'm kicking myself over this one, but I just can't seem to make the argument rigorous. From Axler's Linear Algebra Done Right:  for a vector space $V$ with an underlying field $F$:  Take an element $a$ from $F$ and $\vec{v}$ from $V$. $a\vec{v}=\vec{0}\implies a=0 $ or $ \vec{v}=\vec{0}$  After only being able to come up with half of a direct proof, I tried doing this by proving the contrapositive $a\neq 0 \wedge \vec{v} \neq \vec{0} \implies a\vec{v}\neq \vec{0}$  Say $a\vec{v}=\vec{u}$. Since $a$ is non-zero, we can divide both sides by $a$.  $$\vec{v}=\frac 1 a \vec{u}$$  If $\vec{u}$ were $\vec{0}$ then by  $$\frac 1 a \vec{0}=\frac 1 a (\vec{0}+\vec{0})\implies\frac 1 a \vec{0}=\vec{0}$$ $v$ would be $0$ as well. Since it isn't by assumption, $\frac 1 a \vec{u}$ cannot be zero and so $\vec{u}$ cannot be as well.   Is this fully rigorous? It seems like a very simple question, but I'm not sure about it. Namely, the last step of $\frac 1 a \vec{u}\neq 0 \implies \vec{u}\neq 0$ doesn't seem obvious. I think I need to use the $1\vec{v}=\vec{v}$ axiom, but I'm not sure how.  Is there a more direct proof? This whole contrapositive business seems a bit clunky for something so simple. I won a Samsung Galaxy S6 Edge, signed into it with my Google account, played around with it, wiped it, and sold it.  Now the person can't get past setup because it's saying he must login with an account previously synced to the phone.   Is there anything I can do from my side to remove this lock? Uploading an image from the web can leave paste broken in editor Building a matrix by merging the same row vector multiple times apt-get update, unlinking Can python send text to the Mac clipboard Formatting should not count towards the character limit on comments If we had an airtight container that was large enough, say extended out to the Karman line, would there still be a pressure gradient? If we assume that the pressure at the bottom was $1\ \mathrm{atm}$, would it have the same exact pressure gradient at 6 miles up on the outside? This is assuming that the gravity is still $9.8\ \mathrm{m/s^2}$.