Top Risks Lead to Top Priorities

What should be our top priority in life? Well, it depends on the top risks in life. Depending on whom you ask this question, the answer may vary.

Top priorities

I suspect risk to life comes first. What else can come closer or even be ranked higher? To a large section of the world, it could simply be getting out of poverty. It can be so powerful that individuals may even risk their lives to achieve it for their families and future generations at least. Here, I assume that, at least for the people who read this post, the risk to life is the top one.

Top risks to life

What is the top risk to life? It could be diseases, accidents, extreme weather events, wars, terrorist attacks, etc. Let’s explore this further. According to the World Health Organization (WHO), diseases are the top 10 causes of death and are responsible for 32 out of the 56 million deaths in a year. That is about 60%, according to the 2019 data. And what are they?

Noncommunicable diseases occupied the top seven spots in 2019. Yes, that will change in 2020 and 21, thanks to the COVID-19 pandemic. Deaths due to the current pandemic can reach the top three in 2021, but getting into the top spot is unlikely, at least based on the official records.

The Oscar goes to

The unrivalled winner is cardiovascular diseases (CVDs) – heart attacks and strokes – which cost 18 mln lives in 2019. The risk factors include unhealthy diets, physical inactivity, smoking, and the harmful use of alcohol. And an early warning to watch out for is high blood pressure.

There are three ways to manage the top risk: 1) medication for blood pressure management, 2) regular exercise, and 3) getting into the habit of a healthy diet.

Top 10 causes of death: WHO

Cardiovascular diseases: WHO

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Anscombe’s Trap

What is so special about the following four scatter plots? Do you see any similarities between them?

Well, all four look different from each other. The first is a scatter plot with a linear trend, the second is a curve, the third represents a straight line with one outlier, and the fourth is a collection of points in a cluster with an extreme outlier. And you are right; they represent four different behaviours or x and y.

Beware of statistical summary

Imagine you don’t get to see how they are organised in the x-y plane, but instead, only the summary statistics, and here they are:

PropertyValue
Mean x 9.0
Mean y 7.5
Sample variance of x11
Sample variance of x4.12
Correlation between x and y0.816
By the way, the numbers above represent all four sets!

Not over yet!

Now put linear regression lines to all.

And if you don’t believe me, see all four in one plot with the common linear regression line.

Following is the complete dataset in an R data frame.

Q1 <- data.frame("x" = c(10, 8, 13, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0), "y" = c(8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68))
Q2 <- data.frame("x" = c(10, 8, 13, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0), "y" = c(9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74))
Q3 <- data.frame("x" = c(10, 8, 13, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0), "y" = c(7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73))
Q4 <- data.frame("x" = c(8, 8, 8, 8, 8, 8, 8, 19, 8, 8, 8), "y" = c(6.58, 5.76, 7.71, 8.84, 8.47, 7.04, 5.25, 12.50, 5.56, 7.91, 6.89))

The moral of the story is

Summary statistics are great ways to communicate trends. But, as the reviewer, you must exercise the utmost care in understanding the actual data points.

Anscombe’s quartet: wiki

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Normal – Normal Sensitivity

In this fourth and final episode on the Bayesian inference with normal-normal conjugate pair, we find out how important is the choice for the prior and data collection in arriving at the answer. We will start from the previous set of inputs. But the relationship between the prior and the posterior first.

\\ \mu_{posterior} = \frac{\tau_0 \mu_0 + \tau \Sigma_{i=1}^{n} x_i}{\tau_0 + n*\tau} \\ \tau_{posterior} = \tau_0 + n*\tau \\ \sigma_{posterior} = \sqrt{\frac{1}{\tau_{posterior} }}

The parameters used in the first example are:

\\ \mu = \text {variable}; \sigma = 2 \text { (hypothesis)} \\ \mu_0 = 15; \sigma_0 = 4 \text { (prior)} \\ x = 9 \text{ (data)}

And the output is plotted below. The green curve is the prior probability distribution, blue represents the posterior in case of one data and red for 5 data (same value of 9). The vertical dotted line points at the data (or the average of the data).

It shows that (multiple) data at nine is pulling the distribution to come closer to it. Now let’s change the prior further right, mu0 = 25.

You see that the posterior distributions did not change much. Next, we will make the distribution narrower by defining sigma0 to be 2 and keeping mu0 at 25.

Things are now beginning to separate from the data. This suggests that if you want to be conservative with the estimation of posterior, it is better to keep the prior distribution narrower.

Finally, we check the impact of the standard deviation of the hypothesis. We change the value from 2 to 4, while keeping the original parameters as they are (mu_0 = 15; sigma_0 = 4):

Compare this with the first plot: you will see that making the hypothesis broader did not impact the mean value of the posterior.

Previous Posts on Normal-Normal

Normal-Normal conjugate
Normal-Normal and height distribution
Normal-Normal continued

Normal – Normal Sensitivity Read More »

Normal – Normal Continued

We will do a few sensitivities and see how the posterior distribution modifies. Let’s define the variables before we start.

\\ \mu = \text {variable}; \sigma = 2 \text { (hypothesis)} \\ \\ \mu_0 = 15; \sigma_0 = 4 \text { (prior)} \\ \\ x = 9 \text{ (data)}

The plots of the distributions are

Imagine you collected four more data, and interestingly they are all the same, 9! You have five data points now, although the average remains the same (9) as the previous case.

Note that the vertical dotted line represents the data (or the average of the data).

The relations used in the calculations of the posteriors are:

\\ \mu_{posterior} = \frac{\tau_0 \mu_0 + \tau \Sigma_{i=1}^{n} x_i}{\tau_0 + n\tau} \\ \\ \tau_{posterior} = \tau_0 + n\tau \\ \\ \sigma_{posterior} = \sqrt{\frac{1}{\tau_{posterior} }}

Normal – Normal Continued Read More »

Normal – Normal and Height Distribution

Continuing from the previous post, we will apply the normal-normal Bayesian inference to the height problem. The general format of the Bayes’ rule is:

P(\text{height distribution}|height = 71)= \frac{Likelihood (height = 71|\text{a height distribution}) * P(\text{height distribution})}{\int Likelihood (height = 71|\text{a height distribution}) * P(\text{height distribution})}

Since there are two parameters involved in the normal distribution, we have to have either a double integral in the denominator or choose one of the parameters, say the standard deviations, as known. So, we assume the standard deviation to be 4.0 in the present case. Let’s take the mean for the prior to be 69, the standard deviation to be 3 and see what happens. Therefore,

\\ \mu = variable, unknown \\ \sigma = 4 \\ \tau = \frac{1}{\sigma^2} = 0.0625 \\ \mu_0 = 69 \\ \sigma_0 = 3 \\ \tau_0 = \frac{1}{\sigma_0^2} = 0.11 \\ \text{You will soon find out why I have defined }\tau

Now we have everything to complete the problem.

\\ \mu_{posterior} = \frac{\tau_0 \mu_0 + \tau \Sigma_{i=1}^{n} x_i}{\tau_0 + n\tau} = \frac{0.11 * 69 + 0.0625*71}{0.11 + 1*0.0625} = 69.72 \\ \\ \tau_{posterior} = \tau_0 + n\tau = 0.11 + 1*0.0625 = 0.1725 \\ \\ \sigma_{posterior} = \sqrt{\frac{1}{\tau_{posterior} }} = 2.4

You can see a few things here: 1) the prior has moved a bit in the direction of the data but is still far from it, 2) the posterior is narrower than the prior.

Normal – Normal and Height Distribution Read More »

Normal – Normal Conjugate

We have seen how to come up with estimates of events based on assumed prior data using Bayesian inference. For discrete events that are rare, we have Poisson likelihood, and on such occasions, we use a Gamma prior and get a gamma posterior. In the same manner for Shaq, we found that the best one for estimating his success (or the lack of it) of entering Whitehouse is a binomial distribution. Once a data point is collected (one attempt), you update the chances using a beta distribution as a prior, and you get beta as the posterior.

Here is a new challenge you have a task to estimate the male height distribution of a particular country. We know from the examples of other countries that normal distributions describe the height distributions. A normal distribution has two parameters – mean and standard deviation, and our challenge is to estimate these parameters for our new country. We start with a set of hypotheses, use the available data and apply Bayesian inference to reach the goal.

Assume that I have collected data from the region, and it is 71 inches. Now I assume that the means ranges from 50 to 80 inches and the standard deviation from 1 to 4. Collect five hypotheses (purely random) from them as follows:

N(mean = 60, sd = 2), N(71, 3), N(65, 1.5), N(75, 4), N (80,1)

Now ask the question: what is the likelihood that my distribution of the new country is N(75, 4) given that I have collected data of 71 inches? Same for the rest four of the hypotheses. It can be estimated utilising the standard tables or using the R function dnorm(71, 75,4) = 0.06049268. The Pictorial representation is below.

Before applying the Bayes’ theory, we should realise that 1) we need prior probabilities for each of the above five hypotheses and 2) we can have infinite hypotheses, not just five! Then we can apply the formula as we did before.

\\ P(\text{height distribution}|height = 71)= \frac{Likelihood (height = 71|\text{a height distribution}) * P(\text{height distribution})}{\int Likelihood (height = 71|\text{a height distribution}) * P(\text{height distribution})}

It is a conjugate problem (the prior will be a pdf), and the right prior is a normal distribution, making it a normal-normal problem. How to complete the exercise using normal-normal is something we will see next.

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Binomial to Normal

We have seen it before: what is the chance of getting three heads if you toss a coin five times? The answer is about 30%. You know how to do it: 5C3 (0.5)3(0.5)5-3 = 0.3125. The distribution representing the probability of each success when a coin is tossed five times is given by the PMF of binomial.

We have used the following R code (the dbinom function for PMF) to generate this plot. 

xx <- seq(1,30,1)
p <- 0.5
q = 1 - p
toss <- 5

par(bg = "antiquewhite1")
plot(dbinom(xx,race,p), xlim = c(1,8), ylim = c(0,0.52), xlab="Success #", ylab="Probability", cex = 1, pch = 5, type = "h", main="", col = ifelse(xx >= 24,"#006600",'red'))

If you look at the distribution carefully, you will see that it resembles a normal distribution. For the normal distribution, we need a mean and a standard deviation. The mean is easy: you multiply the total number of tossing with the probability of success in one toss, i.e., 5 x 0.5 = 2.5. The variance of a binomial distribution is N x p x (1-p); the standard deviation is its square root. Let’s try making a normal distribution and superpose.

xx <- seq(1,30,1)
p <- 0.5
q = 1 - p
toss <- 5

par(bg = "antiquewhite1")
plot(dbinom(xx,toss,p), xlim = c(1,8), ylim = c(0,0.52), xlab="Success #", ylab="Probability", cex = 1, pch = 5, type = "h", main="", col = ifelse(xx >= 24,"#006600",'red'))

xx <- seq(1,30,0.1)
mean_i <- toss*p
sd_i <- sqrt(toss*p*q)

lines(xx, dnorm(xx, mean = mean_i , sd = sd_i ), xlim = c(0,30), ylim = c(0,0.2), xlab="Evidence for guilt", ylab="Frequency", col = "springgreen4", cex = 1, pch = 5, type = "l", bg=23, lwd = 2, main="")
grid(nx = 10, ny = 9)

Looks like the green line representing the normal distribution is almost passing through binomial values. We will soon see that when the number of trials (N) is large, the binomial becomes indistinguishable from a normal distribution. For N is 25. Like this:

Binomial to Normal Read More »

A Gene that is Selfish and A Brain that is Not

This post is not a commentary on the book Selfish Gene by Richard Dawkins, although I do recommend the book. But this is around how to understand what is selfishness.

It seems scientists often forget about the common public when it comes to naming their books or concepts. That way, the title, selfish gene, is just another one: survival of the fittest, natural selection, god particle, the list goes on. None of these phrases represents what they truly meant. God particles have nothing to do with god, survival does not correlate with any physical fitness, or there is nobody in nature to select or reject anything.

Selfishness in biology simply means the ability to survive no matter what. It has little to do with a species’ deliberated actions using its brain, such as mating (or not mating), not sharing resources, stealing or killing. It only means making copies and preserving genes by passing from generation to generation.

What a headless virus can do

Take the case of the most popular show in town, the novel coronavirus, SARS-CoV-2 or Covid-19. The virus has two parts, the outer envelope, which has a bunch of proteins, including the famous spikes, all embedded in a lipid membrane and the inside material that contains the genome, a single-stranded RNA, preserved nicely on a protein called the nucleocapsid or N-protein. The genome is long and contains information to make new viruses using someone else’s workshop, the human cells.

The virus thus mobilises human cell machinery (e.g. ribosomes) to replicate. It creates billions of copies that infect millions of people. And the virus does all these without having a body or a brain!

Brain doesn’t need to follow

Human genes, too, want to preserve themselves. They are also selfish and want to be immortal. But it has a master, the human brain, which can overrule the instincts for the greater good. It may have inherent altruism, but more importantly, it is trainable based on a value system. While the brainless gene wants and will long for eternal life, you, as a human, can prefer not to have offspring. You may stop taking sugar, run for hours when no lion is chasing, donate organs to strangers.

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Disease of the Decent

Parkinson’s disease accompanies the loss of dopaminergic neurons responsible for synthesising the neurotransmitter dopamine. Dopamine is often known as the molecule that controls motivation through a reward mechanism. Naturally, the chemical earned the reputation of being responsible for addiction, pleasure etc.

An illness negatively correlated with an addiction-causing chemical is thus an ideal candidate for seeing confounding. Let’s take some well-known addictions, namely coffee, smoking and alcohol.

Coffees’ negative correlation with Parkinson’s is something we have seen earlier. Next up is smoking, and lo and behold, smoking is associated with a lower tendency of Parkinson’s disease! Now, I have no choice but to search for alcoholism. This one appears more complex. Most of the studies showed no association or weak negative correlation.

Confounding, is it?

It has all the ingredients to be a confounding phenomenon. But until a new study came up with a mechanistic explanation – Nicotins effect on neuron survival.

Disease of the Decent Read More »

Risk Perception

Estimation of risk is a very desirable quality to have because it can improve survival rates. Earlier, we have seen the definition of risk as the product of probability and impact. But for most people, it is something far more intuitive and personal. Risk perception, as they are called, may come from recent experiences, news headlines or simply from the lack of knowledge of something. One dominant example is the perception that people live more risky life today than in the past. Data suggest this is incorrect.

Incorrect estimation of risk comes from our mind’s inability to quantify probabilities. That is why when asked about the risk of an even, experts looks at the past (annual incident rate), whereas people consider the future (catastrophic potential). An example is how laypeople perceive the risk due to nuclear power (very high) versus what the experts’ estimate (one of the safest energy technologies)! Wikipedia reports about 30 incidents related to radiations in history, and the deaths were in single digits in 21 of them. Now compare that with a million annually due to coal!

Slovic, Perception of Risk
List of nuclear and radiation accidents
Health and environmental impact of the coal industry

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