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A collection of common probability distributions for stochastic nodes in PyMC.

Continuous uniform log-likelihood.

The pdf of this distribution is

f ( x l o w e r , u p p e r ) = 1 u p p e r l o w e r

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Calculate log-probability of Uniform distribution at specified value.

Draw random values from Uniform distribution.

Uninformative log-likelihood that returns 0 regardless of the passed value.

Calculate log-probability of Flat distribution at specified value.

Raises ValueError as it is not possible to sample from Flat distribution

Improper flat prior over the positive reals.

Calculate log-probability of HalfFlat distribution at specified value.

Raises ValueError as it is not possible to sample from HalfFlat distribution

Univariate normal log-likelihood.

f ( x μ , τ ) = τ 2 π exp { τ 2 ( x μ ) 2 }

Normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by

τ = 1 σ 2


Calculate log-probability of Normal distribution at specified value.

Draw random values from Normal distribution.

Univariate truncated normal log-likelihood.

f ( x ; μ , σ , a , b ) = ϕ ( x μ σ ) σ ( Φ ( b μ σ ) Φ ( a μ σ ) )

Truncated normal distribution can be parameterized either in terms of precision or standard deviation. The link between the two parametrizations is given by

Believeland Ball

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Cleveland Indians: Stars that can replace LeBron on the banner downtown by Carson Ferrell
Cleveland Indians: Previewing the rematch with the Oakland A’s by Carson Ferrell
by Steven Kubitza Follow @StevenKubitza

Trevor Bauer has been a dominant force on the mound for the Cleveland Indians so far in 2018.

He leads the qualified starters on the team in ERA, strikeouts, innings pitched, and hasn’t allowed more than three earned runs in a start since May 27. He allowed four in what ended up being a 10-9 Indians win over the Houston Astros.

After another great start Wednesday, he had this to say:

Trevor Bauer: "I should be an All-Star. I'd imagine I will be and if I'm not, they didn't get it right, whoever they is. I'm glad I'm pitching well. That's been the goal, I feel like I should be an All-Star every year. That's the standard I hold myself to."

— Zack Meisel (@ZackMeisel)

There are several ways to look at this quote.

The lame way is to get mad and say that Bauer is lacking any sense of humility. Based on how some view him based on his past and some of his personal beliefs, there are those out there who think Bauer is bragging and will take this negatively.

The logical and fun way to view this quote is to agree with Bauer because he is right. He does indeed deserve to be in the game and will likely be selected.

I touched on it in the takeaways from Wednesday’s game, but Bauer is all over the AL leaderboards when it comes to starting pitchers. The only thing he isn’t at the top of is the win category, but hopefully no one is citing that as a reason why he shouldn’t be in the All-Star Game.

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Media Events

The holiday season is a good time to think about gifts, and not just the paper-wrapped kind. The people in your life can themselves be gifts—and so can a thousand other things, big and little, many of which you probably overlook day to day. Taking time during the holidays to notice, contemplate, and express gratitude for these people and things can make your holidays far more meaningful. Gratitude is a healing and supportive emotion, too. If you’re struggling with family drama, stressful travel, or disappointments, the practice of gratitude can help you through.

What is gratitude? Gratitude is a state of mind that arises when you affirm a good thing in your life that comes from outside yourself, or when you notice and relish little pleasures. Though some people and things are clear blessings, this state of mind doesn’t actually depend on your life circumstances. Whether it’s the sight of a lovely face or a tasty bite of food or good health, there is always something to be grateful for. Even bad experiences at least teach us something. And gratitude is not just a feeling outside your control that arrives willy-nilly. It’s more like a radio channel: you can choose at any time to tune in.

Gratitude acknowledges connection, and perhaps for this reason it is central to spiritual traditions worldwide, including Islam, Christianity, Buddhism, Hinduism, Judaism, and East Asian religions. When we contemplate our place in the intricate, interdependent network of life, we feel wonder and joy. That realization can lead us to express thanksgiving.

What are the benefits of gratitude? More than any other personality trait, gratitude is strongly linked to mental health and life satisfaction. Grateful people experience more joy, love, and enthusiasm, and they enjoy protection from destructive emotions like envy, greed, and bitterness. Gratitude also reduces lifetime risk for depression, anxiety, and substance abuse disorders, and it helps people entangled with those and other problems to heal and find closure. It can give you a deep and steadfast trust that goodness exists, even in the face of uncertainty or suffering.

Not only is gratitude a warm and uplifting way to feel, it benefits the body as well. People who experience gratitude cope better with stress, recover more quickly from illness, and enjoy more robust physical health, including lower blood pressure and better immune function.

Unlike other positive emotions like hope and happiness, gratitude is inherently relational: it reaches past the person experiencing it and into the social realm. It is gratitude in large measure that inspires people to acts of kindness, since it’s natural to respond to gifts with heartfelt gifts of your own. And that strengthens your bonds with other people. Grateful people are rated by others as more helpful, outgoing, optimistic, and trustworthy.

What is gratitude practice? Gratitude isn’t just an emotion that happens along, but a virtue we can cultivate. Think of it as something you practice as you might meditation or yoga.

Gratitude practice begins by paying attention . Notice all the good things you normally take for granted. Did you sleep well last night? Did someone at work or on the street treat you with courtesy? Have you caught a glimpse of the sky, with its sun and clouds, and had a moment of peace? It also involves acknowledging that difficult and painful moments are instructive and you can be grateful for them as well. Directing our attention this way blocks feelings of victimhood.

Second, consider writing about it in a journal or in a letter. Writing helps you organize thoughts, accept experiences, and put them into context, and gratitude journaling may bring a new and redemptive frame of reference to difficult life situations. It also helps you create meaning when you place everyday experiences within a framework of gifts and gratefulness. By writing, you can magnify and expand on the sources of goodness in your life, and think about what resources you’ve gained from your experiences, even bad ones.

In one study, people randomly assigned to keep weekly gratitude journals exercised more regularly, reported fewer physical symptoms, felt better about their lives, and were more optimistic about the upcoming week compared to people assigned to record hassles or neutral events. In another, young adults who kept a daily gratitude journal reported higher alertness, enthusiasm, determination, attentiveness and energy compared to those who focused on hassles or compared themselves to others less fortunate.

Finally, expressing gratitude completes the feeling of connection. Many people in your life have helped you in one way or another. Have you thanked them? Consider sending a letter to someone telling them what their actions meant to you, even if—especially if—it happened long ago. As for a response to blessings that don’t come from people, the arts and many faith traditions offer countless ways to express our gratitude. It may be as simple as a moment of deliberate reflection. Either way, the practice of gratitude may be the best holiday gift of all.

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Associate Director

Robert Emmons, Ph.D. Professor, University of California, Davis

“Gratitude as a Psychotherapeutic Intervention,” by Robert A Emmons, Ph.D. Dept. of Psychology, University of California, Davis and Robin S. Stern, Associate Director of the Yale Center for Emotional Intelligence, Volume 69, Issue 8, pages 846–855, August 2013.

Learn more about the Yale Center for Emotional Intelligence. Download PDF

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Some of the seminal research on the effects of practicing gratitude on a recurring basis was provided via a series of 3 experiments conducted by Emmons and McCullough. 4 In order to maximize the potential generalizability of the results, the sample participants for the first 2 studies consisted of healthy college students, whereas participants in the third study were adults with various congenital and acquired neuromuscular disorders. Within each study, some participants were instructed to maintain a journal on a weekly basis (for 10 weeks) and others, on a daily basis (for 2 or 3 weeks). The entire sample kept records of both positive and negative affects experienced as well as their coping behaviors, health behaviors, physical symptoms, and overall appraisals of life. Subgroups in each study were directed to focus their journal entries on different things:

Group A recorded things for which they were grateful (ie, they were “counting their blessings’’)

Group B recorded things they found annoying and/or irritating

Group C recorded things that had a major impact on them

When the data from all 3 studies were compiled and analyzed, 2 overarching trends were readily discernable 4 :

Participants in the group monitoring things for which they were grateful evidenced higher levels of well-being than those in either of the 2 comparison groups, but particularly when compared to group B (tabulating things they experienced as annoying or irritating).

The positive effects of a grateful outlook for participants in the longer study duration (10 weeks vs 2 or 3 weeks) included not only overall well-being, but social and physical benefits as well.

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Emmons and McCullough’s 6 conclusion that the habit of noting and savoring gratitude for aspects of one’s life plays a significant role in a person’s sense of well-being is supported by studies that preceded 9 11 and followed (S. Lyubomirsky et al, unpublished data, 2005) their own. This noted effect of gratitude on well-being, moreover, has more recently been distinguished as unique. Carey 09 Womens Distressed Platform Strappy Open Toe Wedges Blue Denim Blue Denim 4sHkt6F9YH
In addition to enhancing one’s general sense of well-being, those who engage in practicing gratitude report significantly greater happiness, optimism, and satisfaction with their lives, greater progress in attaining important life goals, higher frequency of feeling loved, and lower reported incidence of stress and depression compared to their counterparts tallying neutral or negative aspects.

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