Upper hybrid oscillation

In Bayesian probability, the Jeffreys prior, named after Harold Jeffreys, is a non-informative (objective) prior distribution on parameter space that is proportional to the square root of the determinant of the Fisher information:

$p\left({\vec {\theta }}\right)\propto {\sqrt {\det {\mathcal {I}}\left({\vec {\theta }}\right)}}.\,$

It has the key feature that it is invariant under reparameterization of the parameter vector ${\vec {\theta }}$. This makes it of special interest for use with scale parameters.

Reparameterization

One-parameter case

For an alternate parameterization $\varphi$ we can derive

$p(\varphi )\propto {\sqrt {I(\varphi )}}\,$

from

$p(\theta )\propto {\sqrt {I(\theta )}}\,$

using the change of variables theorem and the definition of Fisher information:

${\begin{aligned}p(\varphi )&=p(\theta )\left|{\frac {d\theta }{d\varphi }}\right|\propto {\sqrt {I(\theta )\left({\frac {d\theta }{d\varphi }}\right)^{2}}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\theta }}\right)^{2}\right]\left({\frac {d\theta }{d\varphi }}\right)^{2}}}\\&={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\theta }}{\frac {d\theta }{d\varphi }}\right)^{2}\right]}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d\ln L}{d\varphi }}\right)^{2}\right]}}={\sqrt {I(\varphi )}}.\end{aligned}}$

Multiple-parameter case

For an alternate parameterization ${\vec {\varphi }}$ we can derive

$p({\vec {\varphi }})\propto {\sqrt {\det I({\vec {\varphi }})}}\,$

from

$p({\vec {\theta }})\propto {\sqrt {\det I({\vec {\theta }})}}\,$

using the change of variables theorem, the definition of Fisher information, and that the product of determinants is the determinant of the matrix product:

${\begin{aligned}p({\vec {\varphi }})&=p({\vec {\theta }})\left|\det {\frac {\partial \theta _{i}}{\partial \varphi _{j}}}\right|\\&\propto {\sqrt {\det I({\vec {\theta }})\,{\det }^{2}{\frac {\partial \theta _{i}}{\partial \varphi _{j}}}}}\\&={\sqrt {\det {\frac {\partial \theta _{k}}{\partial \varphi _{i}}}\,\det \operatorname {E} \!\left[{\frac {\partial \ln L}{\partial \theta _{k}}}{\frac {\partial \ln L}{\partial \theta _{l}}}\right]\,\det {\frac {\partial \theta _{l}}{\partial \varphi _{j}}}}}\\&={\sqrt {\det \operatorname {E} \!\left[\sum _{k,l}{\frac {\partial \theta _{k}}{\partial \varphi _{i}}}{\frac {\partial \ln L}{\partial \theta _{k}}}{\frac {\partial \ln L}{\partial \theta _{l}}}{\frac {\partial \theta _{l}}{\partial \varphi _{j}}}\right]}}\\&={\sqrt {\det \operatorname {E} \!\left[{\frac {\partial \ln L}{\partial \varphi _{i}}}{\frac {\partial \ln L}{\partial \varphi _{j}}}\right]}}={\sqrt {\det I({\vec {\varphi }})}}.\end{aligned}}$

Attributes

From a practical and mathematical standpoint, a valid reason to use this non-informative prior instead of others, like the ones obtained through a limit in conjugate families of distributions, is that it is not dependent upon the set of parameter variables that is chosen to describe parameter space.

Sometimes the Jeffreys prior cannot be normalized, and is thus an improper prior. For example, the Jeffreys prior for the distribution mean is uniform over the entire real line in the case of a Gaussian distribution of known variance.

Use of the Jeffreys prior violates the strong version of the likelihood principle, which is accepted by many, but by no means all, statisticians. When using the Jeffreys prior, inferences about ${\vec {\theta }}$ depend not just on the probability of the observed data as a function of ${\vec {\theta }}$, but also on the universe of all possible experimental outcomes, as determined by the experimental design, because the Fisher information is computed from an expectation over the chosen universe. Accordingly, the Jeffreys prior, and hence the inferences made using it, may be different for two experiments involving the same ${\vec {\theta }}$ parameter even when the likelihood functions for the two experiments are the same—a violation of the strong likelihood principle.

Minimum description length

In the minimum description length approach to statistics the goal is to describe data as compactly as possible where the length of a description is measured in bits of the code used. For a parametric family of distributions one compares a code with the best code based on one of the distributions in the parameterized family. The main result is that in exponential families, asymptotically for large sample size, the code based on the distribution that is a mixture of the elements in the exponential family with the Jeffreys prior is optimal. This result holds if one restricts the parameter set to a compact subset in the interior of the full parameter space. If the full parameter is used a modified version of the result should be used.

Examples

The Jeffreys prior for a parameter (or a set of parameters) depends upon the statistical model.

Gaussian distribution with mean parameter

For the Gaussian distribution of the real value $x$

$f(x|\mu )={\frac {e^{-(x-\mu )^{2}/2\sigma ^{2}}}{\sqrt {2\pi \sigma ^{2}}}}$

with $\sigma$ fixed, the Jeffreys prior for the mean $\mu$ is

${\begin{aligned}p(\mu )&\propto {\sqrt {I(\mu )}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d}{d\mu }}\log f(x|\mu )\right)^{2}\right]}}={\sqrt {\operatorname {E} \!\left[\left({\frac {x-\mu }{\sigma ^{2}}}\right)^{2}\right]}}\\&={\sqrt {\int _{-\infty }^{+\infty }f(x|\mu )\left({\frac {x-\mu }{\sigma ^{2}}}\right)^{2}dx}}={\sqrt {\frac {\sigma ^{2}}{\sigma ^{4}}}}\propto 1.\end{aligned}}$

That is, the Jeffreys prior for $\mu$ does not depend upon $\mu$; it is the unnormalized uniform distribution on the real line — the distribution that is 1 (or some other fixed constant) for all points. This is an improper prior, and is, up to the choice of constant, the unique translation-invariant distribution on the reals (the Haar measure with respect to addition of reals), corresponding to the mean being a measure of location and translation-invariance corresponding to no information about location.

Gaussian distribution with standard deviation parameter

For the Gaussian distribution of the real value $x$

$f(x|\sigma )={\frac {e^{-(x-\mu )^{2}/2\sigma ^{2}}}{\sqrt {2\pi \sigma ^{2}}}},$

the Jeffreys prior for the standard deviation σ > 0 is

${\begin{aligned}p(\sigma )&\propto {\sqrt {I(\sigma )}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d}{d\sigma }}\log f(x|\sigma )\right)^{2}\right]}}={\sqrt {\operatorname {E} \!\left[\left({\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{3}}}\right)^{2}\right]}}\\&={\sqrt {\int _{-\infty }^{+\infty }f(x|\sigma )\left({\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{3}}}\right)^{2}dx}}={\sqrt {\frac {2}{\sigma ^{2}}}}\propto {\frac {1}{\sigma }}.\end{aligned}}$

Equivalently, the Jeffreys prior for log σ² (or log σ) is the unnormalized uniform distribution on the real line, and thus this distribution is also known as the Template:Visible anchor. It is the unique (up to a multiple) prior (on the positive reals) that is scale-invariant (the Haar measure with respect to multiplication of positive reals), corresponding to the standard deviation being a measure of scale and scale-invariance corresponding to no information about scale. As with the uniform distribution on the reals, it is an improper prior.

Poisson distribution with rate parameter

For the Poisson distribution of the non-negative integer $n$,

$f(n|\lambda )=e^{-\lambda }{\frac {\lambda ^{n}}{n!}},$

the Jeffreys prior for the rate parameter λ ≥ 0 is

${\begin{aligned}p(\lambda )&\propto {\sqrt {I(\lambda )}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d}{d\lambda }}\log f(x|\lambda )\right)^{2}\right]}}={\sqrt {\operatorname {E} \!\left[\left({\frac {n-\lambda }{\lambda }}\right)^{2}\right]}}\\&={\sqrt {\sum _{n=0}^{+\infty }f(n|\lambda )\left({\frac {n-\lambda }{\lambda }}\right)^{2}}}={\sqrt {\frac {1}{\lambda }}}.\end{aligned}}$

Equivalently, the Jeffreys prior for ${\sqrt {\lambda }}$ is the unnormalized uniform distribution on the non-negative real line.

Bernoulli trial

For a coin that is "heads" with probability γ ∈ [0,1] and is "tails" with probability 1 − γ, for a given (H,T) ∈ {(0,1), (1,0)} the probability is $\gamma ^{H}(1-\gamma )^{T}$. The Jeffreys prior for the parameter $\gamma$ is

${\begin{aligned}p(\gamma )&\propto {\sqrt {I(\gamma )}}={\sqrt {\operatorname {E} \!\left[\left({\frac {d}{d\gamma }}\log f(x|\gamma )\right)^{2}\right]}}={\sqrt {\operatorname {E} \!\left[\left({\frac {H}{\gamma }}-{\frac {T}{1-\gamma }}\right)^{2}\right]}}\\&={\sqrt {\gamma \left({\frac {1}{\gamma }}-{\frac {0}{1-\gamma }}\right)^{2}+(1-\gamma )\left({\frac {0}{\gamma }}-{\frac {1}{1-\gamma }}\right)^{2}}}={\frac {1}{\sqrt {\gamma (1-\gamma )}}}\,.\end{aligned}}$

This is the arcsine distribution and is a beta distribution with $\alpha =\beta =1/2$. Furthermore, if $\gamma =\sin ^{2}(\theta )$ the Jeffreys prior for $\theta$ is uniform in the interval $[0,\pi /2]$. Equivalently, $\theta$ is uniform on the whole circle $[0,2\pi ]$.

N-sided die with biased probabilities

Similarly, for a throw of an $N$-sided die with outcome probabilities ${\vec {\gamma }}=(\gamma _{1},\ldots ,\gamma _{N})$, each non-negative and satisfying $\sum _{i=1}^{N}\gamma _{i}=1$, the Jeffreys prior for ${\vec {\gamma }}$ is the Dirichlet distribution with all (alpha) parameters set to $1/2$. In particular, if we write $\gamma _{i}={\phi _{i}}^{2}$ for each $i$, then the Jeffreys prior for ${\vec {\phi }}$ is uniform on the (N–1)-dimensional unit sphere (i.e., it is uniform on the surface of an N-dimensional unit ball).

References

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534