Ex 1.12 - Pattern Recognition and ML by Christopher Bishop

Pre-read

Question

Solution from Book

Solution Using ChatGPT

Are you aware of the result E[xn, xm] = mu^2 + I . sigma ^ 2

I assume you meant $E[X_n \cdot X_m] = \mu^2 + I \cdot \sigma^2$, where $I$ is the Kronecker delta, which is 1 when $n = m$ and 0 otherwise.

In that case, $E[X_n \cdot X_m]$ simplifies to:

$E[X_n \cdot X_m] = \begin{cases} \mu^2 + \sigma^2 & \text{if } n = m \\ \mu^2 & \text{if } n \neq m \end{cases}$

This is because when $n = m$, you have the product of a variable with itself, and the additional $\sigma^2$ term comes from the covariance term. When $n \neq m$, the covariance term is zero, and you are left with the square of the mean.

If you have further questions or if there's anything specific you'd like to clarify, feel free to ask!

Can you derive this expression?

Certainly! Let's derive the expression for $E[X_n \cdot X_m]$ for two continuous random variables $X_n$ and $X_m$ from the same Gaussian distribution $N(\mu, \sigma^2)$ with mean $\mu$ and variance $\sigma^2$.

The expectation of the product is given by:

$E[X_n \cdot X_m] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x_n \cdot x_m \cdot f(x_n) \cdot f(x_m) \,dx_n \,dx_m$

Since $X_n$ and $X_m$ are from the same distribution, let's denote $f(x)$ as the probability density function (PDF) of $X$. The expression becomes:

$E[X_n \cdot X_m] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x_n \cdot x_m \cdot f(x) \cdot f(x) \,dx_n \,dx_m$

Now, split the integral into two cases: $n = m$ and $n \neq m$.

Case 1: $n = m$

$E[X_n \cdot X_m] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x_n^2 \cdot f(x) \cdot f(x) \,dx_n \,dx_m$

$= \left(\int_{-\infty}^{\infty} x_n^2 \cdot f(x) \,dx_n\right) \cdot \left(\int_{-\infty}^{\infty} f(x) \,dx_m\right)$

$= \mu^2 + \sigma^2$

This result comes from the fact that the integral of $x^2$ over the entire range for a Gaussian distribution with mean $\mu$ is $\mu^2 + \sigma^2$.

Case 2: $n \neq m$

$E[X_n \cdot X_m] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x_n \cdot x_m \cdot f(x) \cdot f(x) \,dx_n \,dx_m$

$= \left(\int_{-\infty}^{\infty} x_n \cdot f(x) \,dx_n\right) \cdot \left(\int_{-\infty}^{\infty} x_m \cdot f(x) \,dx_m\right)$

$= \mu \cdot \mu$

This result comes from the fact that the integral of $x$ over the entire range for a Gaussian distribution with mean $\mu$ is $\mu$.

Therefore, combining both cases:

$E[X_n \cdot X_m] = \mu^2 + I \cdot \sigma^2$

where $I$ is the Kronecker delta, equal to 1 when $n = m$ and 0 otherwise.

References

• give me math expression for: (1/n) * sum_for_n_equals_1_to_N(E[ (1/n^2) * sum_for_m_equals_1_to_N(sum_for_l_equals_1_to_N(xm*xl)) ])
• what is the mathematical expression for (1/N) * sum_n_equal_1_to_N(E[(2/N) * xn * sum_m_equals_1_to_N(xm)])

Tags: Mathematical Foundations for Data Science,Machine Learning,

Solution 1.11 - On Gaussian Distribution and it's Log Likelihood (Pattern Recognition and ML by Christopher Bishop)

Can you give me the equation for Gaussian distribution?

Can you give me it's log likelihood function?

Note: This is already given as part of the problem statement.

Can you show the detailed steps how you arrived at it?

Some paperwork

In the next slide we will solve for the below expression:

Some Logarithm Properties we would use:

Can you maximize the log likelihood expression w.r.t. mu and show the detailed steps?

Some paperwork for derivative of:

We will prove it next:

Some ‘Differentiation’ identities we would see next

And now maximize log likelihood w.r.t. sigma-squared and show the detailed steps

Identity:

References

https://chat.openai.com/share/4278db33-9b76-4d8b-9586-4e1e78844433

Tags: Mathematical Foundations for Data Science,Machine Learning,

Question 1.11 - On Gaussian Distribution and it's Log Likelihood (Pattern Recognition and ML by Christopher Bishop)

Pre-read:

(1)


(2)


(3)

From (1.46) and (1.53), the log likelihood function can be written in the form:



Question:

By setting the derivatives of the log likelihood function (1.54) with respect to µ and σ^2 equal to zero, verify the results (1.55) and (1.56).

Link to Solution

Valprol - CR 500 Tablet

Valprol - CR 500 Tablet

Prescription Required

Marketer: Intas Pharmaceuticals Ltd
SALT COMPOSITION: Sodium Valproate (333mg) + Valproic Acid (145mg)

Product introduction

Valprol -CR 500 Tablet is a combination of two medicines used to treat epilepsy, a neurological disorder in which there are recurrent episodes of seizures or fits. It controls the abnormal activity of the brain, relaxes the nerves and hence, prevents seizures or fits.

Valprol -CR 500 Tablet should be taken with food. Your doctor will decide the correct dose for you. This may increase gradually until your condition is stable. This medicine may take several weeks to work but it is important to take it regularly to get the benefit. Do not stop taking it, even if you feel fine unless your doctor advises you to. You may have more seizures, or your bipolar disorder may get worse.

The most common side effects of this medicine include nausea, vomiting, loss of appetite, swelling of gums, headache, sleepiness, tremor, hair loss and liver injury. Most side effects wear off, but if they bother you or do not go away, tell your doctor. There may be ways of preventing or reducing these effects.

Before taking this medicine, tell your doctor if you have kidney or liver problems, meningitis or depression or suicidal thoughts. Also let your doctor know about all other medications you are using as some may affect, or be affected by, this medicine, including contraceptive pills. If you are pregnant or breastfeeding, Valprol -CR 500 Tablet can be taken if it is clearly needed but the dose may be adjusted. You should avoid driving or riding a bicycle if this medicine makes you drowsy or dizzy. You may need frequent blood tests to check how you are responding to this medicine.

You should avoid alcohol while taking Valprol -CR 500 Tablet, as it may worsen certain side effects. This medicine can also lead to weight gain, eat a healthy balanced diet, avoid snacking with high-calorie food, and exercise regularly. Inform your doctor if you develop any unusual changes in mood or behavior, new or worsening depression, or suicidal thoughts or behavior.
Uses of Valprol Tablet CR

    Epilepsy

Benefits of Valprol Tablet CR

In Epilepsy

Valprol -CR 500 Tablet slows down electrical signals in the brain which cause seizures (fits). It can also help reduce symptoms such as confusion, uncontrollable jerking movements, loss of awareness, and fear or anxiety. The medicine can allow you to do some activities that you would otherwise be forbidden or scared to do (such as swimming and driving). It can take a few weeks for this medicine to work (because the dose has to be increased slowly) and during this time you may still have seizures. Do not stop using this medicine even if you feel well, until your doctor advises you to. Missing doses may trigger a seizure.

Side effects of Valprol Tablet CR

Most side effects do not require any medical attention and disappear as your body adjusts to the medicine. Consult your doctor if they persist or if you’re worried about them
Common side effects of Valprol

    Nausea
    Vomiting
    Weight gain
    Loss of appetite
    Gum swelling
    Headache
    Sleepiness
    Tremors
    Hair loss
    Liver injury
    Increased liver enzymes
    Bleeding gums
    Epigastralgia
    Diarrhea
    Memory impairment
    Decreased sodium level in blood
    Anemia (low number of red blood cells)
    Thrombocythemia
    Pain during periods

How to use Valprol Tablet CR

Take this medicine in the dose and duration as advised by your doctor. Swallow it as a whole. Do not chew, crush or break it. Valprol -CR 500 Tablet is to be taken with food.

How Valprol Tablet CR works

Valprol-CR 500 Tablet is a combination of two antiepileptic medicines: Sodium Valproate and Valproic Acid.

Safety advice

Alcohol
UNSAFE
It is unsafe to consume alcohol with Valprol -CR 500 Tablet.

Pregnancy
CONSULT YOUR DOCTOR
Valprol -CR 500 Tablet is unsafe to use during pregnancy as there is definite evidence of risk to the developing baby. However, the doctor may rarely prescribe it in some life-threatening situations if the benefits are more than the potential risks. Please consult your doctor.

Breast feeding
SAFE IF PRESCRIBED
Valprol -CR 500 Tablet is safe to use during breastfeeding. Human studies suggest that the drug does not pass into the breastmilk in a significant amount and is not harmful to the baby.

Driving
UNSAFE
Valprol -CR 500 Tablet may decrease alertness, affect your vision or make you feel sleepy and dizzy. Do not drive if these symptoms occur.

Kidney
SAFE IF PRESCRIBED
Valprol -CR 500 Tablet is probably safe to use in patients with kidney disease. Limited data available suggests that dose adjustment of Valprol -CR 500 Tablet may not be needed in these patients. Please consult your doctor.
However, talk to your doctor if you have any underlying kidney disease.

Liver
CAUTION
Valprol -CR 500 Tablet should be used with caution in patients with liver disease. Dose adjustment of Valprol -CR 500 Tablet may be needed. Please consult your doctor.
Use of Valprol -CR 500 Tablet is not recommended in patients with severe liver disease and active liver disease.

What if you forget to take Valprol Tablet CR?
If you miss a dose of Valprol -CR 500 Tablet, take it as soon as possible. However, if it is almost time for your next dose, skip the missed dose and go back to your regular schedule. Do not double the dose.

All substitutes

For informational purposes only. Consult a doctor before taking any medicines.
Valprol -CR 500 Tablet
₹9.67/Tablet CR
Valprolex Chrono 500 CR Tablet
Zeelab Pharmacy Pvt Ltd
₹3.42/tablet cr
65% cheaper
Valtec CR 500 Tablet
Cipla Ltd
₹9.07/tablet cr
6% cheaper
Encorate Chrono 500 Tablet CR
Sun Pharmaceutical Industries Ltd
₹9.66/tablet cr
same price

Valparin Chrono 500 Tablet CR
Sanofi India Ltd
₹9.67/tablet cr
same price

Torvate Chrono 500 Tablet CR
Torrent Pharmaceuticals Ltd
₹7.6/tablet cr
21% cheaper

Quick tips

    You have been prescribed Valprol - CR 500 Tablet for treatment and prevention of seizures.
    Take it at the same time every day to maintain consistent levels in your body.
    Do not drive or do anything that requires mental focus until you know how Valprol -CR 500 Tablet affects you.
    Avoid consuming alcohol when taking Valprol -CR 500 Tablet, as it may cause excessive sleepiness or drowsiness.
    Monitor your weight during the treatment as Valprol -CR 500 Tablet can cause weight gain.
    Your doctor may monitor Valprol -CR 500 Tablet level, blood counts and liver function regularly throughout your treatment.  Inform your doctor if you develop fever, stomach pain, nausea or yellowing of your eyes or skin.
    Do not take Valprol -CR 500 Tablet if you are pregnant or planning to conceive or breastfeeding.
    Do not stop taking the medication suddenly without talking to your doctor as it may increase the seizure frequency.

Fact Box

Habit Forming: No
Therapeutic Class: NEURO CNS

Lithosun 300 Tablet

Lithosun 300 Tablet

Prescription Required

Marketer: Sun Pharmaceutical Industries Ltd
SALT COMPOSITION: Lithium carbonate (300mg)

Product introduction

Lithosun 300 Tablet is used in the treatment of bipolar disorder and mania in adults. It is sometimes used to treat other behavioral disorders as well. It reduces the frequency of manic episodes and decreases the intensity of those episodes that might occur.

Lithosun 300 Tablet should be taken with food. However, it is advised to take it at the same time each day as this helps to maintain a consistent level of medicine in the body. Do not skip any doses and finish the full course of treatment even if you feel better. If you have missed a dose, take it as soon as you remember it. You should continue to take this medicine for as long as the doctor advises and do not stop taking it suddenly as it may result in harmful effects.

Side effects associated with the use of this medicine include tremors, slurred speech, uncoordinated body movements, nausea, acne, increased white blood cell count, and memory impairment. It may also cause hair loss, enlarged thyroid gland, skin rash, and increased thirst. This medicine may also cause weight gain and to control it, you should have a balanced diet and exercise regularly. If you have been taking this medicine for a long time then regular monitoring of thyroid and kidney function may be required.

Uses of Lithosun Tablet

    Mania
    Bipolar disorder

Benefits of Lithosun Tablet

In Mania

Mania means extremely excited or elevated mood. Lithosun 300 Tablet helps calm the mood and relax the nerves. This stabilises the mood and prevents the symptoms of mania from recurring again. Taking Lithosun 300 Tablet will ensure that you have a better social life and are able to do daily activities more comfortably.

In Bipolar disorder

Lithosun 300 Tablet works to restore the normal balance of nerve activity in your brain. It can help prevent extreme changes in mood and help you feel less agitated. You may experience fewer episodes of low mood. It can take several weeks for this medicine to work and you may still feel low during this time. It is likely that you will take this medicine for at least six months but possibly much longer. It needs to be taken regularly to work effectively. Do not stop taking it unless your doctor advises you to. You should continue to eat well and keep fit to reduce symptoms of bipolar disorder.

Side effects of Lithosun Tablet

Most side effects do not require any medical attention and disappear as your body adjusts to the medicine. Consult your doctor if they persist or if you’re worried about them

Common side effects of Lithosun

    Tremors
    Slurred speech
    Uncoordinated body movements
    Nausea
    Acne
    Increased white blood cell count
    Memory impairment
    Hair loss
    Goiter (enlarged thyroid gland)
    Skin rash
    Increased thirst
    Weight gain
    Polyuria
    Diarrhea

How to use Lithosun Tablet

Take this medicine in the dose and duration as advised by your doctor. Swallow it as a whole. Do not chew, crush or break it. Lithosun 300 Tablet is to be taken with food.
How Lithosun Tablet works
Lithosun 300 Tablet works by suppressing the abnormal and excessive activity of the nerve cells in the brain.

Safety advice

Alcohol
UNSAFE
It is unsafe to consume alcohol with Lithosun 300 Tablet.


Pregnancy
CONSULT YOUR DOCTOR
Lithosun 300 Tablet is unsafe to use during pregnancy as there is definite evidence of risk to the developing baby. However, the doctor may rarely prescribe it in some life-threatening situations if the benefits are more than the potential risks. Please consult your doctor.

Breast feeding
SAFE IF PRESCRIBED
Lithosun 300 Tablet is probably safe to use during breastfeeding. Limited human data suggests that the drug does not represent any significant risk to the baby.

Driving
UNSAFE
Lithosun 300 Tablet may cause side effects which could affect your ability to drive.
Lithosun 300 Tablet may make you feel sleepy, tired or dizzy. Lithosun 300 Tablet may also cause other effects that can make it dangerous to drive such as numbness, paralysis, weakness or fits (seizures).

Kidney
CAUTION
Lithosun 300 Tablet should be used with caution in patients with kidney disease. Dose adjustment of Lithosun 300 Tablet may be needed. Please consult your doctor.
Use of Lithosun 300 Tablet is not recommended in patients with severe kidney disease.

Liver
SAFE IF PRESCRIBED
Lithosun 300 Tablet is probably safe to use in patients with liver disease. Limited data available suggests that dose adjustment of Lithosun 300 Tablet may not be needed in these patients. Please consult your doctor.

All substitutes

For informational purposes only. Consult a doctor before taking any medicines.
No substitutes found for this medicine

Quick tips

    Lithosun 300 Tablet is one of the most effective long-term treatment for bipolar disorder.
    It may also be used for the treatment of mania.
    Your lithium blood levels, thyroid function, and kidney function may be monitored as long as you take this medicine.
    It may cause diarrhea and dehydration. Drink plenty of water and and use normal amounts of salt. Inform your doctor if you start or stop a low-salt diet.
    Monitor your weight during treatment as Lithosun 300 Tablet can cause weight gain.
    Do not drive or do anything that requires mental focus until you know how it affects you.
    Inform your doctor if you have persistent stomach upset, problems with your speech, or extreme drowsiness. Your dose may need to be adjusted.
    Do not take Lithosun 300 Tablet if you are pregnant or breastfeeding.
    Do not stop taking the medication suddenly without talking to your doctor.

Fact Box

Chemical Class: Organic Carbonic Acids
Habit Forming: No
Therapeutic Class: NEURO CNS
Action Class: Mood stabilizers

Clonotril 0.5 Tablet

Clonotril 0.5 Tablet

Prescription: Required
Marketer: Torrent Pharmaceuticals Ltd
SALT COMPOSITION: Clonazepam (0.5mg)

Product introduction

Clonotril 0.5 Tablet is a prescription medicine used to treat epilepsy (seizures) and anxiety disorder. It helps to decrease the abnormal and excessive activity of the nerve cells and calms the brain.

Clonotril 0.5 Tablet may be taken with or without food. However, take it at the same time each day as this helps to maintain a consistent level of medicine in the body. Take this medicine in the dose and duration as advised by your doctor as it has a high potential of habit-forming. If you have missed a dose, take it as soon as you remember it and finish the full course of treatment even if you feel better. You should not stop this medicine suddenly without talking to your doctor as it may increase the seizure frequency and can cause nausea, anxiety, flu-like symptoms, and muscle pain.

Some common side effects of this medicine include fatigue, depression and impaired coordination. It may also cause dizziness and sleepiness, so do not drive or do anything that requires mental focus until you know how this medicine affects you. It is important to inform your doctor if you develop any unusual changes in mood or depression as this medicine may cause suicidal thoughts.

Uses of Clonotril Tablet

Treatment of Anxiety disorder

Treatment of Epilepsy/Seizures

Benefits of Clonotril Tablet

In Treatment of Anxiety disorder

Clonotril 0.5 Tablet stops your brain from releasing the chemicals that make you feel anxious so it can reduce the symptoms of excessive anxiety and worry. It can also reduce feelings of restlessness, tiredness, difficulty concentrating, feeling irritable and sleep problems that often come with Anxiety Disorder. It will therefore help you go about your daily activities more easily and be more productive. Keep taking this medicine even if you feel well. Stopping suddenly can cause serious problems.

In Treatment of Epilepsy/Seizures

Clonotril 0.5 Tablet slows down electrical signals in the brain which cause seizures (fits). It can also help reduce symptoms such as confusion, uncontrollable jerking movements, loss of awareness, and fear or anxiety. The medicine can allow you to do some activities that you would otherwise be forbidden or scared to do (such as swimming and driving). It can take a few weeks for this medicine to work (because the dose has to be increased slowly) and during this time you may still have seizures. Do not stop using this medicine even if you feel well, until your doctor advises you to. Missing doses may trigger a seizure.

Side effects of Clonotril Tablet

Most side effects do not require any medical attention and disappear as your body adjusts to the medicine. Consult your doctor if they persist or if you’re worried about them

Common side effects of Clonotril

    Depression
    Dizziness
    Drowsiness
    Fatigue
    Impaired coordination
    Memory impairment

How to use Clonotril Tablet

Take this medicine in the dose and duration as advised by your doctor. Swallow it as a whole. Do not chew, crush or break it. Clonotril 0.5 Tablet may be taken with or without food, but it is better to take it at a fixed time.

How Clonotril Tablet works

Clonotril 0.5 Tablet is a benzodiazepine. It works by increasing the action of a chemical messenger (GABA) which suppresses the abnormal and excessive activity of the nerve cells in the brain.

Safety advice

Alcohol
UNSAFE
It is unsafe to consume alcohol with Clonotril 0.5 Tablet.
warnings
Pregnancy
CONSULT YOUR DOCTOR
Clonotril 0.5 Tablet is unsafe to use during pregnancy as there is definite evidence of risk to the developing baby. However, the doctor may rarely prescribe it in some life-threatening situations if the benefits are more than the potential risks. Please consult your doctor.
warnings
Breast feeding
SAFE IF PRESCRIBED
Clonotril 0.5 Tablet is safe to use during breastfeeding. Human studies suggest that the drug does not pass into the breastmilk in a significant amount and is not harmful to the baby.
warnings
Driving
UNSAFE
Clonotril 0.5 Tablet may cause side effects which could affect your ability to drive.
warnings
Kidney
CAUTION
Clonotril 0.5 Tablet should be used with caution in patients with kidney disease. Dose adjustment of Clonotril 0.5 Tablet may be needed. Please consult your doctor.
Use of Clonotril 0.5 Tablet can cause excessive sleepiness in patients with end stage kidney disease.

Liver
CAUTION
Clonotril 0.5 Tablet should be used with caution in patients with liver disease. Dose adjustment of Clonotril 0.5 Tablet may be needed. Please consult your doctor.

What if you forget to take Clonotril Tablet?

If you miss a dose of Clonotril 0.5 Tablet, take it as soon as possible. However, if it is almost time for your next dose, skip the missed dose and go back to your regular schedule. Do not double the dose.

All substitutes

For informational purposes only. Consult a doctor before taking any medicines.
No substitutes found for this medicine

Quick tips

    The addiction / habit-forming potential of this medicine is very high. Take it only as per the dose and duration advised by your doctor
    It may cause dizziness. Do not drive or do anything that requires mental focus until you know how this medicine affects you.
    Avoid consuming alcohol as it may increase dizziness and drowsiness.
    Inform your doctor if you are pregnant, planning to conceive or breastfeeding.
    Inform your doctor if you experience worsen anxiety, depression angry, or violent behavior and mania while taking this medicine.
    Do not stop taking medication suddenly without talking to your doctor as that may lead to nausea, anxiety, agitation, flu-like symptoms, sweating, tremor, and confusion.

Fact Box

Chemical Class: Benzodiazepines Derivative
Habit Forming: Yes
Therapeutic Class: NEURO CNS
Action Class: Benzodiazepines