In practice, this difference is obvious, but how to put it in words?
I have been wondering about the differences between model paramters and model hyperparameters, as well as what their categorization means for a learning problem.  Is the distinction between model parameters and hyperparameters only about reducing the complexity of the problem or are there implications for the 'model quality' to be considered?  Let's consider an example from (1, Eq. 2.43), a linear expansion of basis functions   $$f_\theta(x) = \sum_{m=1}^{M} \theta_m h_m(x), $$  where $h_m\:\forall m=1,...,M$ is some function of $x$, $\theta=[\theta_1,...,\theta_M]^\intercal$ is a vector of the parameters of the model and $M$ is the number of basis functions.   In my understanding, we would consider $\theta$ to contain the parameters of the model, while $M$ is a .  To quote the Wikipedia article of hyperparameter that I linked above:     In machine learning, a hyperparameter is a parameter whose value is   set before the learning process begins. By contrast, the values of   other parameters are derived via training.   It makes sense treating the values in $\theta$ as parameters that should be learned from the data, similar to the coefficients in a linear model - it is surely far more reliable  and  specifying them manually would be quite annoying, although it's not impossible. But what about $M$? It surely is easier to pick some integer value for it. But would it really be terrible to determine $M$ based on the data, too? Probably not, as that is what we would do if we search a grid of possible values for the best performing one.   Let's consider the above basis expansion model and choose Gaussian kernel functions to serve as basis functions. As the Kernel itself has parameters too  this makes the problem a bit more complicated.  $$f_\theta(x) = \sum_{m=1}^{M}\theta_m K_{\lambda_m}(\mu_m,x), $$ with $\mu_1,...,\mu_m$ being the location of the kernels and $\lambda_1,...,\lambda_m$ their scales/bandwidths.  Friedman et al. write in (1, p. 36)        "In general we would like the data to dictate them as well. Including   these as parameters changes the regression problem from a   straightforward linear problem to a combinatorially hard nonlinear   problem. In practice, shortcuts such as greedy algorithms or two stage processes are used."   I see that if we were to treat $M$, $\theta_1 ... \theta_M$, $\mu_1 ... \mu_M$ and $\lambda_1 ... \lambda_M$, as model parameters, this problem would be really complex, especially with the choice of $M$ determining the number of the other parameters. But what if we simplify the model definition - let us for example use the same scale/bandwidth for each kernel, that gives us only one parameter $\lambda$. Furthermore, let's say we specify $\mu_1 ... \mu_M$ based on some heuristic, which would mean we treat it as a hyperparameter.  This gives us   $$f_{\theta,\lambda}(x) = \sum_{m=1}^{M}\theta_m K_\lambda(\mu_m,x), $$  Besides increased complexity of the involved optimization problem, does treating $\lambda$ as a parameter rather than a hyperparameter have other negative or undesired effects?      (1) Friedman, Jerome, Trevor Hastie, and Robert Tibshirani. The   elements of statistical learning. Vol. 1. No. 10. New York: Springer   series in statistics, 2001.
Derive a formula to find the number of trailing zeroes in $n!$ Prove that $||x|-|y||\le |x-y|$ how to write in TeX "computer zero" with slanted vertical line on it? In TeX, how is computing zero sign written with slanted vertical line on it? Are planets actually moving in elliptical orbits around the Sun or do they move in circular orbits around their center of mass? In every derivation of Kepler's Laws that I have seen, we assume that the sun is stationary. However, in other places I have read that celestial bodies move about their barycentre (center of mass). So are planets actually moving in elliptical orbits around the Sun or do they move in circular orbits around their center of mass? Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n Are DSLRs a dying breed, making now the time to switch to a mirrorless camera system? I came across this  on Stuck in Customs today that makes a strong case to hold off too much investment in DSLR gear, and instead switch to mirror-less cameras like the Sony NEX. The author prefers to call such cameras 3rd generation cameras to prevent any bias.  The crux of the argument lies in the smaller size and faster shooting speeds of the new cameras, while the sensor size is an area where they seem to be lacking (Sony &amp; Samsung cameras are APS-C sized though). Currently the big makers - Nikon &amp; Canon - have practically no presence in this market (Nikon seems to be doing something with the V1). However, they seem to have slowed down the introduction of new entry level DSLR models over the last year as well.  So, as DSLR users looking to build and enhance their kit (there are many like me who've jumped the bandwagon over the last couple of years, and are just starting to build their kit), what are the advantages of sticking with the DSLR system rather than switching to the 3rd gen cameras?  IMO, one point that could cause a massive switch in the DSLR base would be if Nikon &amp; Canon announced mirror-less models compatible with their current lens lineups, but that is a hypothetical scenario.  P.S. There is a similar question on mirror-less cameras, but that seems to be from the perspective of a person starting out rather than a person already invested in the DSLR system. How do I search events between two set dates inside WP? Where did folder "merge" or "keep both" go in OS X 10.9 Mavericks? Before Mavericks one could hold the ALT key and move a folder "A" onto a folder "B" where already an "A" existed. In this case Mac OS X asked in Lion if I want to merge - or on Mountain Lion if I want to keep both (which was also a merge).  I can't find this option in Mavericks.  Did Apple remove this useful feature?  Any hints?  (And yes: I own ForkLift, which can do it - I just sometimes still prefer the Finder). On Groups of Order 315 with a unique sylow 3-subgroup . in Dummit and Foote , an exercise asked me to prove that , if $G$ is a group of order $315$ , $G$ has a normal sylow $3$-subgroup then , $G$ is abelian .  this is exercise number $27$ , section $5$ , chapter $4$  my method on proving this is showing that $ |G| = |Z(G)|$ as follows   i showed that $|Z(G)|$ $\in$ ${ 9 , 63 , 45 , 315 } $  and  showed that$ |Z(G)|$ is not $63 , 45$ or $9$   so it must be $315 $  but i showed this under two condition ,   first condition: if $X$ is the set of all elements of $G$ of order $5$ , $H$ is the subgroup generated by $X$  then $|H|$is not $315$  second condition : also , if $Y$ is the set of all elements of $G$ of order $7$ , $N$ is the subgroup generated by $Y$ then $|N|$ is not $315$   so , if those two condition are then my method of proving works .   but i don't know how to prove those conditions .   any help ? Allow linking to specific revision of revision history showing diff Currently, the only way to link to a revision (clicking "link"), you just get  which shows what the post looked like at that revision:        Personally, I rarely use this and haven't ever found myself in a scenario (as far as I remember) where I would prefer the way it currently works to what I'm suggesting here.  Could we:   Add another link, or Change the current link   Such that it:   Displays the diff on a separate page (I'd prefer this, since the highlight and scroll thing doesn't always work properly - although perhaps it has something to do with loading images, so it might not apply here, or, if it doesn't, we could fix that instead), or          Highlights and scrolls to a diff, like what is done when you link to an answer () What were Luke's lessons? Luke tells Rey that      there are three lessons he will teach her as he trains her as a Jedi   What are the lessons? ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$ Suppose $u(x,y)=\ln(x^2+y^2)$  i) Show that $u$ is harmonic on $\mathbb{C} \backslash \lbrace 0 \rbrace$  ii) Show that $u$ is not the real part of a function which analytic on  $\mathbb{C} \backslash \lbrace 0 \rbrace$  I manage to show the first part. For the second part, note that $u(x,y)=\ln(x^2+y^2)=\ln(|z|^2)=2 \Re \log(z)$  But this only show that $u$ is not a real part of $\log(z)$. I don know how to show $u$ cannot be the real part of a function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$  Can anyone guide me? Looking for a title of a sci-fi novel or story where a government department's job was to slow things down Lost USB BlueTooth Dongle for Keyboard and Mouse I lost my USB bluetooth dongle (from HP) which I used for a wireless keyboard and mouse. They came together in a set when I bought my last HP computer.  Are the keyboard and mouse useless now? or will any bluetooth dongle allow them to work? If I can just buy a new one - any recommendations on a replacement (good price, and perhaps works with Windows and Ubuntu out-of-the-box?   Forgive me if this is an easy question. I'm just not that familiar with wireless mice/keyboards. Find   $$\lim _{x\to \infty }x\Big(\ln\left(x+1\right)-\ln x\Big)$$  Here's how I do it:  $$x\Big(\ln(x+1)-\ln x\Big) = x\Bigg(\ln(x(1+\frac{1}{x}) - \ln x\Bigg)$$ $$x\Big(\ln x + \ln(1+\frac{1}{x}\Big)-\ln x =x\ln\Big(1+\frac{1}{x}\Big)$$ $$\ln\Bigg( \Big(1+\frac{1}{x}\Big)^x \Bigg) \rightarrow \ln 1^\infty = 0$$  What am I doing wrong? The answer is supposed to be $1$, but I get $0$. Get full name twice in Bibliography I use biblatex with style=authoryear and when there are two references of the same author in the bibliography there is just a dash. How can I get the full name in the second entry too? I'm curious of how illusions would affect spells or abilities that require a target to be looking at something. A good example of this would be Medusa's Petrifying Gaze which has the following effect:  Petrifying Gaze: When a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't incapacitated and can see the creature. If the saving throw fails by 5 or more, the creature is instantly petrified. [...]  So if someone were to cast Minor Illusion to put a bag over Medusa's head, would other player/creatures who do not realize that the bag is an illusion be affected by the gaze if they were looking in the direction of it? Does the illusion block the effect? Requested value 'CheckBoxes' was not found There's a weird thing happening with my Windows application. I have created a Sharepoint utility to automate the creation of content-types and site-columns. At a point where I try to get my available fields, I get this error:     An unhandled exception of type 'Microsoft.SharePoint.Client.ServerException' occurred in Microsoft.SharePoint.Client.Runtime.dll   Additional information: Requested value 'CheckBoxes' was not found.   It is throwing an exception at:  ClientContext clientContxt = new ClientContext(clientContext.Url);         clientContext.Credentials = clientContext.Credentials;         Web web = clientContext.Web;          clientContext.Load(web, w =&gt; w.AvailableFields);         clientContext.ExecuteQuery(); //**exception thrown here**   Kindly please suggest what is wrong here? Which is correct: "the below information" or "the information below"? I frequently see statements that refer to something later in the text that use a phrase such as "the below information". Is it more correct instead to say "the information below" (or "the following information")? Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing