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# Maximum Likelihood and De Finetti's Theorem

I have a question about whether it is possible to use De Finetti's representation theorem for maximum likelihood estimation.

De Finnetti's theorem states that for any exchangable infinite sequence of random variables $(X_1,...,X_n,...)$, we can represent the probability distribution $f(X_1,...,X_n)$ as a mixture $$f(x_1,...,x_n) = \int \prod_{i=1}^n g(x_i | H) dP(H).$$

where the variables $x_1,...,x_n$ are conditionally independent given $H$ and $P(H)$ is a probability distribution over $H$.

I would like to theoretically show that $P(H)$ can be estimated if we observe a matrix $(x_{ij})_{i,j=1}^n$ where each row $(x_{i1},...,x_{in})$ is drawn independently from the distribution $f(x_1,...,x_n)$.

The idea would be to do a two step estimation:

• For each i, estimate $H_i =_{def} argmax_H \prod_{i=1}^n g(x_i | H)$
• Once we observe $H_1,H_2,H_3,...$ use these to estimate $P^* = argmax_P \prod_{i=1}^n Prob(H_i | P).$

It seems that for this idea to work, each $H_i$ would have to live in a compact set (otherwise, maximum likelihood estimation in the first step does not necessarily work). But since $H$ is drawn from $P(H)$ and $P$ does not necessarily have a compact support, this won't work.

Is there a way to parametrize $P$ to ensure it has compact support?


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# What are some examples of the pluperfect tense?

May 18, 2016

John $\textcolor{b l u e}{\text{had left}}$ for school before his mother got up.
By the time the lullaby finished the baby $\textcolor{b l u e}{\text{had fallen}}$ asleep.

#### Explanation:

The pluperfect tense is also called the past perfect tense.

A perfect tense is one that indicates an action is completed before some other event in time.
The past perfect (or pluperfect) indicates an action completed before some event in the past.
$\textcolor{w h i t e}{\text{XXX}}$I $\textcolor{b l u e}{\text{had been awakened}}$ long before the sun had risen.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Form comparison:

The present perfect indicates an action completed before the present moment.
$\textcolor{w h i t e}{\text{XXX}}$We already $\textcolor{g r e e n}{\text{have walked}}$ ten miles; let's rest.

The future perfect indicates an action which will be completed before some future event.
$\textcolor{w h i t e}{\text{XXX}}$Mary $\textcolor{g r e e n}{\text{will have made}}$ supper before Tom gets home from work this evening.

format: plain


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# How to integrate a bullet physics engine with an ode

Currently, I am in a situation where one of my sensors performs better when using bullet engine. When used under ode the Real Time Factor sinks to 0.05.

I would like to have this sensor in another simulation. However, that simulation seems to only work with ode, otherwise, it would show:

[Err] [BulletHingeJoint.cc:356] Joint must be created before getting low stop

[Err] [BulletHingeJoint.cc:345] Joint must be created before getting upper limit

[Err] [BulletHingeJoint.cc:442] Joint must be created before getting friction

This same simulation uses gazebo_ros_control, gazebo_ros_skid_steer_drive, and a Velodyne. (I don't know if any of those plugins makes the use of bullet engine impeditive).

My question is maybe how to integrate both worlds, bullet and ode. Or whether I should try to improve the sensors plugin to actually work with ode physics.

edit retag close merge delete


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# If a spring is cut in half, does the period change?

by clairez93
Tags: period, spring
 P: 114 1. The problem statement, all variables and given/known data A simple harmonic oscillator consists of a mass m and an ideal spring with spring constant k. Particle oscillates as shown in (i) with preriod T. If the spring is cut in half and used with the same particle, as shown in (ii), the period will be. A) 2T B) $$\sqrt{2}$$T C) T/$$\sqrt{2}$$ D) T E) T/2 2. Relevant equations T = 2$$\pi$$$$\sqrt{m/k}$$ 3. The attempt at a solution I figured that since the length of the spring isn't relevant to the period, the period would stay the same. The answer provided is C. I'm not sure why. Attached Thumbnails
 P: 114 Oh I see. So the first spring: $$k = mg/x$$ Thus, $$T = 2\pi\sqrt{x/g}$$ So in the second spring: $$k = 2mg/x$$ Thus, $$T' = 2\pi\sqrt{x/2g} = T/\sqrt{2}$$ Is this correct?


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# Change of measurement units

1. Oct 27, 2005

### asdf1

how do you change the measurement unit for an electron mass from kg to MeV/c^2?

2. Oct 27, 2005

### andrevdh

use Einstein's famous equation
$$E=mc^2$$
and use the conversion for joules per electron volt.

3. Oct 27, 2005

### asdf1

thanks!!! :)

4. Oct 28, 2005

### andrevdh

It is a pleasure.

5. Oct 30, 2005

### asdf1

@@a i'm still having trouble doing the calculations~
could you write out the whole equation to transfer the mass of an electron from kg to Mev?

6. Oct 30, 2005

### big man

You know the mass of the electron and you know the speed of light. Just plug those into $$E=mc^2$$. You will have an energy value in terms of joules. Since you also know that one electronvolt is equivalent to $$1.602*10^-^1^9$$ joules, you can find the energy value in terms of electronvolts instead. Rearrange the original expression and you can reexpress the mass in terms of $$MeV/c^2$$.

Last edited: Oct 31, 2005
7. Oct 31, 2005

### andrevdh

It is amazing how much confusion such a deceptively simple formula can generate. First calculate the rest mass of an electron:
$$\begin{split*} E_e = m_ec^2 \\ \ = 9.11 \times 10^{-31} \times (3.0 \times 10^8)^2 \\ \ = 8.2 \times 10^{-14} \ joule \\ \ = 512 \ keV \\ \ = 0.512 \ MeV \\ \ m_ec^2=0.512 \ MeV\end{split*}$$
Now get the mass in the "strange" new units:
$$m_e=0.512 \ MeV/{c^2}$$

Last edited: Nov 1, 2005
8. Oct 31, 2005

### asdf1

thank you very much!!! :)


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Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion.(English)Zbl 1351.60112

Summary: The explosion probability before time $$t$$ of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at $$\infty$$.

MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35K59 Quasilinear parabolic equations 60K99 Special processes
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# 4.4 Graphs of logarithmic functions  (Page 3/8)

 Page 3 / 8

Given a logarithmic function with the form $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right),$ graph the function.

1. Draw and label the vertical asymptote, $\text{\hspace{0.17em}}x=0.$
2. Plot the x- intercept, $\text{\hspace{0.17em}}\left(1,0\right).$
3. Plot the key point $\text{\hspace{0.17em}}\left(b,1\right).$
4. Draw a smooth curve through the points.
5. State the domain, $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range, $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote, $\text{\hspace{0.17em}}x=0.$

## Graphing a logarithmic function with the form f ( x ) = log b ( x ).

Graph $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{5}\left(x\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

Before graphing, identify the behavior and key points for the graph.

• Since $\text{\hspace{0.17em}}b=5\text{\hspace{0.17em}}$ is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote $\text{\hspace{0.17em}}x=0,$ and the right tail will increase slowly without bound.
• The x -intercept is $\text{\hspace{0.17em}}\left(1,0\right).$
• The key point $\text{\hspace{0.17em}}\left(5,1\right)\text{\hspace{0.17em}}$ is on the graph.
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see [link] ).

The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

Graph $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{\frac{1}{5}}\left(x\right).\text{\hspace{0.17em}}$ State the domain, range, and asymptote.

The domain is $\text{\hspace{0.17em}}\left(0,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=0.$

## Graphing transformations of logarithmic functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ without loss of shape.

## Graphing a horizontal shift of f ( x ) = log b ( x )

When a constant $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is added to the input of the parent function $\text{\hspace{0.17em}}f\left(x\right)=lo{g}_{b}\left(x\right),$ the result is a horizontal shift     $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units in the opposite direction of the sign on $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ To visualize horizontal shifts, we can observe the general graph of the parent function $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ and for $\text{\hspace{0.17em}}c>0\text{\hspace{0.17em}}$ alongside the shift left, $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{b}\left(x+c\right),$ and the shift right, $\text{\hspace{0.17em}}h\left(x\right)={\mathrm{log}}_{b}\left(x-c\right).$ See [link] .

## Horizontal shifts of the parent function y = log b ( x )

For any constant $\text{\hspace{0.17em}}c,$ the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)$

• shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ left $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}c>0.$
• shifts the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ right $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units if $\text{\hspace{0.17em}}c<0.$
• has the vertical asymptote $\text{\hspace{0.17em}}x=-c.$
• has domain $\text{\hspace{0.17em}}\left(-c,\infty \right).$
• has range $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

Given a logarithmic function with the form $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right),$ graph the translation.

1. Identify the horizontal shift:
1. If $\text{\hspace{0.17em}}c>0,$ shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ left $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units.
2. If $\text{\hspace{0.17em}}c<0,$ shift the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ right $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units.
2. Draw the vertical asymptote $\text{\hspace{0.17em}}x=-c.$
3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ from the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ coordinate.
4. Label the three points.
5. The Domain is $\text{\hspace{0.17em}}\left(-c,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=-c.$

## Graphing a horizontal shift of the parent function y = log b ( x )

Sketch the horizontal shift $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x-2\right)\text{\hspace{0.17em}}$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Since the function is $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x-2\right),$ we notice $\text{\hspace{0.17em}}x+\left(-2\right)=x–2.$

Thus $\text{\hspace{0.17em}}c=-2,$ so $\text{\hspace{0.17em}}c<0.\text{\hspace{0.17em}}$ This means we will shift the function $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{3}\left(x\right)\text{\hspace{0.17em}}$ right 2 units.

The vertical asymptote is $\text{\hspace{0.17em}}x=-\left(-2\right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=2.$

Consider the three key points from the parent function, $\text{\hspace{0.17em}}\left(\frac{1}{3},-1\right),$ $\left(1,0\right),$ and $\text{\hspace{0.17em}}\left(3,1\right).$

The new coordinates are found by adding 2 to the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ coordinates.

Label the points $\text{\hspace{0.17em}}\left(\frac{7}{3},-1\right),$ $\left(3,0\right),$ and $\text{\hspace{0.17em}}\left(5,1\right).$

The domain is $\text{\hspace{0.17em}}\left(2,\infty \right),$ the range is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ and the vertical asymptote is $\text{\hspace{0.17em}}x=2.$

How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture


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905 views

### Sum of N random variables from the same distributions [duplicate]

Given $n$ independent random variables from the same distribution, how to obtain the distribution of their sum? For example, the distribution of $n$ normal distribution is $N(n\mu, n\delta^2)$. ...
If I have two independent discrete random variables, say, $$X \in \{1,3,10,20\}$$ and $$Y \in \{2,3,5,9,11,15\}$$ and let $$Z = X + Y$$ be the sum of two variables. Also, each value taken by ...


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SEARCH HOME
 Math Central Quandaries & Queries
 Question from Nina: Change 253 in base 6 to a number in base 3

Hi Nina,

I would first convert $253_6$ to base ten and then convert this base ten number to base three.

$253_6 = 2 \times 6^2 + 5 \times 6^1 + 3 \times 6^0 = 72 + 30 + 3 = 105.$

Can you now convert 105 to base three?

Write back if you ned more assistance,
Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.


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POSITION PAPER

# Challenges to curing primary brain tumours

## Abstract

Despite decades of research, brain tumours remain among the deadliest of all forms of cancer. The ability of these tumours to resist almost all conventional and novel treatments relates, in part, to the unique cell-intrinsic and microenvironmental properties of neural tissues. In an attempt to encourage progress in our understanding and ability to successfully treat patients with brain tumours, Cancer Research UK convened an international panel of clinicians and laboratory-based scientists to identify challenges that must be overcome if we are to cure all patients with a brain tumour. The seven key challenges summarized in this Position Paper are intended to serve as foci for future research and investment.

## Introduction

Brain tumours are among the most feared of all forms of cancer. More than two-thirds of adults diagnosed with glioblastoma — the most aggressive type of brain cancer — will die within 2 years of diagnosis1,2. Brain cancers are also the most common and most lethal of all paediatric solid tumours3. Furthermore, children with these tumours who survive and enter adulthood will often be affected by the long-term consequences of exposing the developing brain to medical interventions, including surgery, radiotherapy and/or chemotherapy4,5.

Brain tumours have proved challenging to treat, largely owing to the biological characteristics of these cancers, which often conspire to limit progress. First, by infiltrating one of the body’s most crucial organs, these tumours are often located beyond the reach of even the most skilled neurosurgeon. These tumours are also located behind the blood–brain barrier (BBB) — a system of tight junctions and transport proteins that protect delicate neural tissues from exposure to factors in the general circulation, thus also impeding exposure to systemic chemotherapy6,7. Furthermore, the unique developmental, genetic, epigenetic and microenvironmental features of the brain frequently render these cancers resistant to conventional and novel treatments alike8,9,10. These challenges are compounded by the rarity of brain tumours relative to many other forms of cancer, which limits the level of funding and interest from the pharmaceutical industry and attracts a relatively small and fragmented research community.

To begin to address these issues and improve the outcomes of patients with brain tumours, Cancer Research UK (CRUK) convened an international panel of brain cancer researchers with interests in neurobiology, preclinical tumour modelling, genomics, pharmacology, drug discovery and/or development, neuropathology, neurosurgery, imaging, radiotherapy and medical oncology, with the task of identifying the most important challenges that must be overcome if we are to eventually be in the position to cure all patients with a brain tumour. In this Position Paper, we summarize seven key challenges identified by the panel that should serve as the foci for future research and investment (Fig. 1). Each of these challenges is worthy of extensive discussion and review, which are beyond the scope of this manuscript. Therefore, we highlight these challenges as a ‘call-to-arms’, summarizing the nature and importance of each challenge rather than providing an exhaustive review of the current level of understanding.

## Challenge 1: redesign research pipeline

Clinical trials have yet to reveal an effective therapy for most brain tumours. This harsh reality stems, in part, from an incomplete understanding of brain tumour biology and the existence of a disconnect between preclinical drug development and rigorous testing in the clinic. Each element of the brain tumour research pipeline, from basic neurobiology to clinical trials, requires careful scrutiny and increased investment, although the development of an overarching strategy that facilitates and promotes interdisciplinary research is equally important (Fig. 1). This strategy would bring an end to the previous ‘siloed’ organization of working practices, in which basic and clinical researchers performed their studies independently and collaborated only when laboratory research was judged to be ready for the clinic or when the laboratory is engaged to understand the reasons why a promising drug failed to achieve the expected level of efficacy in clinical trials. Much deeper, longitudinal collaboration is essential in order to drive progress as rapidly as possible.

Nascent attempts to unify the brain tumour research pipeline are underway. The international paediatric brain tumour community has demonstrated remarkable levels of collaborative activity over many years and is now working to discover and define the genomic subtypes of paediatric brain tumours and form multidisciplinary teams in order to design preclinical studies that better inform the design of clinical trials11,12. However, the community must go further, by forming international collaborations that extend beyond the borders of traditional research disciplines and by engaging the entire breadth of expertise available within the biological and physical sciences (see Challenge 2). Such congregations of experts could provide a more comprehensive understanding of the workings of the brain and how these processes are subverted during malignant transformation.

The community should also explore innovative methods of designing and delivering clinical trials. Interest among the brain tumour research community in the use of adaptive trial designs is particularly welcomed13,14. Such trials could provide prospective opportunities to modify ongoing studies and integrate the investigation of new hypotheses as data are accumulated and analysed (Fig. 2). This approach has distinct advantages over the less flexible, traditional trial designs, which are typically designed to test one or two primary hypotheses, typically in heavily pretreated patients over a number of years. However, several hurdles must be overcome before novel trial designs can be implemented successfully, including identifying a cadre of rational therapies to feed into these new trials; developing robust biomarker-based patient selection criteria; constructing systems that enable the provision of real-time, in-trial, biomarker and end point data; and formulating optimal mechanisms of generating and sharing complex research data among centres worldwide (Fig. 2). Notwithstanding these challenges, improving the current approach to clinical trials is an urgent objective for the community. In the context of small, genomically defined patient subgroups, international collaborative clinical trials are crucial. A comprehensive review of the current logistical, regulatory and financial hurdles that inhibit the initiation of such trials is therefore warranted.

Greater integration of research disciplines will also require a change in the current research culture. The principal metrics used to reward success in most academic environments are focused on competitive activities, such as publication in high-impact journals and the acquisition of grant funding. These metrics have some value in enabling the assessment and quantification of the extent of innovative thought and discovery, although they arguably also work against the development of the deeper levels of collaborative activity that will ultimately be required to cure patients with brain tumours. This change in academic culture must go beyond a cursory recognition of the value of ‘team science’. Rather, akin to the complex and multidisciplinary groups that operate in the pharmaceutical industry, new reward and promotion structures should be developed for laboratory and clinical researchers alike that inspire and encourage their participation in collaborative academic brain tumour research.

## Challenge 2: use neuroscience research

We now have a considerable level of understanding of the cellular hierarchies and signals that generate and maintain brain activity in the absence of cancer15. Specialized glial cells serve as neural stem cells (NSCs) in both the embryonic and adult brain, in which they reside in neurogenic niches and produce daughter cells that differentiate into neurons, astrocytes and oligodendrocytes (Fig. 1). These niches also include supporting cells and secreted factors that are critical for the close regulation of both NSC self-renewal and the early stages of daughter cell differentiation: an intimate relationship between NSCs and blood vessels seems to be especially important in the neurogenic niche16.

Several lines of evidence suggest that brain tumours arise within, or are driven by, cells that recapitulate the neurogenic niche (Fig. 3). Stem-like cells have been isolated from paediatric and adult brain tumours17,18,19, and brain tumours have been shown to contain malignant niches that seem to recapitulate the micro-architectural features and signalling properties of the nonmalignant neurogenic niche20,21. Recurrent mutations in brain tumours can also perturb the signalling pathways that regulate brain development12, and subgroups of brain tumours have been shown to contain the transcriptomes and epigenomes of their originating parental NSCs that generated these tumours in the mouse brain22,23,24,25,26,27,28,29.

Evidence that brain tumours are a consequence of aberrant brain development or repair underscores the importance of closer integration between neurobiological and cancer research; however, clear examples of such collaborations are rare. Only ~1% (US$260 million) of the$21 billion invested in neuroscience research by the US NIH in 2018 is aimed specifically at brain tumour research. Furthermore, although the PubMed database currently contains ~320,000 and 198,000 articles on neuroscience and brain tumour research, respectively, only 6,000 (~2% and ~3.5%, respectively) overlap. These observations are crude indicators of the level of interaction, but they suggest that the brain tumour research community is not capitalizing fully on the available research data. An increased focus on areas of common interest, such as immune dysregulation in patients with cancer and in those with dementia, should drive joint funding initiatives and ultimately result in progress for patients30,31.

## Challenge 3: understand the TME

A thorough understanding of the properties and functions of the tumour microenvironment (TME) is required in order to obtain a complete understanding of brain tumour biology and treatment9,32 (Fig. 4). As discussed with regard to Challenge 2, closer interactions with scientists investigating nonmalignant brain diseases will be especially important to this effort. Over many decades, research groups studying neurodegeneration and other nonmalignant brain diseases have generated a considerable level of understanding of the biology of the extracellular matrix, of specific brain-resident cellular populations (such as microglia, astrocytes and neurons) and of blood vessels (including those forming the BBB) that comprise the TME.

The immune and vascular components of the TME are likely to have particular relevance for improving the treatment of brain tumours6,33. For example, the discovery of the immune or glymphatic system of the brain and evidence that immunotherapies are effective treatments for an increasing number of cancers have prompted studies of the efficacy of such therapies in patients with brain tumours31,34,35,36. The major myeloid cell populations of the brain TME comprise macrophages and microglia — together referred to as tumour-associated macrophages (TAMs)37,38. Glioblastoma cells stimulate TAMs to produce immunosuppressive, tumour-promoting cytokines and enhanced apoptosis of T cells39. Glioblastoma cells can also inhibit the production of immunostimulatory cytokines and activate the recruitment of regulatory T cells, thus inhibiting antitumour immune responses40. These data suggest that inhibiting the activity of TAMs might provide therapeutic benefit in patients with brain tumours, although the clinical reality is likely to be more complex. Research conducted over the past decade suggests that re-educating TAMs to adopt phenotypes that are likely to prevent or inhibit the progression of brain tumours might be more effective than depleting all TAM populations32,41. A range of additional immune-based therapies are also currently in development, including vaccines, cellular therapies (such as chimeric antigen receptor (CAR) T cells) and immune-checkpoint inhibitors9,31,34,40,41,42.

Regardless of the type of immunotherapy that is being developed, several key challenges must be addressed if we are to deploy these agents effectively in the treatment of brain tumours (Fig. 4). Once again, a better understanding of tumour biology will be important to the selection of patients who are most likely to benefit from therapies and to improve the level of understanding of treatment-related toxicities (see Challenge 1). Addressing these challenges will not be straightforward for immunotherapies, not least because even the imprecise predictors of a response to immune-checkpoint inhibition in other cancers, such as tumour mutational burden and/or DNA methylation, remain to be proved in patients with brain tumours43. Furthermore, the identity of the optimal targets of other immunotherapies, such as CAR T cells, which might be patient specific, also remains unclear. Other important and unanswered questions surrounding the use of immunotherapies in patients with brain tumours include the degree to which the BBB will impede the access of cellular and molecular immunotherapies to tumours; whether immunotherapies will synergize with or counteract the effects of existing treatments, such as temozolomide, steroids and/or radiotherapy; and the type and severity of neurological adverse effects associated with immunotherapies and how best to prevent, monitor and manage these. Thus, although immunotherapy holds great promise as a new therapeutic approach for the treatment of patients with brain tumours, many aspects of the unique immunological and microenvironmental status of the brain present challenges that need to be overcome in order to develop and validate these therapies efficiently9.

As previously alluded to, the BBB is a major hurdle to the successful treatment of brain tumours44. This dynamic structure of tight junctions and molecular pumps comprises endothelial, ependymal and tanycytic cells that communicate with other brain cells and with circulating immune cells45,46,47,48. Single-cell transcriptomics has revealed a remarkable level of regional heterogeneity along the arteriovenous axis of the BBB, including a seamless continuum of endothelial cells versus a punctuated continuum of mural cells49. These insights could prove important in determining the functionality of the BBB in tumours located in different regions of the brain.

Discrete subtypes of brain tumours might also alter the BBB (Fig. 4). In this regard, the WNT subtype of medulloblastomas, which are highly sensitive to treatment, has been shown to secrete WNT antagonists that ‘re-educate’ the BBB, thus causing it to adopt a phenotype that is highly permeable to systemic chemotherapy6. Similarly, the therapeutic response of gliomas to temozolomide might be improved by use of this agent in combination with small-molecule inhibitors of WNT signalling33. Greater research efforts should be invested in cataloguing the integrity and other alterations of the BBB in each brain tumour subtype. This information might enable the more widespread use of systemic therapies that are currently regarded as BBB non-penetrating in those tumours that lack an intact BBB and support the use of WNT inhibitors and other approaches to enhance the delivery of chemotherapy to those tumours in which the BBB is intact50.

## Challenge 4: develop preclinical models

The limited progress made in the treatment of brain tumours relates, in part, to the inaccuracy of preclinical models that have thus far failed to consistently show responses to agents with therapeutic activity in patients. Preclinical drug development pipelines that enable the accurate prediction of effective drugs are especially important for rare cancers, including brain tumours, owing to the low numbers of patients available for participation in clinical research12,51. Current pipelines are limited in their ability to identify new, more effective treatments of brain tumours for several reasons: they typically involve poorly characterized in vitro systems or subcutaneous tumour xenografts, rather than more accurate orthotopic models of brain tumours; they do not enable the assessment of benefit from new treatments in terms of survival relative to that provided by the existing combinations of neurosurgery, radiotherapy and/or chemotherapy; and finally, they usually lack rigorous characterization of other clinically important features, such as those of the BBB. Thus, curing all patients with a brain tumour will require a new approach that leverages an improved understanding of neuroscience and brain tumour biology to develop and deploy more accurate preclinical models.

A better understanding of the processes that drive the development and progression of brain tumours should promote the development of more accurate genetically engineered mouse models (GEMMs) and patient-derived xenografts (PDXs). We recommend that the brain tumour community maintains a centralized catalogue of all existing and newly developed preclinical models that have been subjected to a minimum agreed set of rigorous histological, genomic and imaging evaluations. Models from this catalogue should be made readily available and would provide the community with universally accessible, validated models that closely mimic the characteristics of human tumours. Initiatives such as the ITCC Paediatric Preclinical Proof of Concept Platform are already underway, with the intention of creating such a resource.

In addition to minimum standards of model characterization, the community should also adopt minimum standards for preclinical studies, thus enabling more effective comparisons of the results of different studies (Fig. 5). Of particular note, although patients with brain tumours receive complex, multimodality therapy, mice in preclinical studies typically receive monotherapies that are rarely comparable to the standard of care for the malignancy that is being modelled. Thus, efforts to establish preclinical ‘mouse hospitals’, in which potential new therapies are tested in the context of combination with neurosurgery, fractionated radiotherapy and/or conventional chemotherapy, should become the new standard11. Researchers with access to such preclinical hospitals should also perform robust pharmacokinetic (with particular attention to BBB penetration), pharmacodynamic and biomarker (tumour, liquid biopsy and imaging) studies for novel agents, thus maximizing the speed and likelihood of success of clinical translation (Fig. 5).

The failure of many drugs to penetrate the BBB to any clinically meaningful extent is an important reason for the poor treatment responses of patients with brain tumours to systemic therapies52. Functional imaging studies in particular can provide evidence of early treatment responses, although a failure to detect a response cannot distinguish between a drug that is ineffective and one that simply does not cross the BBB. Contrast enhancement of tumours on MRI, using various agents, can reveal disruption of the BBB, for example, following ultrasonography-mediated opening of the BBB53; however, this approach does not provide direct evidence of drug delivery. More direct evidence can be provided by observing tumour accumulation on PET imaging of drugs labelled with a positron-emitting isotope, such as [methyl-11C] temozolomide in patients with gliomas54. However, conjugating all new drugs with such isotopes — particularly with 11C, which will not perturb drug chemistry and thus drug–target interactions but has a sufficiently short half-life (20.3 minutes) — would be a formidable undertaking. Thus, the community should agree on acceptable preclinical and clinical standards for demonstrating successful drug delivery to brain tumours.

In addition to being grown as PDXs, patient-derived brain tumour cells can be consistently and successfully encouraged to propagate under NSC culture conditions as organoids55,56. These short-term culture systems have enormous potential for studying interactions between different cells and drug sensitivities and for interrogation using genome-wide interference and/or editing techniques20,55,56,57,58. Preclinical pipelines that deploy these novel screening approaches, together with the use of matched GEMMs and PDX models (ideally in humanized mice), should provide a more comprehensive and more accurate means of delivering optimized drugs and their associated biomarkers into clinical trials11.

Preclinical and clinical research efforts to evaluate the value of new treatments, relative to that of the conventional treatments of brain tumours, should also consider re-evaluating some long-established treatment approaches that might be of questionable value. For example, corticosteroids, such as dexamethasone, continue to be used perioperatively and might be continued throughout subsequent treatment. The original reasons for using dexamethasone in this way probably reflect antiproliferative activity that confers protection of nonmalignant cells from radiotherapy-induced and chemotherapy-induced genotoxic stress. However, both clinical data and preclinical data from mouse models suggest that corticosteroids might decrease the effectiveness of treatments and shorten the survival durations of patients with glioblastoma. Thus, greater efforts should be invested in identifying alternative agents, on the basis of robust preclinical and clinical evidence, such as anti-VEGF agents for the treatment of brain tumour-associated oedema59.

## Challenge 5: drugging complex cancers

The so-called omics revolution is providing a comprehensive view of the genomes, epigenomes and transcriptomes of brain tumours; this additional information has enabled the segregation of histologically similar subsets of tumours into clinically and molecularly distinct subgroups. However, despite a few notable exceptions, such as inhibitors of mTOR or BRAF in selected patients60,61, genomics has yet to deliver on the promise of identifying new therapeutic targets in patients with brain tumours. Discovering new treatments of brain tumours will require a much deeper level of understanding of the biology of potential drug targets, combined with the application of cutting-edge drug discovery approaches to inhibit these targets.

Progress in drug development will not come easily. Alterations identified in brain tumours in the past decade include poorly understood targets that are frequently regarded as undruggable, including amplification of genes encoding transcription factors, such as MYC and MYCN; gene translocations, such as FGFR–TACC and C11OF95–RELA27,62; oncogene ‘enhancer hijacking’, for example, by GFI1 (ref.63); and mutated histones, such as H3F3A and HIST1H3B variants64,65. Furthermore, these alterations exist in a shifting landscape of tumour heterogeneity, in which the populations of clones carrying each alteration fluctuate as the disease evolves in response to therapy66,67. This clonal evolution means that the proportion of tumour cells carrying a specific drug target can vary markedly during the course of disease, thus complicating efforts to target specific alterations. This heterogeneity can be measured in terms of mutational load, epigenetic alterations, rewiring of transcriptional circuits and microenvironmental influences and is widely considered to underpin the diversity of responses to treatment. The serial analysis of tumour biology using additional measures, such as advanced metabolic imaging and analyses involving serial sampling of circulating tumour DNA, is discussed with regard to Challenge 6 and might provide a more complete and accurate picture of tumour evolution.

Although daunting, the data provided by these novel approaches reflect an important principle that must be adopted by brain tumour researchers and clinicians — the notion that brain tumours are monogenetic and monoclonal must be dispelled. Comprehensive mapping of genomic evolution before, during and following treatment, including the use of single-cell sequencing approaches, should be used to better identify and prioritize treatment targets68,69. This objective will be best achieved in the context of prospective clinical trials, in which molecular imaging and other diagnostic approaches can be used to provide real-time assessments of disease evolution70. Prioritized targets should also be interrogated using approaches, such as deep mutational scanning, that can reveal information on the intrinsic properties of proteins and their function and how this changes with mutation71. Reverse translation efforts involving detailed mechanistic studies after a clinical trial of appropriately prioritized targets should include functional studies that enable the accurate definition of target function in tumours. Such serial analyses will likely be complex and resource-intensive and are probably best organized as part of a community-wide approach. This approach might be promoted in several ways: development of international tumour subtype-specific trials; the widespread availability of brain tumour stem cell banks (such as the UK glioma cellular genetics resource (GCGR), the human glioblastoma cell culture (HGCC) resource in Sweden and the Stand Up to Cancer (SU2C) Canada Cancer Stem Cell Dream Team); established efforts to understand complex cancer targets (such as the US National Cancer Institute RAS Initiative); and closer collaborations with industry.

## Challenge 6: precision medicine

The current classifications of brain tumours are based primarily on microscopic morphology and immunohistochemistry72. This approach enables the broad characterization of tumour type and a certain level of prognostication, although it fails to capitalize on the wealth of clinically relevant insights produced by the genomic subtyping of brain tumours. This dilemma is exemplified by medulloblastoma and ependymoma, both of which consist of clinically relevant subtypes with discrete cells of origin, driver mutations and global transcriptomic and epigenomic patterns that are yet to be incorporated into the WHO classification of central nervous system tumours10,12,72,73,74,75.

Concerns that genome-wide classification tools might prove impractical for the routine diagnosis of brain tumours have been dispelled by the successful large-scale methylome subtyping of formalin-fixed, paraffin-embedded tumours75,76. Therefore, the challenge to the brain tumour community is not whether but how to deploy diagnostic genomic profiling. In the UK, the centralization of genomic profiling through the National Health Service provides an enormous opportunity to begin this process at a population level. In doing so, it will be important to note that not all brain tumours are created equally: paediatric medulloblastomas and ependymomas comprise robust subgroups with distinct characteristics. However, the degree of diagnostic separation of other brain tumours, including gliomas in adults, is less clear.

The genomic classification of brain tumours will increase the level of diagnostic precision that is currently achievable, although intratumoural heterogeneity and the often limited amounts of tumour material available for analysis are likely to confound the accurate diagnosis of some tumours68,77. Therefore, the research community should aim to urgently investigate the use of more advanced imaging techniques, such as 13C-hyperpolarized MRI, radiomics and sequencing of cell-free DNA obtained from plasma and/or cerebrospinal fluid, as ancillary approaches to the diagnosis and classification of brain tumours70,78,79,80. The challenge of validating and combining these new diagnostic modalities is enormous but also holds the potential to greatly improve the accuracy of brain tumour classification. Implementation of such approaches might also improve the assessment of novel therapies as part of the federal drug approval process. In this regard, the current standards, such as the Response Assessment in Neuro-Oncology criteria81, which describe complete and partial drug responses, are of limited value for assessing new treatments of slow-growing tumours, such as low-grade glioma. Early molecular and/or metabolic responses that are predictive of ultimate clinical benefit might, therefore, provide a more accurate means of evaluating the efficacy of treatments.

Once validated, new classification tools, such as those outlined above, must be adopted within the WHO guidelines in order to enable use by the wider community. A comprehensive strategy to achieve this goal should formally include genomic metrics, such as transcriptomic analysis, in WHO diagnosis and classification schemes; involve appropriate training of experienced and newly qualified pathologists in genomic methodologies; be accompanied by appropriate levels of investment in the resources (both capital and personnel) needed to deliver genomic assays routinely in diagnostic pathology labs; and incorporate standardized genomic characterization within both conventional and clinical trial management strategies. Histology will always have a place in the diagnosis of patients, although only when used in concert with advanced molecular and imaging-based diagnostics will this provide a comprehensive prediction of tumour behaviour.

In addition to developing advanced diagnostic approaches, we must also learn how best to integrate these rich data streams if they are to revolutionize the treatment of patients with cancer (Fig. 5). Continual, iterative and integrated analysis of these complementary and complex data streams should be made possible by machine learning, artificial intelligence and/or other mathematical and computational approaches. This will require a concerted effort to invent and deploy new analytical approaches and visualization technologies in order to generate a single flow of information that travels with a patient throughout his or her treatment journey, enabling better-informed management decisions and guiding the patient towards the maximum opportunity for cure (Fig. 6). Reporting the clinically relevant findings from such large-scale data sets to health-care professionals and patients will require a multidisciplinary approach, supported by appropriate information technology infrastructure, genetic counsellors, medical geneticists and clinical scientists. This longitudinal and integrated approach will also enable the development of the next generation of adaptive precision medicine clinical trials.

## Challenge 7: reduce treatment for some

Certain brain tumours, particularly those arising in children, can be cured with aggressive surgery, radiotherapy and chemotherapy, although these cures often carry a considerable cost, particularly for young children who have lifelong neurocognitive and endocrine adverse effects4,82. Reducing the intensity of treatment in children with other cancers, such as acute lymphoblastic leukaemia, has enabled a reduction in the incidence of adverse effects while maintaining high cure rates; however, this approach has proved challenging to implement in patients with brain tumours, the majority of whom have disease progression following surgery alone. Identifying which patients might be cured with less-intense adjuvant radiotherapy and chemotherapy and to what extent the intensity of these modalities can be safely reduced remains a major challenge.

Medulloblastomas provide a clear example of how a detailed understanding of tumour biology can guide adjustments of treatment intensity. Historical attempts to limit the damaging effects of craniospinal radiotherapy on the developing brain by reducing the radiation dose have failed, largely owing to an inadequate understanding of medulloblastoma biology, which precluded the accurate selection of patients with truly low-risk disease83,84,85. A breakthrough was achieved with the recognition that almost all patients with the WNT subtype of medulloblastoma are cured12. This discovery has led to a series of highly selective ongoing studies testing reduced-intensity radiotherapy in patients with this disease subtype (NCT02724579, NCT02066220, NCT02212574 and NCT01878617). Evidence that WNT-subtype tumours (but not other forms of medulloblastoma) lack a BBB and are therefore remarkably vulnerable to systemic chemotherapy provides an explanation for the curability of these tumours, thus opening further options for alternatives to radiation-based treatments6. Paediatric low-grade gliomas provide another example of how molecularly guided treatments (such as inhibitors of activated BRAF and MEK) are enabling reductions in radiation dose86,87. Studies involving patients with other brain tumours are likely to identify similar approaches for the rational de-escalation of therapy.

A comprehensive approach to preventing and ameliorating the adverse effects associated with long-term treatment should include prospective functional and patient-reported outcome measures, as well as innovative predictive assessments of risk. Novel ways of supplementing these data are also emerging in the form of genetic polymorphisms and somatic genomic characteristics that might predict the risk of subsequent intellectual disability and/or hearing loss88,89.

## Conclusions

The past 15 years have witnessed a revolution in our understanding of cancer. The integration of genomic and developmental biology has shown that morphologically similar cancers comprise discrete subtypes, driven by different genetic alterations, which likely arise from distinct cell lineages. These data help to explain why cancers once regarded as histologically homogeneous diseases have a discrepant range of characteristics. Improved understanding is also leading to the development of completely new treatment approaches for cancer, such as immunotherapies, and novel ways to test such therapies, such as adaptive trial designs. However, the successes achieved with these improvements have not occurred equally across all forms of cancer. Of particular note, the treatment of most childhood and adult brain tumours is at an impasse, with no new, more effective therapies being developed in the past 30 years. Thus, all available evidence suggests that the current preclinical and clinical research approaches to curing brain tumours are ineffective.

Herein, we have summarized seven major challenges to progress that must be overcome if we are to produce the sea change in brain tumour therapy that has long been awaited by patients and their families. Some of these challenges might seem to be self-evident; however, each will require a profound change in research and clinical practice and investment. In order to truly redesign the brain tumour research and treatment pipeline and to leverage the full spectrum of neuroscience research, modest improvements in the level of interdisciplinary collaborations among otherwise siloed research groups will not be sufficient. We envisage a much more substantial congregation of experts and infrastructure, focused on the task of curing brain tumours. This call reflects the need to practise brain tumour research differently and to identify, convene and support teams of scientists and clinicians who focus specifically and urgently on the need to improve the lives of patients with brain tumours. This approach must be different from the largely ad hoc and predominantly poorly structured brain tumour collaborations and programmes that currently exist in most academic centres. The harsh reality is that the current efforts at various universities and aligned clinical environments around the world have failed to adequately improve the understanding and treatment of patients with brain tumours. Nevertheless, this approach would probably be easiest to implement within the context of existing academic settings in order to engage with and closely incorporate disciplines not traditionally involved in brain tumour research, such as engineering, chemistry, physics and mathematics.

As we seek to better understand the role of the TME and aim to develop more predictive preclinical model systems, we need to take a long, hard look at the current model systems. Many of these clearly do not adequately reflect all the nonmalignant and malignant cell types that compose brain tumours and have failed to accurately predict treatments that will be effective for testing in clinical trials. Only systems that are proven to model clinically relevant aspects of tumour biology should be used as evidence to guide the initiation of clinical trials. Furthermore, through reverse translation, we should refine the use of or entirely remove models that fail to predict clinical efficacy.

Concerns relating to safety and patient acceptability limit the ability to obtain repeat biopsy samples from brain tumours in most patients, thus preventing longitudinal studies of tumour biology and treatment response and the deployment of a precision medicine approach. The use of liquid biopsies and advanced imaging approaches should be explored immediately to provide additional approaches to determine tumour response earlier in the course of treatment. Finally, the challenge most likely to yield patient benefit in the immediate future will be the use of advanced stratification approaches to reduce the risk of treatment-related toxicities, especially among children.

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## Acknowledgements

R.J.G. gratefully acknowledges financial support from the US NIH (grants P01CA96832 and R0CA1129541), Cancer Research UK, the Mathile Family Foundation, Cure Search and the Sohn Foundation.

### Reviewer information

Nature Reviews Clinical Oncology thanks D. Reardon and other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

Authors

### Contributions

All authors made substantial contributions to researching data for the article, to discussions of content and to writing the manuscript. R.J.G. reviewed and edited the manuscript before submission.

### Corresponding author

Correspondence to Richard J. Gilbertson.

## Ethics declarations

### Competing interests

P.W. and R.C. are employees of The Institute of Cancer Research (ICR), which has a commercial interest in a range of drug targets. The ICR operates a Rewards to Inventors scheme whereby employees of the ICR may receive financial benefit following commercial licensing of a project. P.W. is a consultant/scientific advisory board member for NextechInvest, Storm Therapeutics, Astex Pharmaceuticals and CV6 and holds stock in Chroma Therapeutics, NextInvest and Storm Therapeutics; he is also a Non-Executive Director of Storm Therapeutics and the Royal Marsden NHS Trust and a Director of the non-profit Chemical Probes Portal. R.C. is an advisor to Syncona Limited and holds equity in Celgene Corporation, e-Therapeutics and Monte Rosa Therapeutics. The other authors declare no competing interests.

### Publisher’s note

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Human glioblastoma cell culture (HGCC) resource: http://www.hgcc.se/#about

ITCC Paediatric Preclinical Proof of concept Platform: http://www.itccp4.eu

Stand Up to Cancer (SU2C) Canada Cancer Stem Cell Dream Team: http://www.standuptocancer.ca/en/dream_teams/view/cancer_stem_cell_dream_team

UK glioma cellular genetics resource (GCGR): https://www.ed.ac.uk/cancer-centre/research/s-pollard-group/edinburghbraincancer/about-us

US National Cancer Institute RAS Initiative: https://www.cancer.gov/research/key-initiatives/ras

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Aldape, K., Brindle, K.M., Chesler, L. et al. Challenges to curing primary brain tumours. Nat Rev Clin Oncol 16, 509–520 (2019). https://doi.org/10.1038/s41571-019-0177-5

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text:
Department of

# Mathematics

Seminar Calendar
for events the day of Saturday, October 11, 2014.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2014          October 2014          November 2014
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1  2  3  4  5  6             1  2  3  4                      1
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28 29 30               26 27 28 29 30 31      23 24 25 26 27 28 29
30

Saturday, October 11, 2014

1:00 pm in 239 Altgeld Hall,Saturday, October 11, 2014

#### Gathering for Gardner

Abstract: The Department of Mathematics will celebrate the 100th Gardner Day with a Mathematics Fair on October 11 from 1-4 p.m. in 239 Altgeld Hall. Events include math activities, puzzles, games, mathematical art and several short talks. Light refreshments will be served. Learn more http://www.math.uiuc.edu/~phierony/gardner100.html


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## Tag: aic

181 Is there any reason to prefer the AIC or BIC over the other? 2010-07-23T20:49:12.340

149 Algorithms for automatic model selection 2012-01-09T18:22:23.617

58 Do all interactions terms need their individual terms in regression model? 2012-05-04T02:10:29.487

55 What do the residuals in a logistic regression mean? 2010-08-09T07:32:32.767

44 AIC,BIC,CIC,DIC,EIC,FIC,GIC,HIC,IIC --- Can I use them interchangeably? 2014-02-16T14:11:45.150

33 What does the Akaike Information Criterion (AIC) score of a model mean? 2016-06-01T00:04:04.030

31 Negative values for AICc (corrected Akaike Information Criterion) 2010-07-22T10:11:15.210

27 Is it possible to calculate AIC and BIC for lasso regression models? 2012-04-04T03:33:22.513

25 What is the difference in what AIC and c-statistic (AUC) actually measure for model fit? 2013-03-04T04:40:20.520

23 Negative values for AIC in General Mixed Model 2014-02-01T12:14:46.577

22 What is the upside of treating a factor as random in a mixed model? 2014-08-14T18:20:37.657

22 Logistic Regression: Bernoulli vs. Binomial Response Variables 2015-03-30T20:25:41.043

21 AIC guidelines in model selection 2014-01-06T20:55:32.177

21 Model selection with Firth logistic regression 2014-03-01T18:24:16.793

20 Can AIC compare across different types of model? 2010-11-29T16:08:13.153

20 AIC or p-value: which one to choose for model selection? 2011-04-04T18:25:56.523

20 How should mixed effects models be compared and or validated? 2012-02-28T03:33:59.977

20 AIC & BIC number interpretation 2012-08-24T14:20:23.220

19 Meaning of 'number of parameters' in AIC 2014-06-16T12:16:33.513

17 Prerequisites for AIC model comparison 2013-01-28T19:55:21.987

15 Comparing non nested models with AIC 2014-09-26T22:14:03.793

15 When to stop refining a model? 2015-10-27T14:15:02.057

13 Understanding AIC and Schwarz criterion 2010-10-26T07:28:19.323

13 AIC/BIC: how many parameters does a permutation count for? 2013-01-30T01:15:07.707

13 Can the Burnham-Anderson book on multimodel inference be recommended? 2014-02-15T04:49:33.850

13 Paradox in model selection (AIC, BIC, to explain or to predict?) 2015-10-17T15:50:42.493

12 Testing the difference in AIC of two non-nested models 2011-03-21T04:57:51.157

12 AIC for non-nested models: normalizing constant 2012-09-20T12:40:32.583

12 Problem with comparing GLM models having a different link function 2013-01-02T16:04:05.927

12 Comparing AIC of a model and its log-transformed version 2013-06-10T11:03:34.667

12 AIC, BIC and GCV: what is best for making decision in penalized regression methods? 2014-07-20T03:44:52.977

12 Does BIC try to find a true model? 2016-04-03T08:54:28.580

12 Are $\frac18$-th root transformations recommended? 2017-11-23T05:40:30.633

11 Comparing regression models on count data 2010-12-16T18:04:08.740

11 Non-nested model selection 2012-01-01T12:15:03.760

11 Compare fits of model to transformed and untransformed response 2012-03-15T03:37:29.403

11 Why applying model selection using AIC gives me non-significant p-values for the variables 2012-08-30T08:44:14.677

11 What is the difference between AIC() and extractAIC() in R? 2012-11-16T11:01:22.530

11 Criteria for selecting the "best" model in a Hidden Markov Model 2013-07-23T17:11:42.493

11 Are models identified by auto.arima() parsimonious? 2013-09-26T14:23:23.857

11 Different AIC definitions 2016-01-20T08:59:53.700

11 AIC of ridge regression: degrees of freedom vs. number of parameters 2016-12-02T11:34:16.093

10 Simple linear regression, p-values and the AIC 2016-06-13T12:43:50.360

9 Can you compare AIC values as long as the models are based on the same dataset? 2013-05-15T15:54:00.170

9 Calculating AIC “by hand” in R 2014-02-20T19:46:41.197

9 Variable selection vs Model selection 2016-04-06T20:17:03.323

9 Why isn't Akaike information criterion used more in machine learning? 2017-11-15T14:12:21.557

8 One sentence explanation of the AIC for non-technical types 2012-03-05T11:23:28.730

8 What does an infinite AIC mean and what can be done about it? 2013-08-22T15:59:41.190

8 What do I do when values of AIC are low and approximately equal? 2014-01-08T01:36:37.813

8 AIC versus cross validation in time series: the small sample case 2015-02-25T11:36:06.437

8 Number of parameters in an artificial neural network for AIC 2015-09-28T16:24:05.800

8 AIC formula in Introduction to Statistical Learning 2015-11-13T00:13:16.677

7 Model-selection for linear mixed models over alternative sets of parameters (nlme function in R) 2012-06-19T20:25:56.980

7 REML vs ML stepAIC 2012-10-24T15:33:18.153

7 Relative variable importance with AIC 2012-12-20T10:26:31.873

7 When is it appropriate to select models by minimising the AIC? 2013-12-08T13:32:11.523

7 Comparing AIC and adjusted $R^2$ 2014-01-23T16:42:37.387

7 Equivalence of AIC and p-values in model selection 2014-03-07T20:46:36.763

7 Does Box-Cox parameter estimation count towards parameters for AIC? 2014-06-16T13:57:45.170

7 On Negative AIC Values 2015-01-25T08:09:20.867

7 Stepwise Model Selection in Logistic Regression in R 2015-02-02T20:40:09.237

7 Selecting regularization penalty: cross validation or information criteria? 2015-10-04T14:00:06.750

7 An Information Criterion that considers how many variables we can choose from 2016-08-15T20:31:36.520

7 What exactly is "stepwise model selection"? 2017-02-28T18:08:47.663

6 What is AIC? Looking for a formal but intuitive answer 2010-07-27T17:46:25.223

6 Why does AIC formula in R appear to use one extra parameter than expected? 2010-07-31T09:39:47.323

6 Comparing model fits across a set of nonlinear regression models 2010-11-02T06:26:26.320

6 Why do AIC and BIC show inversed outputs? 2011-11-17T08:26:53.817

6 Why does log likelihood function for a model use SSE/n and not SSE/df? 2012-10-21T12:33:28.863

6 Correct definition of number of parameters $K$ in Akaike Information Criterion 2013-09-26T12:02:07.347

6 Generalized linear mixed models: model selection 2013-10-31T08:48:04.587

6 Correct number of parameters of AR models for AIC / BIC ? 2013-12-25T13:49:14.520

6 Is the Akaike information criterion inversely proportional to the chi-squared statistic? 2015-11-16T16:28:38.863

6 Are there circumstances in which BIC is useful and AIC is not? 2016-09-29T18:50:04.493

6 Comparing differences of AIC of different data sets 2016-12-12T08:40:29.323

6 Conflicting approaches to variable selection: AIC, p-values or both? 2017-03-05T16:48:19.537

5 Why do I get equal AIC, BIC and log likelihood for different models in LME framework? 2010-11-23T09:38:30.733

5 BIC or AIC to determine the optimal number of clusters in a scale-free graph? 2012-07-09T15:16:56.090

5 Analogous measure of AIC which uses the posterior distribution for model selection? 2013-03-22T21:27:56.770

5 Sparse parameters when computing AIC, BIC, etc 2014-03-04T19:49:34.293

5 How to do external validation of logistic regression models and perform model benchmarking 2014-03-17T14:37:27.457

5 Cross-Validation vs. AICc for LASSO 2014-07-02T13:56:32.990

5 Assuming a probability density for MLE to do model selection 2014-10-28T14:27:27.930

5 Gamma vs tweedie distribution for large productivity dataset 2014-11-16T22:46:27.620

5 What exactly is the Box-Jenkins method for ARIMA processes? 2015-01-04T01:37:40.303

5 lme4: Why is AIC no longer displayed when using REML 2015-01-05T14:40:33.320

5 When AIC and Adjusted $R^2$ lead to different conclusions 2015-03-09T10:49:07.700

5 Model selection between parametric nonparametric methods 2015-10-16T07:19:11.330

5 Time series model selection: AIC vs. out-of-sample SSE and their equivalence 2016-03-08T11:34:55.707

5 Using the drop1 command in R and AIC 2016-05-22T09:23:09.387


text:
# Representation of a semi-group

$S$ in a class of semi-groups $\Sigma$

A homomorphism of $S$ into a semi-group from $\Sigma$ (in the case of an injective homomorphism one speaks of a faithful representation). One usually has in mind a class of concrete semi-groups. Representations in a class of transformation semi-groups (for short — representations by transformations, cf. also Transformation semi-group), in a class of semi-groups of partial transformations or binary relations, or in a class of matrix semi-groups (so-called matrix, or linear, representations of a semi-group) have been studied most completely. In automata theory, with each automaton one associates a representation of the free semi-group of transformations of the set of its internal states. Representations of a semi-group by transformations that are in some way or other related to properties of the elements of the transformed set, endowed with some structure, (endomorphisms, continuous transformations, etc.) have special significance. Every semi-group with an identity element can be isomorphically represented as the semi-group of all endomorphisms of an oriented or non-oriented graph, as the semi-group of all endomorphisms of an algebra with unary operations, etc. A few constructions for obtaining all representations of a semi-group by partial transformations are known (1983). These representations are constructed from certain simple representations by union, multiple repetition, restriction to a subset, and by the operation of imbedding of semi-groups.

By adjoining an identity element to a semi-group $S$, $S^1 = S \cup \{1\}$, and by extending the regular representation of $S$ by left shifts onto a Cartesian power $S^J$ of $S^1$, one obtains a representation $\phi_J$ — the $J$-fold repetition of the regular representation of $S$. Every representation $\psi$ of $S$ by transformations of a set $\Omega$ can be obtained (cf. [2]) from $\phi_J$ by means of a mapping $\theta:S^J\rightarrow \Omega$ such that $$\psi a(\theta\alpha) = \theta(\phi_J a(\alpha))\,\ \ a\in S\,,\ \alpha\in S^J \ .$$

Transitive representations (i.e. representations $\phi$ by transformations of a set $\Omega$ such that for any $\alpha,\beta \in \Omega$ there is an $a \in S$ for which $\phi a(\alpha) = \beta$) play a special role.

The representations of a semi-group are, by one-to-one partial transformations, related to the notion and properties of inverse semi-groups (cf. Inversion semi-group).

For the study of matrix representations of a semi-group one invokes semi-group algebras (cf. Semi-group algebra). The problem of reducibility of matrix representations has been studied. For a number of semi-groups (including finite semi-groups) the irreducible representations (cf. Irreducible representation) have been found. The matrix representations of completely-simple and completely $0$-simple semi-groups (cf. Completely-simple semi-group) can be constructed as extensions of representations of their subgroups. The matrix representations of arbitrary semi-groups can be described by means of representations of their factors, which are simple and $0$-simple semi-groups.

#### References

 [1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) [2] V.V. Vagner, "Representations of ordered semi-groups" Mat. Sb. , 38 : 2 (1956) pp. 203–240 (In Russian) [3] E.S. Lyapin, "Representations of semi-groups by partial mappings" Mat. Sb. , 52 : 1 (1960) pp. 589–596 (In Russian) [4] B.M. Shain, "Representations of semi-groups by binary relations" Mat. Sb. , 60 : 3 (1963) pp. 293–303 (In Russian) [5a] D.B. McAlister, "Representations of semigroups by linear transformations I" Semi-group Forum , 2 : 3 (1971) pp. 189–263 [5b] D.B. McAlister, "Representations of semigroups by linear transformations II" Semi-group Forum , 2 : 4 (1971) pp. 283–320 [6] B. Jónsson, "Topics in universal algebra" , Springer (1972)
How to Cite This Entry:
Representation of a semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_a_semi-group&oldid=42995
This article was adapted from an original article by L.M. GluskinE.S. Lyapin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article


text:
# How do I describe this negative binomial regression with offsets in math and words?

I've been using a negative binomial model to compare the number of particles in the ocean of different sizes to their abundance.

My gam, in r code looks like this

gam(TotalParticles ~ log(lb), offset = log(binsize * vol), data = df, family = "nb")

I'm using gam, rather than glm, because the former can handle negative binomial regressions and poisson regressions with the same syntax and I was comparing the two at one point

Essentially I'm interested in the slope of the relationship between the log of size lb and the log of particle number Total Paricles. The particle sizes have been binned by size, and so I normalize to the width of that bin. The particls are also collected in different volumes of water, so I also normalize to that volume vol.

I'm trying to express this in math and words in a corresponding manuscript. Right now, based on this post, I'm writing the equation as:

$$ln(\frac{E(Total\,Particles)}{Volume *Binsize}) = b_0 + b_1\,ln(Size)$$

My co-authors keep asking me what E means in this context. My understanding is that we are predicting an expected "negative binomial" distribution of total particle numbers from size.

So far I have written:

The term on the left describes the expected volume and bin-size normalized count data, assuming a negative binomial distribution of residuals, with E referring to the conditional expectation of total particle numbers, assuming a negative binomial distribution.

I have two questions:

1. Does my equation actually correctly represent the linear model written in code? If not, is there a better way of writing it?

2. Is there a better way I can express in words what is going on on the left hand side of the equation?

• Related answer in the context of Poisson model, but that idea is identical. – Dimitriy V. Masterov May 4 at 20:51
• Thanks. That gives me confidence that the equation that I am using is ok, unless of course NB are written differently than poisson models. That explanation also uses the E term, though I don't fully understand what that means, even from the provided post. – ohnoplus May 5 at 1:48
• The expectations have the same form for NB and Poisson (though the parameters will of course be different). E stands for expected value here, which is a conditional mean. Technically you are modeling the mean of the number of particles given some covariates, not the number of particles itself. – Dimitriy V. Masterov May 5 at 1:54


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# Clifford algebra

by PeteKH
Tags: basics, clifford algebra
 P: 295 Where a vector's tail sits has to be represented separately from the vector, and I don't think that is necessarily any different when using Clifford Algebra unless you introduce some mechanism for encoding that translation. Are you asking about representations of lines in Clifford algebras? For example, a standard parametric representation for points x on a line with points a and b on the line is: $$x = a + \alpha (b-a)$$ you can rewrite this as $$(x - a) = \alpha (b-a)$$ Then wedge both sides with (b-a): $$(x - a) \wedge (b-a) = 0$$ This produces a bivector representation of the line equivalent to the original parameterized equation. Points x on the line will be colinear with the direction vector (b-a), so the wedge product is zero.
 P: 295 Clifford algebra Hi Pete, Again I think that your representation issue probably has to be thought through independent of any Clifford Algebra context. If you wanted to represent a vector and it's origin throughout space, then a pair of vectors is not really enough information. Suppose you represented the base of a 2D vector as a complex number B and it's length and magnetude as another complex number P. Then is a vector formed out of this pairing: $$V = (B, P)$$ really a good representation? I don't think so (consider addition, and what it does to the B values). You probably want a representation something like the computer graphics matrix representation of a point. For this 2D case you could probably use $$V = \begin{bmatrix} B & 0 \\ P & 1 \end{bmatrix}$$ Points with a representation like this can be added. Rotation and scaling are matrix operations of the form $$\begin{bmatrix} T & 0 \\ 0 & 1 \end{bmatrix}$$ And since you have the 1 value in the corner you can rescale your origin after any arbitrary sequence of operations (there's a name for this sort of representation in CG but its been 10 years since school where I used it so I forget that part). I'd conclude that you need at least five real coordinates for your 2D vector field representation problem (one to encode the scale factor). Once you get as far as picking a representation for this sort of vector field space, if desired I'd imagine that there would be a number of possible multivector representations that you could pick from to encode it, just as you have freedom to pick how you want to do so with a matrix representation (could use three by three matrixes with a 2x2 point representation, and 2x1 origin, plus a 1 and two zeros).